Added trace operation + test and renamed base2k to backend

backend/.vscode/settings.json

@@ -0,0 +1,11 @@
+{
+    "github.copilot.enable": {
+        "*": false,
+        "plaintext": false,
+        "markdown": false,
+        "scminput": false
+    },
+    "files.associations": {
+        "random": "c"
+    }
+}

backend/Cargo.toml

@@ -0,0 +1,18 @@
+[package]
+name = "backend"
+version = "0.1.0"
+edition = "2024"
+
+[dependencies]
+rug = {workspace = true}
+criterion = {workspace = true}
+itertools = {workspace = true}
+rand = {workspace = true}
+rand_distr = {workspace = true}
+rand_core = {workspace = true}
+sampling = { path = "../sampling" }
+utils = { path = "../utils" }
+
+[[bench]]
+name = "fft"
+harness = false

backend/README.md

@@ -0,0 +1,12 @@
+
+## WSL/Ubuntu
+To use this crate you need to build spqlios-arithmetic, which is provided a as a git submodule:
+1) Initialize the sub-module
+2) $ cd backend/spqlios-arithmetic
+3) mdkir build
+4) cd build
+5) cmake ..
+6) make
+
+## Others
+Steps 3 to 6 might change depending of your platform. See [spqlios-arithmetic/wiki/build](https://github.com/tfhe/spqlios-arithmetic/wiki/build) for additional information and build options.

backend/benches/fft.rs

@@ -0,0 +1,56 @@
+use backend::ffi::reim::*;
+use criterion::{BenchmarkId, Criterion, criterion_group, criterion_main};
+use std::ffi::c_void;
+
+fn fft(c: &mut Criterion) {
+    fn forward<'a>(m: u32, log_bound: u32, reim_fft_precomp: *mut reim_fft_precomp, a: &'a [i64]) -> Box<dyn FnMut() + 'a> {
+        unsafe {
+            let buf_a: *mut f64 = reim_fft_precomp_get_buffer(reim_fft_precomp, 0);
+            reim_from_znx64_simple(m as u32, log_bound as u32, buf_a as *mut c_void, a.as_ptr());
+            Box::new(move || reim_fft(reim_fft_precomp, buf_a))
+        }
+    }
+
+    fn backward<'a>(m: u32, log_bound: u32, reim_ifft_precomp: *mut reim_ifft_precomp, a: &'a [i64]) -> Box<dyn FnMut() + 'a> {
+        Box::new(move || unsafe {
+            let buf_a: *mut f64 = reim_ifft_precomp_get_buffer(reim_ifft_precomp, 0);
+            reim_from_znx64_simple(m as u32, log_bound as u32, buf_a as *mut c_void, a.as_ptr());
+            reim_ifft(reim_ifft_precomp, buf_a);
+        })
+    }
+
+    let mut b: criterion::BenchmarkGroup<'_, criterion::measurement::WallTime> = c.benchmark_group("fft");
+
+    for log_n in 10..17 {
+        let n: usize = 1 << log_n;
+        let m: usize = n >> 1;
+        let log_bound: u32 = 19;
+
+        let mut a: Vec<i64> = vec![i64::default(); n];
+        a.iter_mut().enumerate().for_each(|(i, x)| *x = i as i64);
+
+        unsafe {
+            let reim_fft_precomp: *mut reim_fft_precomp = new_reim_fft_precomp(m as u32, 1);
+            let reim_ifft_precomp: *mut reim_ifft_precomp = new_reim_ifft_precomp(m as u32, 1);
+
+            let runners: [(String, Box<dyn FnMut()>); 2] = [
+                (format!("forward"), {
+                    forward(m as u32, log_bound, reim_fft_precomp, &a)
+                }),
+                (format!("backward"), {
+                    backward(m as u32, log_bound, reim_ifft_precomp, &a)
+                }),
+            ];
+
+            for (name, mut runner) in runners {
+                let id: BenchmarkId = BenchmarkId::new(name, format!("n={}", 1 << log_n));
+                b.bench_with_input(id, &(), |b: &mut criterion::Bencher<'_>, _| {
+                    b.iter(&mut runner)
+                });
+            }
+        }
+    }
+}
+
+criterion_group!(benches, fft,);
+criterion_main!(benches);

backend/build.rs

@@ -0,0 +1,13 @@
+use std::path::absolute;
+
+fn main() {
+    println!(
+        "cargo:rustc-link-search=native={}",
+        absolute("spqlios-arithmetic/build/spqlios")
+            .unwrap()
+            .to_str()
+            .unwrap()
+    );
+    println!("cargo:rustc-link-lib=static=spqlios");
+    // println!("cargo:rustc-link-lib=dylib=spqlios")
+}

backend/examples/fft.rs

@@ -0,0 +1,56 @@
+use backend::ffi::reim::*;
+use std::ffi::c_void;
+use std::time::Instant;
+
+fn main() {
+    let log_bound: usize = 19;
+
+    let n: usize = 2048;
+    let m: usize = n >> 1;
+
+    let mut a: Vec<i64> = vec![i64::default(); n];
+    let mut b: Vec<i64> = vec![i64::default(); n];
+    let mut c: Vec<i64> = vec![i64::default(); n];
+
+    a.iter_mut().enumerate().for_each(|(i, x)| *x = i as i64);
+    b[1] = 1;
+
+    println!("{:?}", b);
+
+    unsafe {
+        let reim_fft_precomp = new_reim_fft_precomp(m as u32, 2);
+        let reim_ifft_precomp = new_reim_ifft_precomp(m as u32, 1);
+
+        let buf_a = reim_fft_precomp_get_buffer(reim_fft_precomp, 0);
+        let buf_b = reim_fft_precomp_get_buffer(reim_fft_precomp, 1);
+        let buf_c = reim_ifft_precomp_get_buffer(reim_ifft_precomp, 0);
+
+        let now = Instant::now();
+        (0..1024).for_each(|_| {
+            reim_from_znx64_simple(m as u32, log_bound as u32, buf_a as *mut c_void, a.as_ptr());
+            reim_fft(reim_fft_precomp, buf_a);
+
+            reim_from_znx64_simple(m as u32, log_bound as u32, buf_b as *mut c_void, b.as_ptr());
+            reim_fft(reim_fft_precomp, buf_b);
+
+            reim_fftvec_mul_simple(
+                m as u32,
+                buf_c as *mut c_void,
+                buf_a as *mut c_void,
+                buf_b as *mut c_void,
+            );
+            reim_ifft(reim_ifft_precomp, buf_c);
+
+            reim_to_znx64_simple(
+                m as u32,
+                m as f64,
+                log_bound as u32,
+                c.as_mut_ptr(),
+                buf_c as *mut c_void,
+            )
+        });
+
+        println!("time: {}us", now.elapsed().as_micros());
+        println!("{:?}", &c[..16]);
+    }
+}

backend/examples/rlwe_encrypt.rs

@@ -0,0 +1,133 @@
+use backend::{
+    AddNormal, Decoding, Encoding, FFT64, FillUniform, Module, ScalarZnx, ScalarZnxAlloc, ScalarZnxDft, ScalarZnxDftAlloc,
+    ScalarZnxDftOps, ScratchOwned, VecZnx, VecZnxAlloc, VecZnxBig, VecZnxBigAlloc, VecZnxBigOps, VecZnxBigScratch, VecZnxDft,
+    VecZnxDftAlloc, VecZnxDftOps, VecZnxOps, ZnxInfos,
+};
+use itertools::izip;
+use sampling::source::Source;
+
+fn main() {
+    let n: usize = 16;
+    let basek: usize = 18;
+    let ct_size: usize = 3;
+    let msg_size: usize = 2;
+    let log_scale: usize = msg_size * basek - 5;
+    let module: Module<FFT64> = Module::<FFT64>::new(n);
+
+    let mut scratch: ScratchOwned = ScratchOwned::new(module.vec_znx_big_normalize_tmp_bytes());
+
+    let seed: [u8; 32] = [0; 32];
+    let mut source: Source = Source::new(seed);
+
+    // s <- Z_{-1, 0, 1}[X]/(X^{N}+1)
+    let mut s: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
+    s.fill_ternary_prob(0, 0.5, &mut source);
+
+    // Buffer to store s in the DFT domain
+    let mut s_dft: ScalarZnxDft<Vec<u8>, FFT64> = module.new_scalar_znx_dft(s.cols());
+
+    // s_dft <- DFT(s)
+    module.svp_prepare(&mut s_dft, 0, &s, 0);
+
+    // Allocates a VecZnx with two columns: ct=(0, 0)
+    let mut ct: VecZnx<Vec<u8>> = module.new_vec_znx(
+        2,       // Number of columns
+        ct_size, // Number of small poly per column
+    );
+
+    // Fill the second column with random values: ct = (0, a)
+    ct.fill_uniform(basek, 1, ct_size, &mut source);
+
+    let mut buf_dft: VecZnxDft<Vec<u8>, FFT64> = module.new_vec_znx_dft(1, ct_size);
+
+    module.vec_znx_dft(&mut buf_dft, 0, &ct, 1);
+
+    // Applies DFT(ct[1]) * DFT(s)
+    module.svp_apply_inplace(
+        &mut buf_dft, // DFT(ct[1] * s)
+        0,            // Selects the first column of res
+        &s_dft,       // DFT(s)
+        0,            // Selects the first column of s_dft
+    );
+
+    // Alias scratch space (VecZnxDft<B> is always at least as big as VecZnxBig<B>)
+
+    // BIG(ct[1] * s) <- IDFT(DFT(ct[1] * s)) (not normalized)
+    let mut buf_big: VecZnxBig<Vec<u8>, FFT64> = module.new_vec_znx_big(1, ct_size);
+    module.vec_znx_idft_tmp_a(&mut buf_big, 0, &mut buf_dft, 0);
+
+    // Creates a plaintext: VecZnx with 1 column
+    let mut m = module.new_vec_znx(
+        1,        // Number of columns
+        msg_size, // Number of small polynomials
+    );
+    let mut want: Vec<i64> = vec![0; n];
+    want.iter_mut()
+        .for_each(|x| *x = source.next_u64n(16, 15) as i64);
+    m.encode_vec_i64(0, basek, log_scale, &want, 4);
+    module.vec_znx_normalize_inplace(basek, &mut m, 0, scratch.borrow());
+
+    // m - BIG(ct[1] * s)
+    module.vec_znx_big_sub_small_b_inplace(
+        &mut buf_big,
+        0, // Selects the first column of the receiver
+        &m,
+        0, // Selects the first column of the message
+    );
+
+    // Normalizes back to VecZnx
+    // ct[0] <- m - BIG(c1 * s)
+    module.vec_znx_big_normalize(
+        basek,
+        &mut ct,
+        0, // Selects the first column of ct (ct[0])
+        &buf_big,
+        0, // Selects the first column of buf_big
+        scratch.borrow(),
+    );
+
+    // Add noise to ct[0]
+    // ct[0] <- ct[0] + e
+    ct.add_normal(
+        basek,
+        0,                    // Selects the first column of ct (ct[0])
+        basek * ct_size, // Scaling of the noise: 2^{-basek * limbs}
+        &mut source,
+        3.2,  // Standard deviation
+        19.0, // Truncatation bound
+    );
+
+    // Final ciphertext: ct = (-a * s + m + e, a)
+
+    // Decryption
+
+    // DFT(ct[1] * s)
+    module.vec_znx_dft(&mut buf_dft, 0, &ct, 1);
+    module.svp_apply_inplace(
+        &mut buf_dft,
+        0, // Selects the first column of res.
+        &s_dft,
+        0,
+    );
+
+    // BIG(c1 * s) = IDFT(DFT(c1 * s))
+    module.vec_znx_idft_tmp_a(&mut buf_big, 0, &mut buf_dft, 0);
+
+    // BIG(c1 * s) + ct[0]
+    module.vec_znx_big_add_small_inplace(&mut buf_big, 0, &ct, 0);
+
+    // m + e <- BIG(ct[1] * s + ct[0])
+    let mut res = module.new_vec_znx(1, ct_size);
+    module.vec_znx_big_normalize(basek, &mut res, 0, &buf_big, 0, scratch.borrow());
+
+    // have = m * 2^{log_scale} + e
+    let mut have: Vec<i64> = vec![i64::default(); n];
+    res.decode_vec_i64(0, basek, res.size() * basek, &mut have);
+
+    let scale: f64 = (1 << (res.size() * basek - log_scale)) as f64;
+    izip!(want.iter(), have.iter())
+        .enumerate()
+        .for_each(|(i, (a, b))| {
+            println!("{}: {} {}", i, a, (*b as f64) / scale);
+        });
+}

backend/src/encoding.rs

@@ -0,0 +1,333 @@
+use crate::ffi::znx::znx_zero_i64_ref;
+use crate::znx_base::{ZnxView, ZnxViewMut};
+use crate::{VecZnx, znx_base::ZnxInfos};
+use itertools::izip;
+use rug::{Assign, Float};
+use std::cmp::min;
+
+pub trait Encoding {
+    /// encode a vector of i64 on the receiver.
+    ///
+    /// # Arguments
+    ///
+    /// * `col_i`: the index of the poly where to encode the data.
+    /// * `basek`: base two negative logarithm decomposition of the receiver.
+    /// * `k`: base two negative logarithm of the scaling of the data.
+    /// * `data`: data to encode on the receiver.
+    /// * `log_max`: base two logarithm of the infinity norm of the input data.
+    fn encode_vec_i64(&mut self, col_i: usize, basek: usize, k: usize, data: &[i64], log_max: usize);
+
+    /// encodes a single i64 on the receiver at the given index.
+    ///
+    /// # Arguments
+    ///
+    /// * `col_i`: the index of the poly where to encode the data.
+    /// * `basek`: base two negative logarithm decomposition of the receiver.
+    /// * `k`: base two negative logarithm of the scaling of the data.
+    /// * `i`: index of the coefficient on which to encode the data.
+    /// * `data`: data to encode on the receiver.
+    /// * `log_max`: base two logarithm of the infinity norm of the input data.
+    fn encode_coeff_i64(&mut self, col_i: usize, basek: usize, k: usize, i: usize, data: i64, log_max: usize);
+}
+
+pub trait Decoding {
+    /// decode a vector of i64 from the receiver.
+    ///
+    /// # Arguments
+    ///
+    /// * `col_i`: the index of the poly where to encode the data.
+    /// * `basek`: base two negative logarithm decomposition of the receiver.
+    /// * `k`: base two logarithm of the scaling of the data.
+    /// * `data`: data to decode from the receiver.
+    fn decode_vec_i64(&self, col_i: usize, basek: usize, k: usize, data: &mut [i64]);
+
+    /// decode a vector of Float from the receiver.
+    ///
+    /// # Arguments
+    /// * `col_i`: the index of the poly where to encode the data.
+    /// * `basek`: base two negative logarithm decomposition of the receiver.
+    /// * `data`: data to decode from the receiver.
+    fn decode_vec_float(&self, col_i: usize, basek: usize, data: &mut [Float]);
+
+    /// decode a single of i64 from the receiver at the given index.
+    ///
+    /// # Arguments
+    ///
+    /// * `col_i`: the index of the poly where to encode the data.
+    /// * `basek`: base two negative logarithm decomposition of the receiver.
+    /// * `k`: base two negative logarithm of the scaling of the data.
+    /// * `i`: index of the coefficient to decode.
+    /// * `data`: data to decode from the receiver.
+    fn decode_coeff_i64(&self, col_i: usize, basek: usize, k: usize, i: usize) -> i64;
+}
+
+impl<D: AsMut<[u8]> + AsRef<[u8]>> Encoding for VecZnx<D> {
+    fn encode_vec_i64(&mut self, col_i: usize, basek: usize, k: usize, data: &[i64], log_max: usize) {
+        encode_vec_i64(self, col_i, basek, k, data, log_max)
+    }
+
+    fn encode_coeff_i64(&mut self, col_i: usize, basek: usize, k: usize, i: usize, value: i64, log_max: usize) {
+        encode_coeff_i64(self, col_i, basek, k, i, value, log_max)
+    }
+}
+
+impl<D: AsRef<[u8]>> Decoding for VecZnx<D> {
+    fn decode_vec_i64(&self, col_i: usize, basek: usize, k: usize, data: &mut [i64]) {
+        decode_vec_i64(self, col_i, basek, k, data)
+    }
+
+    fn decode_vec_float(&self, col_i: usize, basek: usize, data: &mut [Float]) {
+        decode_vec_float(self, col_i, basek, data)
+    }
+
+    fn decode_coeff_i64(&self, col_i: usize, basek: usize, k: usize, i: usize) -> i64 {
+        decode_coeff_i64(self, col_i, basek, k, i)
+    }
+}
+
+fn encode_vec_i64<D: AsMut<[u8]> + AsRef<[u8]>>(
+    a: &mut VecZnx<D>,
+    col_i: usize,
+    basek: usize,
+    k: usize,
+    data: &[i64],
+    log_max: usize,
+) {
+    let size: usize = (k + basek - 1) / basek;
+
+    #[cfg(debug_assertions)]
+    {
+        assert!(
+            size <= a.size(),
+            "invalid argument k: (k + a.basek - 1)/a.basek={} > a.size()={}",
+            size,
+            a.size()
+        );
+        assert!(col_i < a.cols());
+        assert!(data.len() <= a.n())
+    }
+
+    let data_len: usize = data.len();
+    let k_rem: usize = basek - (k % basek);
+
+    // Zeroes coefficients of the i-th column
+    (0..a.size()).for_each(|i| unsafe {
+        znx_zero_i64_ref(a.n() as u64, a.at_mut_ptr(col_i, i));
+    });
+
+    // If 2^{basek} * 2^{k_rem} < 2^{63}-1, then we can simply copy
+    // values on the last limb.
+    // Else we decompose values base2k.
+    if log_max + k_rem < 63 || k_rem == basek {
+        a.at_mut(col_i, size - 1)[..data_len].copy_from_slice(&data[..data_len]);
+    } else {
+        let mask: i64 = (1 << basek) - 1;
+        let steps: usize = min(size, (log_max + basek - 1) / basek);
+        (size - steps..size)
+            .rev()
+            .enumerate()
+            .for_each(|(i, i_rev)| {
+                let shift: usize = i * basek;
+                izip!(a.at_mut(col_i, i_rev).iter_mut(), data.iter()).for_each(|(y, x)| *y = (x >> shift) & mask);
+            })
+    }
+
+    // Case where self.prec % self.k != 0.
+    if k_rem != basek {
+        let steps: usize = min(size, (log_max + basek - 1) / basek);
+        (size - steps..size).rev().for_each(|i| {
+            a.at_mut(col_i, i)[..data_len]
+                .iter_mut()
+                .for_each(|x| *x <<= k_rem);
+        })
+    }
+}
+
+fn decode_vec_i64<D: AsRef<[u8]>>(a: &VecZnx<D>, col_i: usize, basek: usize, k: usize, data: &mut [i64]) {
+    let size: usize = (k + basek - 1) / basek;
+    #[cfg(debug_assertions)]
+    {
+        assert!(
+            data.len() >= a.n(),
+            "invalid data: data.len()={} < a.n()={}",
+            data.len(),
+            a.n()
+        );
+        assert!(col_i < a.cols());
+    }
+    data.copy_from_slice(a.at(col_i, 0));
+    let rem: usize = basek - (k % basek);
+    (1..size).for_each(|i| {
+        if i == size - 1 && rem != basek {
+            let k_rem: usize = basek - rem;
+            izip!(a.at(col_i, i).iter(), data.iter_mut()).for_each(|(x, y)| {
+                *y = (*y << k_rem) + (x >> rem);
+            });
+        } else {
+            izip!(a.at(col_i, i).iter(), data.iter_mut()).for_each(|(x, y)| {
+                *y = (*y << basek) + x;
+            });
+        }
+    })
+}
+
+fn decode_vec_float<D: AsRef<[u8]>>(a: &VecZnx<D>, col_i: usize, basek: usize, data: &mut [Float]) {
+    let size: usize = a.size();
+    #[cfg(debug_assertions)]
+    {
+        assert!(
+            data.len() >= a.n(),
+            "invalid data: data.len()={} < a.n()={}",
+            data.len(),
+            a.n()
+        );
+        assert!(col_i < a.cols());
+    }
+
+    let prec: u32 = (basek * size) as u32;
+
+    // 2^{basek}
+    let base = Float::with_val(prec, (1 << basek) as f64);
+
+    // y[i] = sum x[j][i] * 2^{-basek*j}
+    (0..size).for_each(|i| {
+        if i == 0 {
+            izip!(a.at(col_i, size - i - 1).iter(), data.iter_mut()).for_each(|(x, y)| {
+                y.assign(*x);
+                *y /= &base;
+            });
+        } else {
+            izip!(a.at(col_i, size - i - 1).iter(), data.iter_mut()).for_each(|(x, y)| {
+                *y += Float::with_val(prec, *x);
+                *y /= &base;
+            });
+        }
+    });
+}
+
+fn encode_coeff_i64<D: AsMut<[u8]> + AsRef<[u8]>>(
+    a: &mut VecZnx<D>,
+    col_i: usize,
+    basek: usize,
+    k: usize,
+    i: usize,
+    value: i64,
+    log_max: usize,
+) {
+    let size: usize = (k + basek - 1) / basek;
+
+    #[cfg(debug_assertions)]
+    {
+        assert!(i < a.n());
+        assert!(
+            size <= a.size(),
+            "invalid argument k: (k + a.basek - 1)/a.basek={} > a.size()={}",
+            size,
+            a.size()
+        );
+        assert!(col_i < a.cols());
+    }
+
+    let k_rem: usize = basek - (k % basek);
+    (0..a.size()).for_each(|j| a.at_mut(col_i, j)[i] = 0);
+
+    // If 2^{basek} * 2^{k_rem} < 2^{63}-1, then we can simply copy
+    // values on the last limb.
+    // Else we decompose values base2k.
+    if log_max + k_rem < 63 || k_rem == basek {
+        a.at_mut(col_i, size - 1)[i] = value;
+    } else {
+        let mask: i64 = (1 << basek) - 1;
+        let steps: usize = min(size, (log_max + basek - 1) / basek);
+        (size - steps..size)
+            .rev()
+            .enumerate()
+            .for_each(|(j, j_rev)| {
+                a.at_mut(col_i, j_rev)[i] = (value >> (j * basek)) & mask;
+            })
+    }
+
+    // Case where prec % k != 0.
+    if k_rem != basek {
+        let steps: usize = min(size, (log_max + basek - 1) / basek);
+        (size - steps..size).rev().for_each(|j| {
+            a.at_mut(col_i, j)[i] <<= k_rem;
+        })
+    }
+}
+
+fn decode_coeff_i64<D: AsRef<[u8]>>(a: &VecZnx<D>, col_i: usize, basek: usize, k: usize, i: usize) -> i64 {
+    #[cfg(debug_assertions)]
+    {
+        assert!(i < a.n());
+        assert!(col_i < a.cols())
+    }
+
+    let cols: usize = (k + basek - 1) / basek;
+    let data: &[i64] = a.raw();
+    let mut res: i64 = data[i];
+    let rem: usize = basek - (k % basek);
+    let slice_size: usize = a.n() * a.size();
+    (1..cols).for_each(|i| {
+        let x = data[i * slice_size];
+        if i == cols - 1 && rem != basek {
+            let k_rem: usize = basek - rem;
+            res = (res << k_rem) + (x >> rem);
+        } else {
+            res = (res << basek) + x;
+        }
+    });
+    res
+}
+
+#[cfg(test)]
+mod tests {
+    use crate::vec_znx_ops::*;
+    use crate::znx_base::*;
+    use crate::{Decoding, Encoding, FFT64, Module, VecZnx, znx_base::ZnxInfos};
+    use itertools::izip;
+    use sampling::source::Source;
+
+    #[test]
+    fn test_set_get_i64_lo_norm() {
+        let n: usize = 8;
+        let module: Module<FFT64> = Module::<FFT64>::new(n);
+        let basek: usize = 17;
+        let size: usize = 5;
+        let k: usize = size * basek - 5;
+        let mut a: VecZnx<_> = module.new_vec_znx(2, size);
+        let mut source: Source = Source::new([0u8; 32]);
+        let raw: &mut [i64] = a.raw_mut();
+        raw.iter_mut().enumerate().for_each(|(i, x)| *x = i as i64);
+        (0..a.cols()).for_each(|col_i| {
+            let mut have: Vec<i64> = vec![i64::default(); n];
+            have.iter_mut()
+                .for_each(|x| *x = (source.next_i64() << 56) >> 56);
+            a.encode_vec_i64(col_i, basek, k, &have, 10);
+            let mut want: Vec<i64> = vec![i64::default(); n];
+            a.decode_vec_i64(col_i, basek, k, &mut want);
+            izip!(want, have).for_each(|(a, b)| assert_eq!(a, b, "{} != {}", a, b));
+        });
+    }
+
+    #[test]
+    fn test_set_get_i64_hi_norm() {
+        let n: usize = 8;
+        let module: Module<FFT64> = Module::<FFT64>::new(n);
+        let basek: usize = 17;
+        let size: usize = 5;
+        let k: usize = size * basek - 5;
+        let mut a: VecZnx<_> = module.new_vec_znx(2, size);
+        let mut source = Source::new([0u8; 32]);
+        let raw: &mut [i64] = a.raw_mut();
+        raw.iter_mut().enumerate().for_each(|(i, x)| *x = i as i64);
+        (0..a.cols()).for_each(|col_i| {
+            let mut have: Vec<i64> = vec![i64::default(); n];
+            have.iter_mut().for_each(|x| *x = source.next_i64());
+            a.encode_vec_i64(col_i, basek, k, &have, 64);
+            let mut want = vec![i64::default(); n];
+            a.decode_vec_i64(col_i, basek, k, &mut want);
+            izip!(want, have).for_each(|(a, b)| assert_eq!(a, b, "{} != {}", a, b));
+        })
+    }
+}

backend/src/ffi/cnv.rs

@@ -0,0 +1,7 @@
+pub type CNV_PVEC_L = cnv_pvec_l_t;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct cnv_pvec_r_t {
+    _unused: [u8; 0],
+}
+pub type CNV_PVEC_R = cnv_pvec_r_t;

backend/src/ffi/mod.rs

@@ -0,0 +1,8 @@
+pub mod module;
+pub mod reim;
+pub mod svp;
+pub mod vec_znx;
+pub mod vec_znx_big;
+pub mod vec_znx_dft;
+pub mod vmp;
+pub mod znx;

backend/src/ffi/module.rs

@@ -0,0 +1,18 @@
+pub struct module_info_t {
+    _unused: [u8; 0],
+}
+
+pub type module_type_t = ::std::os::raw::c_uint;
+pub use self::module_type_t as MODULE_TYPE;
+
+pub type MODULE = module_info_t;
+
+unsafe extern "C" {
+    pub unsafe fn new_module_info(N: u64, mode: MODULE_TYPE) -> *mut MODULE;
+}
+unsafe extern "C" {
+    pub unsafe fn delete_module_info(module_info: *mut MODULE);
+}
+unsafe extern "C" {
+    pub unsafe fn module_get_n(module: *const MODULE) -> u64;
+}

backend/src/ffi/reim.rs

@@ -0,0 +1,172 @@
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_fft_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_FFT_PRECOMP = reim_fft_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_ifft_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_IFFT_PRECOMP = reim_ifft_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_mul_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_FFTVEC_MUL_PRECOMP = reim_mul_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_addmul_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_FFTVEC_ADDMUL_PRECOMP = reim_addmul_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_from_znx32_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_FROM_ZNX32_PRECOMP = reim_from_znx32_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_from_znx64_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_FROM_ZNX64_PRECOMP = reim_from_znx64_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_from_tnx32_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_FROM_TNX32_PRECOMP = reim_from_tnx32_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_to_tnx32_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_TO_TNX32_PRECOMP = reim_to_tnx32_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_to_tnx_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_TO_TNX_PRECOMP = reim_to_tnx_precomp;
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct reim_to_znx64_precomp {
+    _unused: [u8; 0],
+}
+pub type REIM_TO_ZNX64_PRECOMP = reim_to_znx64_precomp;
+unsafe extern "C" {
+    pub unsafe fn new_reim_fft_precomp(m: u32, num_buffers: u32) -> *mut REIM_FFT_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fft_precomp_get_buffer(tables: *const REIM_FFT_PRECOMP, buffer_index: u32) -> *mut f64;
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_fft_buffer(m: u32) -> *mut f64;
+}
+unsafe extern "C" {
+    pub unsafe fn delete_reim_fft_buffer(buffer: *mut f64);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fft(tables: *const REIM_FFT_PRECOMP, data: *mut f64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_ifft_precomp(m: u32, num_buffers: u32) -> *mut REIM_IFFT_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_ifft_precomp_get_buffer(tables: *const REIM_IFFT_PRECOMP, buffer_index: u32) -> *mut f64;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_ifft(tables: *const REIM_IFFT_PRECOMP, data: *mut f64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_fftvec_mul_precomp(m: u32) -> *mut REIM_FFTVEC_MUL_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fftvec_mul(tables: *const REIM_FFTVEC_MUL_PRECOMP, r: *mut f64, a: *const f64, b: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_fftvec_addmul_precomp(m: u32) -> *mut REIM_FFTVEC_ADDMUL_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fftvec_addmul(tables: *const REIM_FFTVEC_ADDMUL_PRECOMP, r: *mut f64, a: *const f64, b: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_from_znx32_precomp(m: u32, log2bound: u32) -> *mut REIM_FROM_ZNX32_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_from_znx32(tables: *const REIM_FROM_ZNX32_PRECOMP, r: *mut ::std::os::raw::c_void, a: *const i32);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_from_znx64(tables: *const REIM_FROM_ZNX64_PRECOMP, r: *mut ::std::os::raw::c_void, a: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_from_znx64_precomp(m: u32, maxbnd: u32) -> *mut REIM_FROM_ZNX64_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_from_znx64_simple(m: u32, log2bound: u32, r: *mut ::std::os::raw::c_void, a: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_from_tnx32_precomp(m: u32) -> *mut REIM_FROM_TNX32_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_from_tnx32(tables: *const REIM_FROM_TNX32_PRECOMP, r: *mut ::std::os::raw::c_void, a: *const i32);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_to_tnx32_precomp(m: u32, divisor: f64, log2overhead: u32) -> *mut REIM_TO_TNX32_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_to_tnx32(tables: *const REIM_TO_TNX32_PRECOMP, r: *mut i32, a: *const ::std::os::raw::c_void);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_to_tnx_precomp(m: u32, divisor: f64, log2overhead: u32) -> *mut REIM_TO_TNX_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_to_tnx(tables: *const REIM_TO_TNX_PRECOMP, r: *mut f64, a: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_to_tnx_simple(m: u32, divisor: f64, log2overhead: u32, r: *mut f64, a: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn new_reim_to_znx64_precomp(m: u32, divisor: f64, log2bound: u32) -> *mut REIM_TO_ZNX64_PRECOMP;
+}
+unsafe extern "C" {
+    pub unsafe fn reim_to_znx64(precomp: *const REIM_TO_ZNX64_PRECOMP, r: *mut i64, a: *const ::std::os::raw::c_void);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_to_znx64_simple(m: u32, divisor: f64, log2bound: u32, r: *mut i64, a: *const ::std::os::raw::c_void);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fft_simple(m: u32, data: *mut ::std::os::raw::c_void);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_ifft_simple(m: u32, data: *mut ::std::os::raw::c_void);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fftvec_mul_simple(
+        m: u32,
+        r: *mut ::std::os::raw::c_void,
+        a: *const ::std::os::raw::c_void,
+        b: *const ::std::os::raw::c_void,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn reim_fftvec_addmul_simple(
+        m: u32,
+        r: *mut ::std::os::raw::c_void,
+        a: *const ::std::os::raw::c_void,
+        b: *const ::std::os::raw::c_void,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn reim_from_znx32_simple(m: u32, log2bound: u32, r: *mut ::std::os::raw::c_void, x: *const i32);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_from_tnx32_simple(m: u32, r: *mut ::std::os::raw::c_void, x: *const i32);
+}
+unsafe extern "C" {
+    pub unsafe fn reim_to_tnx32_simple(m: u32, divisor: f64, log2overhead: u32, r: *mut i32, x: *const ::std::os::raw::c_void);
+}

backend/src/ffi/svp.rs

@@ -0,0 +1,48 @@
+use crate::ffi::module::MODULE;
+use crate::ffi::vec_znx_dft::VEC_ZNX_DFT;
+
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct svp_ppol_t {
+    _unused: [u8; 0],
+}
+pub type SVP_PPOL = svp_ppol_t;
+
+unsafe extern "C" {
+    pub unsafe fn bytes_of_svp_ppol(module: *const MODULE) -> u64;
+}
+unsafe extern "C" {
+    pub unsafe fn new_svp_ppol(module: *const MODULE) -> *mut SVP_PPOL;
+}
+unsafe extern "C" {
+    pub unsafe fn delete_svp_ppol(res: *mut SVP_PPOL);
+}
+
+unsafe extern "C" {
+    pub unsafe fn svp_prepare(module: *const MODULE, ppol: *mut SVP_PPOL, pol: *const i64);
+}
+
+unsafe extern "C" {
+    pub unsafe fn svp_apply_dft(
+        module: *const MODULE,
+        res: *const VEC_ZNX_DFT,
+        res_size: u64,
+        ppol: *const SVP_PPOL,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn svp_apply_dft_to_dft(
+        module: *const MODULE,
+        res: *const VEC_ZNX_DFT,
+        res_size: u64,
+        res_cols: u64,
+        ppol: *const SVP_PPOL,
+        a: *const VEC_ZNX_DFT,
+        a_size: u64,
+        a_cols: u64,
+    );
+}

backend/src/ffi/vec_znx.rs

@@ -0,0 +1,101 @@
+use crate::ffi::module::MODULE;
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_add(
+        module: *const MODULE,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        b: *const i64,
+        b_size: u64,
+        b_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_automorphism(
+        module: *const MODULE,
+        p: i64,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_negate(
+        module: *const MODULE,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_rotate(
+        module: *const MODULE,
+        p: i64,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_sub(
+        module: *const MODULE,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        b: *const i64,
+        b_size: u64,
+        b_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_zero(module: *const MODULE, res: *mut i64, res_size: u64, res_sl: u64);
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_copy(
+        module: *const MODULE,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_normalize_base2k(
+        module: *const MODULE,
+        base2k: u64,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        tmp_space: *mut u8,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_normalize_base2k_tmp_bytes(module: *const MODULE) -> u64;
+}

backend/src/ffi/vec_znx_big.rs

@@ -0,0 +1,161 @@
+use crate::ffi::module::MODULE;
+
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct vec_znx_big_t {
+    _unused: [u8; 0],
+}
+pub type VEC_ZNX_BIG = vec_znx_big_t;
+
+unsafe extern "C" {
+    pub unsafe fn bytes_of_vec_znx_big(module: *const MODULE, size: u64) -> u64;
+}
+unsafe extern "C" {
+    pub unsafe fn new_vec_znx_big(module: *const MODULE, size: u64) -> *mut VEC_ZNX_BIG;
+}
+unsafe extern "C" {
+    pub unsafe fn delete_vec_znx_big(res: *mut VEC_ZNX_BIG);
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_add(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+        b: *const VEC_ZNX_BIG,
+        b_size: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_add_small(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+        b: *const i64,
+        b_size: u64,
+        b_sl: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_add_small2(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        b: *const i64,
+        b_size: u64,
+        b_sl: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_sub(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+        b: *const VEC_ZNX_BIG,
+        b_size: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_sub_small_b(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+        b: *const i64,
+        b_size: u64,
+        b_sl: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_sub_small_a(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        b: *const VEC_ZNX_BIG,
+        b_size: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_sub_small2(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        b: *const i64,
+        b_size: u64,
+        b_sl: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_normalize_base2k_tmp_bytes(module: *const MODULE) -> u64;
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_normalize_base2k(
+        module: *const MODULE,
+        log2_base2k: u64,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_range_normalize_base2k(
+        module: *const MODULE,
+        log2_base2k: u64,
+        res: *mut i64,
+        res_size: u64,
+        res_sl: u64,
+        a: *const VEC_ZNX_BIG,
+        a_range_begin: u64,
+        a_range_xend: u64,
+        a_range_step: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_range_normalize_base2k_tmp_bytes(module: *const MODULE) -> u64;
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_automorphism(
+        module: *const MODULE,
+        p: i64,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_big_rotate(
+        module: *const MODULE,
+        p: i64,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a: *const VEC_ZNX_BIG,
+        a_size: u64,
+    );
+}

backend/src/ffi/vec_znx_dft.rs

@@ -0,0 +1,86 @@
+use crate::ffi::module::MODULE;
+use crate::ffi::vec_znx_big::VEC_ZNX_BIG;
+
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct vec_znx_dft_t {
+    _unused: [u8; 0],
+}
+pub type VEC_ZNX_DFT = vec_znx_dft_t;
+
+unsafe extern "C" {
+    pub unsafe fn bytes_of_vec_znx_dft(module: *const MODULE, size: u64) -> u64;
+}
+unsafe extern "C" {
+    pub unsafe fn new_vec_znx_dft(module: *const MODULE, size: u64) -> *mut VEC_ZNX_DFT;
+}
+unsafe extern "C" {
+    pub unsafe fn delete_vec_znx_dft(res: *mut VEC_ZNX_DFT);
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_dft_zero(module: *const MODULE, res: *mut VEC_ZNX_DFT, res_size: u64);
+}
+unsafe extern "C" {
+    pub unsafe fn vec_dft_add(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a: *const VEC_ZNX_DFT,
+        a_size: u64,
+        b: *const VEC_ZNX_DFT,
+        b_size: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_dft_sub(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a: *const VEC_ZNX_DFT,
+        a_size: u64,
+        b: *const VEC_ZNX_DFT,
+        b_size: u64,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_dft(module: *const MODULE, res: *mut VEC_ZNX_DFT, res_size: u64, a: *const i64, a_size: u64, a_sl: u64);
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_idft(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a_dft: *const VEC_ZNX_DFT,
+        a_size: u64,
+        tmp: *mut u8,
+    );
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_idft_tmp_bytes(module: *const MODULE) -> u64;
+}
+unsafe extern "C" {
+    pub unsafe fn vec_znx_idft_tmp_a(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        res_size: u64,
+        a_dft: *mut VEC_ZNX_DFT,
+        a_size: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_dft_automorphism(
+        module: *const MODULE,
+        d: i64,
+        res_dft: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a_dft: *const VEC_ZNX_DFT,
+        a_size: u64,
+        tmp: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vec_znx_dft_automorphism_tmp_bytes(module: *const MODULE) -> u64;
+}

backend/src/ffi/vmp.rs

@@ -0,0 +1,165 @@
+use crate::ffi::module::MODULE;
+use crate::ffi::vec_znx_big::VEC_ZNX_BIG;
+use crate::ffi::vec_znx_dft::VEC_ZNX_DFT;
+
+#[repr(C)]
+#[derive(Debug, Copy, Clone)]
+pub struct vmp_pmat_t {
+    _unused: [u8; 0],
+}
+
+// [rows][cols] = [#Decomposition][#Limbs]
+pub type VMP_PMAT = vmp_pmat_t;
+
+unsafe extern "C" {
+    pub unsafe fn bytes_of_vmp_pmat(module: *const MODULE, nrows: u64, ncols: u64) -> u64;
+}
+unsafe extern "C" {
+    pub unsafe fn new_vmp_pmat(module: *const MODULE, nrows: u64, ncols: u64) -> *mut VMP_PMAT;
+}
+unsafe extern "C" {
+    pub unsafe fn delete_vmp_pmat(res: *mut VMP_PMAT);
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_apply_dft(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        pmat: *const VMP_PMAT,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_apply_dft_add(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a: *const i64,
+        a_size: u64,
+        a_sl: u64,
+        pmat: *const VMP_PMAT,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_apply_dft_tmp_bytes(module: *const MODULE, res_size: u64, a_size: u64, nrows: u64, ncols: u64) -> u64;
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_apply_dft_to_dft(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a_dft: *const VEC_ZNX_DFT,
+        a_size: u64,
+        pmat: *const VMP_PMAT,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_apply_dft_to_dft_add(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        res_size: u64,
+        a_dft: *const VEC_ZNX_DFT,
+        a_size: u64,
+        pmat: *const VMP_PMAT,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_apply_dft_to_dft_tmp_bytes(
+        module: *const MODULE,
+        res_size: u64,
+        a_size: u64,
+        nrows: u64,
+        ncols: u64,
+    ) -> u64;
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_prepare_contiguous(
+        module: *const MODULE,
+        pmat: *mut VMP_PMAT,
+        mat: *const i64,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_prepare_dblptr(
+        module: *const MODULE,
+        pmat: *mut VMP_PMAT,
+        mat: *const *const i64,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_prepare_row(
+        module: *const MODULE,
+        pmat: *mut VMP_PMAT,
+        row: *const i64,
+        row_i: u64,
+        nrows: u64,
+        ncols: u64,
+        tmp_space: *mut u8,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_prepare_row_dft(
+        module: *const MODULE,
+        pmat: *mut VMP_PMAT,
+        row: *const VEC_ZNX_DFT,
+        row_i: u64,
+        nrows: u64,
+        ncols: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_extract_row_dft(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_DFT,
+        pmat: *const VMP_PMAT,
+        row_i: u64,
+        nrows: u64,
+        ncols: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_extract_row(
+        module: *const MODULE,
+        res: *mut VEC_ZNX_BIG,
+        pmat: *const VMP_PMAT,
+        row_i: u64,
+        nrows: u64,
+        ncols: u64,
+    );
+}
+
+unsafe extern "C" {
+    pub unsafe fn vmp_prepare_tmp_bytes(module: *const MODULE, nrows: u64, ncols: u64) -> u64;
+}

backend/src/ffi/znx.rs

@@ -0,0 +1,76 @@
+use crate::ffi::module::MODULE;
+
+unsafe extern "C" {
+    pub unsafe fn znx_add_i64_ref(nn: u64, res: *mut i64, a: *const i64, b: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_add_i64_avx(nn: u64, res: *mut i64, a: *const i64, b: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_sub_i64_ref(nn: u64, res: *mut i64, a: *const i64, b: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_sub_i64_avx(nn: u64, res: *mut i64, a: *const i64, b: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_negate_i64_ref(nn: u64, res: *mut i64, a: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_negate_i64_avx(nn: u64, res: *mut i64, a: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_copy_i64_ref(nn: u64, res: *mut i64, a: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_zero_i64_ref(nn: u64, res: *mut i64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_divide_by_m_ref(nn: u64, m: f64, res: *mut f64, a: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_divide_by_m_avx(nn: u64, m: f64, res: *mut f64, a: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_rotate_f64(nn: u64, p: i64, res: *mut f64, in_: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_rotate_i64(nn: u64, p: i64, res: *mut i64, in_: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_rotate_inplace_f64(nn: u64, p: i64, res: *mut f64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_rotate_inplace_i64(nn: u64, p: i64, res: *mut i64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_automorphism_f64(nn: u64, p: i64, res: *mut f64, in_: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_automorphism_i64(nn: u64, p: i64, res: *mut i64, in_: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_automorphism_inplace_f64(nn: u64, p: i64, res: *mut f64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_automorphism_inplace_i64(nn: u64, p: i64, res: *mut i64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_mul_xp_minus_one(nn: u64, p: i64, res: *mut f64, in_: *const f64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_mul_xp_minus_one(nn: u64, p: i64, res: *mut i64, in_: *const i64);
+}
+unsafe extern "C" {
+    pub unsafe fn rnx_mul_xp_minus_one_inplace(nn: u64, p: i64, res: *mut f64);
+}
+unsafe extern "C" {
+    pub unsafe fn znx_normalize(nn: u64, base_k: u64, out: *mut i64, carry_out: *mut i64, in_: *const i64, carry_in: *const i64);
+}
+
+unsafe extern "C" {
+    pub unsafe fn znx_small_single_product(module: *const MODULE, res: *mut i64, a: *const i64, b: *const i64, tmp: *mut u8);
+}
+
+unsafe extern "C" {
+    pub unsafe fn znx_small_single_product_tmp_bytes(module: *const MODULE) -> u64;
+}

backend/src/lib.rs

@@ -0,0 +1,251 @@
+pub mod encoding;
+#[allow(non_camel_case_types, non_snake_case, non_upper_case_globals, dead_code, improper_ctypes)]
+// Other modules and exports
+pub mod ffi;
+pub mod mat_znx_dft;
+pub mod mat_znx_dft_ops;
+pub mod module;
+pub mod sampling;
+pub mod scalar_znx;
+pub mod scalar_znx_dft;
+pub mod scalar_znx_dft_ops;
+pub mod stats;
+pub mod vec_znx;
+pub mod vec_znx_big;
+pub mod vec_znx_big_ops;
+pub mod vec_znx_dft;
+pub mod vec_znx_dft_ops;
+pub mod vec_znx_ops;
+pub mod znx_base;
+
+pub use encoding::*;
+pub use mat_znx_dft::*;
+pub use mat_znx_dft_ops::*;
+pub use module::*;
+pub use sampling::*;
+pub use scalar_znx::*;
+pub use scalar_znx_dft::*;
+pub use scalar_znx_dft_ops::*;
+pub use stats::*;
+pub use vec_znx::*;
+pub use vec_znx_big::*;
+pub use vec_znx_big_ops::*;
+pub use vec_znx_dft::*;
+pub use vec_znx_dft_ops::*;
+pub use vec_znx_ops::*;
+pub use znx_base::*;
+
+pub const GALOISGENERATOR: u64 = 5;
+pub const DEFAULTALIGN: usize = 64;
+
+fn is_aligned_custom<T>(ptr: *const T, align: usize) -> bool {
+    (ptr as usize) % align == 0
+}
+
+pub fn is_aligned<T>(ptr: *const T) -> bool {
+    is_aligned_custom(ptr, DEFAULTALIGN)
+}
+
+pub fn assert_alignement<T>(ptr: *const T) {
+    assert!(
+        is_aligned(ptr),
+        "invalid alignement: ensure passed bytes have been allocated with [alloc_aligned_u8] or [alloc_aligned]"
+    )
+}
+
+pub fn cast<T, V>(data: &[T]) -> &[V] {
+    let ptr: *const V = data.as_ptr() as *const V;
+    let len: usize = data.len() / size_of::<V>();
+    unsafe { std::slice::from_raw_parts(ptr, len) }
+}
+
+pub fn cast_mut<T, V>(data: &[T]) -> &mut [V] {
+    let ptr: *mut V = data.as_ptr() as *mut V;
+    let len: usize = data.len() / size_of::<V>();
+    unsafe { std::slice::from_raw_parts_mut(ptr, len) }
+}
+
+/// Allocates a block of bytes with a custom alignement.
+/// Alignement must be a power of two and size a multiple of the alignement.
+/// Allocated memory is initialized to zero.
+fn alloc_aligned_custom_u8(size: usize, align: usize) -> Vec<u8> {
+    assert!(
+        align.is_power_of_two(),
+        "Alignment must be a power of two but is {}",
+        align
+    );
+    assert_eq!(
+        (size * size_of::<u8>()) % align,
+        0,
+        "size={} must be a multiple of align={}",
+        size,
+        align
+    );
+    unsafe {
+        let layout: std::alloc::Layout = std::alloc::Layout::from_size_align(size, align).expect("Invalid alignment");
+        let ptr: *mut u8 = std::alloc::alloc(layout);
+        if ptr.is_null() {
+            panic!("Memory allocation failed");
+        }
+        assert!(
+            is_aligned_custom(ptr, align),
+            "Memory allocation at {:p} is not aligned to {} bytes",
+            ptr,
+            align
+        );
+        // Init allocated memory to zero
+        std::ptr::write_bytes(ptr, 0, size);
+        Vec::from_raw_parts(ptr, size, size)
+    }
+}
+
+/// Allocates a block of T aligned with [DEFAULTALIGN].
+/// Size of T * size msut be a multiple of [DEFAULTALIGN].
+pub fn alloc_aligned_custom<T>(size: usize, align: usize) -> Vec<T> {
+    assert_eq!(
+        (size * size_of::<T>()) % align,
+        0,
+        "size={} must be a multiple of align={}",
+        size,
+        align
+    );
+    let mut vec_u8: Vec<u8> = alloc_aligned_custom_u8(size_of::<T>() * size, align);
+    let ptr: *mut T = vec_u8.as_mut_ptr() as *mut T;
+    let len: usize = vec_u8.len() / size_of::<T>();
+    let cap: usize = vec_u8.capacity() / size_of::<T>();
+    std::mem::forget(vec_u8);
+    unsafe { Vec::from_raw_parts(ptr, len, cap) }
+}
+
+/// Allocates an aligned vector of size equal to the smallest multiple
+/// of [DEFAULTALIGN]/size_of::<T>() that is equal or greater to `size`.
+pub fn alloc_aligned<T>(size: usize) -> Vec<T> {
+    alloc_aligned_custom::<T>(
+        size + (size % (DEFAULTALIGN / size_of::<T>())),
+        DEFAULTALIGN,
+    )
+}
+
+// Scratch implementation below
+
+pub struct ScratchOwned(Vec<u8>);
+
+impl ScratchOwned {
+    pub fn new(byte_count: usize) -> Self {
+        let data: Vec<u8> = alloc_aligned(byte_count);
+        Self(data)
+    }
+
+    pub fn borrow(&mut self) -> &mut Scratch {
+        Scratch::new(&mut self.0)
+    }
+}
+
+pub struct Scratch {
+    data: [u8],
+}
+
+impl Scratch {
+    fn new(data: &mut [u8]) -> &mut Self {
+        unsafe { &mut *(data as *mut [u8] as *mut Self) }
+    }
+
+    pub fn available(&self) -> usize {
+        let ptr: *const u8 = self.data.as_ptr();
+        let self_len: usize = self.data.len();
+        let aligned_offset: usize = ptr.align_offset(DEFAULTALIGN);
+        self_len.saturating_sub(aligned_offset)
+    }
+
+    fn take_slice_aligned(data: &mut [u8], take_len: usize) -> (&mut [u8], &mut [u8]) {
+        let ptr: *mut u8 = data.as_mut_ptr();
+        let self_len: usize = data.len();
+
+        let aligned_offset: usize = ptr.align_offset(DEFAULTALIGN);
+        let aligned_len: usize = self_len.saturating_sub(aligned_offset);
+
+        if let Some(rem_len) = aligned_len.checked_sub(take_len) {
+            unsafe {
+                let rem_ptr: *mut u8 = ptr.add(aligned_offset).add(take_len);
+                let rem_slice: &mut [u8] = &mut *std::ptr::slice_from_raw_parts_mut(rem_ptr, rem_len);
+
+                let take_slice: &mut [u8] = &mut *std::ptr::slice_from_raw_parts_mut(ptr.add(aligned_offset), take_len);
+
+                return (take_slice, rem_slice);
+            }
+        } else {
+            panic!(
+                "Attempted to take {} from scratch with {} aligned bytes left",
+                take_len,
+                aligned_len,
+                // type_name::<T>(),
+                // aligned_len
+            );
+        }
+    }
+
+    pub fn tmp_slice<T>(&mut self, len: usize) -> (&mut [T], &mut Self) {
+        let (take_slice, rem_slice) = Self::take_slice_aligned(&mut self.data, len * std::mem::size_of::<T>());
+
+        unsafe {
+            (
+                &mut *(std::ptr::slice_from_raw_parts_mut(take_slice.as_mut_ptr() as *mut T, len)),
+                Self::new(rem_slice),
+            )
+        }
+    }
+
+    pub fn tmp_scalar_znx<B: Backend>(&mut self, module: &Module<B>, cols: usize) -> (ScalarZnx<&mut [u8]>, &mut Self) {
+        let (take_slice, rem_slice) = Self::take_slice_aligned(&mut self.data, bytes_of_scalar_znx(module, cols));
+
+        (
+            ScalarZnx::from_data(take_slice, module.n(), cols),
+            Self::new(rem_slice),
+        )
+    }
+
+    pub fn tmp_scalar_znx_dft<B: Backend>(&mut self, module: &Module<B>, cols: usize) -> (ScalarZnxDft<&mut [u8], B>, &mut Self) {
+        let (take_slice, rem_slice) = Self::take_slice_aligned(&mut self.data, bytes_of_scalar_znx_dft(module, cols));
+
+        (
+            ScalarZnxDft::from_data(take_slice, module.n(), cols),
+            Self::new(rem_slice),
+        )
+    }
+
+    pub fn tmp_vec_znx_dft<B: Backend>(
+        &mut self,
+        module: &Module<B>,
+        cols: usize,
+        size: usize,
+    ) -> (VecZnxDft<&mut [u8], B>, &mut Self) {
+        let (take_slice, rem_slice) = Self::take_slice_aligned(&mut self.data, bytes_of_vec_znx_dft(module, cols, size));
+
+        (
+            VecZnxDft::from_data(take_slice, module.n(), cols, size),
+            Self::new(rem_slice),
+        )
+    }
+
+    pub fn tmp_vec_znx_big<B: Backend>(
+        &mut self,
+        module: &Module<B>,
+        cols: usize,
+        size: usize,
+    ) -> (VecZnxBig<&mut [u8], B>, &mut Self) {
+        let (take_slice, rem_slice) = Self::take_slice_aligned(&mut self.data, bytes_of_vec_znx_big(module, cols, size));
+
+        (
+            VecZnxBig::from_data(take_slice, module.n(), cols, size),
+            Self::new(rem_slice),
+        )
+    }
+
+    pub fn tmp_vec_znx<B: Backend>(&mut self, module: &Module<B>, cols: usize, size: usize) -> (VecZnx<&mut [u8]>, &mut Self) {
+        let (take_slice, rem_slice) = Self::take_slice_aligned(&mut self.data, module.bytes_of_vec_znx(cols, size));
+        (
+            VecZnx::from_data(take_slice, module.n(), cols, size),
+            Self::new(rem_slice),
+        )
+    }
+}

backend/src/mat_znx_dft.rs

@@ -0,0 +1,232 @@
+use crate::znx_base::ZnxInfos;
+use crate::{Backend, DataView, DataViewMut, FFT64, Module, ZnxSliceSize, ZnxView, alloc_aligned};
+use std::marker::PhantomData;
+
+/// Vector Matrix Product Prepared Matrix: a vector of [VecZnx],
+/// stored as a 3D matrix in the DFT domain in a single contiguous array.
+/// Each col of the [MatZnxDft] can be seen as a collection of [VecZnxDft].
+///
+/// [MatZnxDft] is used to permform a vector matrix product between a [VecZnx]/[VecZnxDft] and a [MatZnxDft].
+/// See the trait [MatZnxDftOps] for additional information.
+pub struct MatZnxDft<D, B: Backend> {
+    data: D,
+    n: usize,
+    size: usize,
+    rows: usize,
+    cols_in: usize,
+    cols_out: usize,
+    _phantom: PhantomData<B>,
+}
+
+impl<D, B: Backend> ZnxInfos for MatZnxDft<D, B> {
+    fn cols(&self) -> usize {
+        self.cols_in
+    }
+
+    fn rows(&self) -> usize {
+        self.rows
+    }
+
+    fn n(&self) -> usize {
+        self.n
+    }
+
+    fn size(&self) -> usize {
+        self.size
+    }
+}
+
+impl<D> ZnxSliceSize for MatZnxDft<D, FFT64> {
+    fn sl(&self) -> usize {
+        self.n() * self.cols_out()
+    }
+}
+
+impl<D, B: Backend> DataView for MatZnxDft<D, B> {
+    type D = D;
+    fn data(&self) -> &Self::D {
+        &self.data
+    }
+}
+
+impl<D, B: Backend> DataViewMut for MatZnxDft<D, B> {
+    fn data_mut(&mut self) -> &mut Self::D {
+        &mut self.data
+    }
+}
+
+impl<D: AsRef<[u8]>> ZnxView for MatZnxDft<D, FFT64> {
+    type Scalar = f64;
+}
+
+impl<D, B: Backend> MatZnxDft<D, B> {
+    pub fn cols_in(&self) -> usize {
+        self.cols_in
+    }
+
+    pub fn cols_out(&self) -> usize {
+        self.cols_out
+    }
+}
+
+impl<D: From<Vec<u8>>, B: Backend> MatZnxDft<D, B> {
+    pub(crate) fn bytes_of(module: &Module<B>, rows: usize, cols_in: usize, cols_out: usize, size: usize) -> usize {
+        unsafe {
+            crate::ffi::vmp::bytes_of_vmp_pmat(
+                module.ptr,
+                (rows * cols_in) as u64,
+                (size * cols_out) as u64,
+            ) as usize
+        }
+    }
+
+    pub(crate) fn new(module: &Module<B>, rows: usize, cols_in: usize, cols_out: usize, size: usize) -> Self {
+        let data: Vec<u8> = alloc_aligned(Self::bytes_of(module, rows, cols_in, cols_out, size));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            size,
+            rows,
+            cols_in,
+            cols_out,
+            _phantom: PhantomData,
+        }
+    }
+
+    pub(crate) fn new_from_bytes(
+        module: &Module<B>,
+        rows: usize,
+        cols_in: usize,
+        cols_out: usize,
+        size: usize,
+        bytes: impl Into<Vec<u8>>,
+    ) -> Self {
+        let data: Vec<u8> = bytes.into();
+        assert!(data.len() == Self::bytes_of(module, rows, cols_in, cols_out, size));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            size,
+            rows,
+            cols_in,
+            cols_out,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D: AsRef<[u8]>> MatZnxDft<D, FFT64> {
+    /// Returns a copy of the backend array at index (i, j) of the [MatZnxDft].
+    ///
+    /// # Arguments
+    ///
+    /// * `row`: row index (i).
+    /// * `col`: col index (j).
+    #[allow(dead_code)]
+    fn at(&self, row: usize, col: usize) -> Vec<f64> {
+        let n: usize = self.n();
+
+        let mut res: Vec<f64> = alloc_aligned(n);
+
+        if n < 8 {
+            res.copy_from_slice(&self.raw()[(row + col * self.rows()) * n..(row + col * self.rows()) * (n + 1)]);
+        } else {
+            (0..n >> 3).for_each(|blk| {
+                res[blk * 8..(blk + 1) * 8].copy_from_slice(&self.at_block(row, col, blk)[..8]);
+            });
+        }
+
+        res
+    }
+
+    #[allow(dead_code)]
+    fn at_block(&self, row: usize, col: usize, blk: usize) -> &[f64] {
+        let nrows: usize = self.rows();
+        let nsize: usize = self.size();
+        if col == (nsize - 1) && (nsize & 1 == 1) {
+            &self.raw()[blk * nrows * nsize * 8 + col * nrows * 8 + row * 8..]
+        } else {
+            &self.raw()[blk * nrows * nsize * 8 + (col / 2) * (2 * nrows) * 8 + row * 2 * 8 + (col % 2) * 8..]
+        }
+    }
+}
+
+pub type MatZnxDftOwned<B> = MatZnxDft<Vec<u8>, B>;
+
+pub trait MatZnxDftToRef<B: Backend> {
+    fn to_ref(&self) -> MatZnxDft<&[u8], B>;
+}
+
+pub trait MatZnxDftToMut<B: Backend> {
+    fn to_mut(&mut self) -> MatZnxDft<&mut [u8], B>;
+}
+
+impl<B: Backend> MatZnxDftToMut<B> for MatZnxDft<Vec<u8>, B> {
+    fn to_mut(&mut self) -> MatZnxDft<&mut [u8], B> {
+        MatZnxDft {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            rows: self.rows,
+            cols_in: self.cols_in,
+            cols_out: self.cols_out,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> MatZnxDftToRef<B> for MatZnxDft<Vec<u8>, B> {
+    fn to_ref(&self) -> MatZnxDft<&[u8], B> {
+        MatZnxDft {
+            data: self.data.as_slice(),
+            n: self.n,
+            rows: self.rows,
+            cols_in: self.cols_in,
+            cols_out: self.cols_out,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> MatZnxDftToMut<B> for MatZnxDft<&mut [u8], B> {
+    fn to_mut(&mut self) -> MatZnxDft<&mut [u8], B> {
+        MatZnxDft {
+            data: self.data,
+            n: self.n,
+            rows: self.rows,
+            cols_in: self.cols_in,
+            cols_out: self.cols_out,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> MatZnxDftToRef<B> for MatZnxDft<&mut [u8], B> {
+    fn to_ref(&self) -> MatZnxDft<&[u8], B> {
+        MatZnxDft {
+            data: self.data,
+            n: self.n,
+            rows: self.rows,
+            cols_in: self.cols_in,
+            cols_out: self.cols_out,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> MatZnxDftToRef<B> for MatZnxDft<&[u8], B> {
+    fn to_ref(&self) -> MatZnxDft<&[u8], B> {
+        MatZnxDft {
+            data: self.data,
+            n: self.n,
+            rows: self.rows,
+            cols_in: self.cols_in,
+            cols_out: self.cols_out,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}

backend/src/mat_znx_dft_ops.rs

@@ -0,0 +1,488 @@
+use crate::ffi::vec_znx_dft::vec_znx_dft_t;
+use crate::ffi::vmp;
+use crate::znx_base::{ZnxInfos, ZnxView, ZnxViewMut};
+use crate::{
+    Backend, FFT64, MatZnxDft, MatZnxDftOwned, MatZnxDftToMut, MatZnxDftToRef, Module, Scratch, VecZnxDft, VecZnxDftToMut,
+    VecZnxDftToRef,
+};
+
+pub trait MatZnxDftAlloc<B: Backend> {
+    /// Allocates a new [MatZnxDft] with the given number of rows and columns.
+    ///
+    /// # Arguments
+    ///
+    /// * `rows`: number of rows (number of [VecZnxDft]).
+    /// * `size`: number of size (number of size of each [VecZnxDft]).
+    fn new_mat_znx_dft(&self, rows: usize, cols_in: usize, cols_out: usize, size: usize) -> MatZnxDftOwned<B>;
+
+    fn bytes_of_mat_znx_dft(&self, rows: usize, cols_in: usize, cols_out: usize, size: usize) -> usize;
+
+    fn new_mat_znx_dft_from_bytes(
+        &self,
+        rows: usize,
+        cols_in: usize,
+        cols_out: usize,
+        size: usize,
+        bytes: Vec<u8>,
+    ) -> MatZnxDftOwned<B>;
+}
+
+pub trait MatZnxDftScratch {
+    /// Returns the size of the stratch space necessary for [MatZnxDftOps::vmp_apply_dft_to_dft].
+    fn vmp_apply_tmp_bytes(
+        &self,
+        res_size: usize,
+        a_size: usize,
+        b_rows: usize,
+        b_cols_in: usize,
+        b_cols_out: usize,
+        b_size: usize,
+    ) -> usize;
+}
+
+/// This trait implements methods for vector matrix product,
+/// that is, multiplying a [VecZnx] with a [MatZnxDft].
+pub trait MatZnxDftOps<BACKEND: Backend> {
+    /// Prepares the ith-row of [MatZnxDft] from a [VecZnxDft].
+    ///
+    /// # Arguments
+    ///
+    /// * `b`: [MatZnxDft] on which the values are encoded.
+    /// * `a`: the [VecZnxDft] to encode on the [MatZnxDft].
+    /// * `row_i`: the index of the row to prepare.
+    ///
+    /// The size of buf can be obtained with [MatZnxDftOps::vmp_prepare_tmp_bytes].
+    fn vmp_prepare_row<R, A>(&self, res: &mut R, res_row: usize, res_col_in: usize, a: &A)
+    where
+        R: MatZnxDftToMut<BACKEND>,
+        A: VecZnxDftToRef<BACKEND>;
+
+    /// Extracts the ith-row of [MatZnxDft] into a [VecZnxDft].
+    ///
+    /// # Arguments
+    ///
+    /// * `b`: the [VecZnxDft] to on which to extract the row of the [MatZnxDft].
+    /// * `a`: [MatZnxDft] on which the values are encoded.
+    /// * `row_i`: the index of the row to extract.
+    fn vmp_extract_row<R, A>(&self, res: &mut R, a: &A, a_row: usize, a_col_in: usize)
+    where
+        R: VecZnxDftToMut<BACKEND>,
+        A: MatZnxDftToRef<BACKEND>;
+
+    /// Applies the vector matrix product [VecZnxDft] x [MatZnxDft].
+    /// The size of `buf` is given by [MatZnxDftOps::vmp_apply_dft_to_dft_tmp_bytes].
+    ///
+    /// A vector matrix product is equivalent to a sum of [crate::SvpPPolOps::svp_apply_dft]
+    /// where each [crate::Scalar] is a limb of the input [VecZnxDft] (equivalent to an [crate::SvpPPol])
+    /// and each vector a [VecZnxDft] (row) of the [MatZnxDft].
+    ///
+    /// As such, given an input [VecZnx] of `i` size and a [MatZnxDft] of `i` rows and
+    /// `j` size, the output is a [VecZnx] of `j` size.
+    ///
+    /// If there is a mismatch between the dimensions the largest valid ones are used.
+    ///
+    /// ```text
+    /// |a b c d| x |e f g| = (a * |e f g| + b * |h i j| + c * |k l m|) = |n o p|
+    ///             |h i j|
+    ///             |k l m|
+    /// ```
+    /// where each element is a [VecZnxDft].
+    ///
+    /// # Arguments
+    ///
+    /// * `c`: the output of the vector matrix product, as a [VecZnxDft].
+    /// * `a`: the left operand [VecZnxDft] of the vector matrix product.
+    /// * `b`: the right operand [MatZnxDft] of the vector matrix product.
+    /// * `buf`: scratch space, the size can be obtained with [MatZnxDftOps::vmp_apply_dft_to_dft_tmp_bytes].
+    fn vmp_apply<R, A, B>(&self, res: &mut R, a: &A, b: &B, scratch: &mut Scratch)
+    where
+        R: VecZnxDftToMut<BACKEND>,
+        A: VecZnxDftToRef<BACKEND>,
+        B: MatZnxDftToRef<BACKEND>;
+
+    // Same as [MatZnxDftOps::vmp_apply] except result is added on R instead of overwritting R.
+    fn vmp_apply_add<R, A, B>(&self, res: &mut R, a: &A, b: &B, scratch: &mut Scratch)
+    where
+        R: VecZnxDftToMut<BACKEND>,
+        A: VecZnxDftToRef<BACKEND>,
+        B: MatZnxDftToRef<BACKEND>;
+}
+
+impl<B: Backend> MatZnxDftAlloc<B> for Module<B> {
+    fn bytes_of_mat_znx_dft(&self, rows: usize, cols_in: usize, cols_out: usize, size: usize) -> usize {
+        MatZnxDftOwned::bytes_of(self, rows, cols_in, cols_out, size)
+    }
+
+    fn new_mat_znx_dft(&self, rows: usize, cols_in: usize, cols_out: usize, size: usize) -> MatZnxDftOwned<B> {
+        MatZnxDftOwned::new(self, rows, cols_in, cols_out, size)
+    }
+
+    fn new_mat_znx_dft_from_bytes(
+        &self,
+        rows: usize,
+        cols_in: usize,
+        cols_out: usize,
+        size: usize,
+        bytes: Vec<u8>,
+    ) -> MatZnxDftOwned<B> {
+        MatZnxDftOwned::new_from_bytes(self, rows, cols_in, cols_out, size, bytes)
+    }
+}
+
+impl<BACKEND: Backend> MatZnxDftScratch for Module<BACKEND> {
+    fn vmp_apply_tmp_bytes(
+        &self,
+        res_size: usize,
+        a_size: usize,
+        b_rows: usize,
+        b_cols_in: usize,
+        b_cols_out: usize,
+        b_size: usize,
+    ) -> usize {
+        unsafe {
+            vmp::vmp_apply_dft_to_dft_tmp_bytes(
+                self.ptr,
+                (res_size * b_cols_out) as u64,
+                (a_size * b_cols_in) as u64,
+                (b_rows * b_cols_in) as u64,
+                (b_size * b_cols_out) as u64,
+            ) as usize
+        }
+    }
+}
+
+impl MatZnxDftOps<FFT64> for Module<FFT64> {
+    fn vmp_prepare_row<R, A>(&self, res: &mut R, res_row: usize, res_col_in: usize, a: &A)
+    where
+        R: MatZnxDftToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+    {
+        let mut res: MatZnxDft<&mut [u8], _> = res.to_mut();
+        let a: VecZnxDft<&[u8], _> = a.to_ref();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(res.n(), self.n());
+            assert_eq!(a.n(), self.n());
+            assert_eq!(
+                a.cols(),
+                res.cols_out(),
+                "a.cols(): {} != res.cols_out(): {}",
+                a.cols(),
+                res.cols_out()
+            );
+            assert!(
+                res_row < res.rows(),
+                "res_row: {} >= res.rows(): {}",
+                res_row,
+                res.rows()
+            );
+            assert!(
+                res_col_in < res.cols_in(),
+                "res_col_in: {} >= res.cols_in(): {}",
+                res_col_in,
+                res.cols_in()
+            );
+            assert_eq!(
+                res.size(),
+                a.size(),
+                "res.size(): {} != a.size(): {}",
+                res.size(),
+                a.size()
+            );
+        }
+
+        unsafe {
+            vmp::vmp_prepare_row_dft(
+                self.ptr,
+                res.as_mut_ptr() as *mut vmp::vmp_pmat_t,
+                a.as_ptr() as *const vec_znx_dft_t,
+                (res_row * res.cols_in() + res_col_in) as u64,
+                (res.rows() * res.cols_in()) as u64,
+                (res.size() * res.cols_out()) as u64,
+            );
+        }
+    }
+
+    fn vmp_extract_row<R, A>(&self, res: &mut R, a: &A, a_row: usize, a_col_in: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: MatZnxDftToRef<FFT64>,
+    {
+        let mut res: VecZnxDft<&mut [u8], _> = res.to_mut();
+        let a: MatZnxDft<&[u8], _> = a.to_ref();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(res.n(), self.n());
+            assert_eq!(a.n(), self.n());
+            assert_eq!(
+                res.cols(),
+                a.cols_out(),
+                "res.cols(): {} != a.cols_out(): {}",
+                res.cols(),
+                a.cols_out()
+            );
+            assert!(
+                a_row < a.rows(),
+                "a_row: {} >= a.rows(): {}",
+                a_row,
+                a.rows()
+            );
+            assert!(
+                a_col_in < a.cols_in(),
+                "a_col_in: {} >= a.cols_in(): {}",
+                a_col_in,
+                a.cols_in()
+            );
+            assert_eq!(
+                res.size(),
+                a.size(),
+                "res.size(): {} != a.size(): {}",
+                res.size(),
+                a.size()
+            );
+        }
+        unsafe {
+            vmp::vmp_extract_row_dft(
+                self.ptr,
+                res.as_mut_ptr() as *mut vec_znx_dft_t,
+                a.as_ptr() as *const vmp::vmp_pmat_t,
+                (a_row * a.cols_in() + a_col_in) as u64,
+                (a.rows() * a.cols_in()) as u64,
+                (a.size() * a.cols_out()) as u64,
+            );
+        }
+    }
+
+    fn vmp_apply<R, A, B>(&self, res: &mut R, a: &A, b: &B, scratch: &mut Scratch)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+        B: MatZnxDftToRef<FFT64>,
+    {
+        let mut res: VecZnxDft<&mut [u8], _> = res.to_mut();
+        let a: VecZnxDft<&[u8], _> = a.to_ref();
+        let b: MatZnxDft<&[u8], _> = b.to_ref();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(res.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(a.n(), self.n());
+            assert_eq!(
+                res.cols(),
+                b.cols_out(),
+                "res.cols(): {} != b.cols_out: {}",
+                res.cols(),
+                b.cols_out()
+            );
+            assert_eq!(
+                a.cols(),
+                b.cols_in(),
+                "a.cols(): {} != b.cols_in: {}",
+                a.cols(),
+                b.cols_in()
+            );
+        }
+
+        let (tmp_bytes, _) = scratch.tmp_slice(self.vmp_apply_tmp_bytes(
+            res.size(),
+            a.size(),
+            b.rows(),
+            b.cols_in(),
+            b.cols_out(),
+            b.size(),
+        ));
+        unsafe {
+            vmp::vmp_apply_dft_to_dft(
+                self.ptr,
+                res.as_mut_ptr() as *mut vec_znx_dft_t,
+                (res.size() * res.cols()) as u64,
+                a.as_ptr() as *const vec_znx_dft_t,
+                (a.size() * a.cols()) as u64,
+                b.as_ptr() as *const vmp::vmp_pmat_t,
+                (b.rows() * b.cols_in()) as u64,
+                (b.size() * b.cols_out()) as u64,
+                tmp_bytes.as_mut_ptr(),
+            )
+        }
+    }
+
+    fn vmp_apply_add<R, A, B>(&self, res: &mut R, a: &A, b: &B, scratch: &mut Scratch)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+        B: MatZnxDftToRef<FFT64>,
+    {
+        let mut res: VecZnxDft<&mut [u8], _> = res.to_mut();
+        let a: VecZnxDft<&[u8], _> = a.to_ref();
+        let b: MatZnxDft<&[u8], _> = b.to_ref();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(res.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(a.n(), self.n());
+            assert_eq!(
+                res.cols(),
+                b.cols_out(),
+                "res.cols(): {} != b.cols_out: {}",
+                res.cols(),
+                b.cols_out()
+            );
+            assert_eq!(
+                a.cols(),
+                b.cols_in(),
+                "a.cols(): {} != b.cols_in: {}",
+                a.cols(),
+                b.cols_in()
+            );
+        }
+
+        let (tmp_bytes, _) = scratch.tmp_slice(self.vmp_apply_tmp_bytes(
+            res.size(),
+            a.size(),
+            b.rows(),
+            b.cols_in(),
+            b.cols_out(),
+            b.size(),
+        ));
+        unsafe {
+            vmp::vmp_apply_dft_to_dft_add(
+                self.ptr,
+                res.as_mut_ptr() as *mut vec_znx_dft_t,
+                (res.size() * res.cols()) as u64,
+                a.as_ptr() as *const vec_znx_dft_t,
+                (a.size() * a.cols()) as u64,
+                b.as_ptr() as *const vmp::vmp_pmat_t,
+                (b.rows() * b.cols_in()) as u64,
+                (b.size() * b.cols_out()) as u64,
+                tmp_bytes.as_mut_ptr(),
+            )
+        }
+    }
+}
+#[cfg(test)]
+mod tests {
+    use crate::{
+        Decoding, FFT64, FillUniform, MatZnxDft, MatZnxDftOps, Module, ScratchOwned, VecZnx, VecZnxAlloc, VecZnxBig,
+        VecZnxBigAlloc, VecZnxBigOps, VecZnxBigScratch, VecZnxDft, VecZnxDftAlloc, VecZnxDftOps, ZnxInfos, ZnxView, ZnxViewMut,
+    };
+    use sampling::source::Source;
+
+    use super::{MatZnxDftAlloc, MatZnxDftScratch};
+
+    #[test]
+    fn vmp_prepare_row() {
+        let module: Module<FFT64> = Module::<FFT64>::new(16);
+        let basek: usize = 8;
+        let mat_rows: usize = 4;
+        let mat_cols_in: usize = 2;
+        let mat_cols_out: usize = 2;
+        let mat_size: usize = 5;
+        let mut a: VecZnx<Vec<u8>> = module.new_vec_znx(mat_cols_out, mat_size);
+        let mut a_dft: VecZnxDft<Vec<u8>, FFT64> = module.new_vec_znx_dft(mat_cols_out, mat_size);
+        let mut b_dft: VecZnxDft<Vec<u8>, FFT64> = module.new_vec_znx_dft(mat_cols_out, mat_size);
+        let mut mat: MatZnxDft<Vec<u8>, FFT64> = module.new_mat_znx_dft(mat_rows, mat_cols_in, mat_cols_out, mat_size);
+
+        for col_in in 0..mat_cols_in {
+            for row_i in 0..mat_rows {
+                let mut source: Source = Source::new([0u8; 32]);
+                (0..mat_cols_out).for_each(|col_out| {
+                    a.fill_uniform(basek, col_out, mat_size, &mut source);
+                    module.vec_znx_dft(&mut a_dft, col_out, &a, col_out);
+                });
+                module.vmp_prepare_row(&mut mat, row_i, col_in, &a_dft);
+                module.vmp_extract_row(&mut b_dft, &mat, row_i, col_in);
+                assert_eq!(a_dft.raw(), b_dft.raw());
+            }
+        }
+    }
+
+    #[test]
+    fn vmp_apply() {
+        let log_n: i32 = 5;
+        let n: usize = 1 << log_n;
+
+        let module: Module<FFT64> = Module::<FFT64>::new(n);
+        let basek: usize = 15;
+        let a_size: usize = 5;
+        let mat_size: usize = 6;
+        let res_size: usize = 5;
+
+        [1, 2].iter().for_each(|in_cols| {
+            [1, 2].iter().for_each(|out_cols| {
+                let a_cols: usize = *in_cols;
+                let res_cols: usize = *out_cols;
+
+                let mat_rows: usize = a_size;
+                let mat_cols_in: usize = a_cols;
+                let mat_cols_out: usize = res_cols;
+                let res_cols: usize = mat_cols_out;
+
+                let mut scratch: ScratchOwned = ScratchOwned::new(
+                    module.vmp_apply_tmp_bytes(
+                        res_size,
+                        a_size,
+                        mat_rows,
+                        mat_cols_in,
+                        mat_cols_out,
+                        mat_size,
+                    ) | module.vec_znx_big_normalize_tmp_bytes(),
+                );
+
+                let mut a: VecZnx<Vec<u8>> = module.new_vec_znx(a_cols, a_size);
+
+                (0..a_cols).for_each(|i| {
+                    a.at_mut(i, 2)[i + 1] = 1;
+                });
+
+                let mut mat_znx_dft: MatZnxDft<Vec<u8>, FFT64> =
+                    module.new_mat_znx_dft(mat_rows, mat_cols_in, mat_cols_out, mat_size);
+
+                let mut c_dft: VecZnxDft<Vec<u8>, FFT64> = module.new_vec_znx_dft(mat_cols_out, mat_size);
+                let mut c_big: VecZnxBig<Vec<u8>, FFT64> = module.new_vec_znx_big(mat_cols_out, mat_size);
+
+                let mut tmp: VecZnx<Vec<u8>> = module.new_vec_znx(mat_cols_out, mat_size);
+
+                // Construts a [VecZnxMatDft] that performs cyclic rotations on each submatrix.
+                (0..a.size()).for_each(|row_i| {
+                    (0..mat_cols_in).for_each(|col_in_i| {
+                        (0..mat_cols_out).for_each(|col_out_i| {
+                            let idx = 1 + col_in_i * mat_cols_out + col_out_i;
+                            tmp.at_mut(col_out_i, row_i)[idx] = 1 as i64; // X^{idx}
+                            module.vec_znx_dft(&mut c_dft, col_out_i, &tmp, col_out_i);
+                            tmp.at_mut(col_out_i, row_i)[idx] = 0 as i64;
+                        });
+                        module.vmp_prepare_row(&mut mat_znx_dft, row_i, col_in_i, &c_dft);
+                    });
+                });
+
+                let mut a_dft: VecZnxDft<Vec<u8>, FFT64> = module.new_vec_znx_dft(a_cols, a_size);
+                (0..a_cols).for_each(|i| {
+                    module.vec_znx_dft(&mut a_dft, i, &a, i);
+                });
+
+                module.vmp_apply(&mut c_dft, &a_dft, &mat_znx_dft, scratch.borrow());
+
+                let mut res_have_vi64: Vec<i64> = vec![i64::default(); n];
+
+                let mut res_have: VecZnx<Vec<u8>> = module.new_vec_znx(res_cols, res_size);
+                (0..mat_cols_out).for_each(|i| {
+                    module.vec_znx_idft_tmp_a(&mut c_big, i, &mut c_dft, i);
+                    module.vec_znx_big_normalize(basek, &mut res_have, i, &c_big, i, scratch.borrow());
+                });
+
+                (0..mat_cols_out).for_each(|col_i| {
+                    let mut res_want_vi64: Vec<i64> = vec![i64::default(); n];
+                    (0..a_cols).for_each(|i| {
+                        res_want_vi64[(i + 1) + (1 + i * mat_cols_out + col_i)] = 1;
+                    });
+                    res_have.decode_vec_i64(col_i, basek, basek * 3, &mut res_have_vi64);
+                    assert_eq!(res_have_vi64, res_want_vi64);
+                });
+            });
+        });
+    }
+}

backend/src/module.rs

@@ -0,0 +1,104 @@
+use crate::GALOISGENERATOR;
+use crate::ffi::module::{MODULE, delete_module_info, module_info_t, new_module_info};
+use std::marker::PhantomData;
+
+#[derive(Copy, Clone)]
+#[repr(u8)]
+pub enum BACKEND {
+    FFT64,
+    NTT120,
+}
+
+pub trait Backend {
+    const KIND: BACKEND;
+    fn module_type() -> u32;
+}
+
+pub struct FFT64;
+pub struct NTT120;
+
+impl Backend for FFT64 {
+    const KIND: BACKEND = BACKEND::FFT64;
+    fn module_type() -> u32 {
+        0
+    }
+}
+
+impl Backend for NTT120 {
+    const KIND: BACKEND = BACKEND::NTT120;
+    fn module_type() -> u32 {
+        1
+    }
+}
+
+pub struct Module<B: Backend> {
+    pub ptr: *mut MODULE,
+    n: usize,
+    _marker: PhantomData<B>,
+}
+
+impl<B: Backend> Module<B> {
+    // Instantiates a new module.
+    pub fn new(n: usize) -> Self {
+        unsafe {
+            let m: *mut module_info_t = new_module_info(n as u64, B::module_type());
+            if m.is_null() {
+                panic!("Failed to create module.");
+            }
+            Self {
+                ptr: m,
+                n: n,
+                _marker: PhantomData,
+            }
+        }
+    }
+
+    pub fn n(&self) -> usize {
+        self.n
+    }
+
+    pub fn log_n(&self) -> usize {
+        (usize::BITS - (self.n() - 1).leading_zeros()) as _
+    }
+
+    pub fn cyclotomic_order(&self) -> u64 {
+        (self.n() << 1) as _
+    }
+
+    // Returns GALOISGENERATOR^|generator| * sign(generator)
+    pub fn galois_element(&self, generator: i64) -> i64 {
+        if generator == 0 {
+            return 1;
+        }
+        ((mod_exp_u64(GALOISGENERATOR, generator.abs() as usize) & (self.cyclotomic_order() - 1)) as i64) * generator.signum()
+    }
+
+    // Returns gen^-1
+    pub fn galois_element_inv(&self, gal_el: i64) -> i64 {
+        if gal_el == 0 {
+            panic!("cannot invert 0")
+        }
+        ((mod_exp_u64(gal_el.abs() as u64, (self.cyclotomic_order() - 1) as usize) & (self.cyclotomic_order() - 1)) as i64)
+            * gal_el.signum()
+    }
+}
+
+impl<B: Backend> Drop for Module<B> {
+    fn drop(&mut self) {
+        unsafe { delete_module_info(self.ptr) }
+    }
+}
+
+fn mod_exp_u64(x: u64, e: usize) -> u64 {
+    let mut y: u64 = 1;
+    let mut x_pow: u64 = x;
+    let mut exp = e;
+    while exp > 0 {
+        if exp & 1 == 1 {
+            y = y.wrapping_mul(x_pow);
+        }
+        x_pow = x_pow.wrapping_mul(x_pow);
+        exp >>= 1;
+    }
+    y
+}

backend/src/sampling.rs

@@ -0,0 +1,365 @@
+use crate::znx_base::ZnxViewMut;
+use crate::{FFT64, VecZnx, VecZnxBig, VecZnxBigToMut, VecZnxToMut};
+use rand_distr::{Distribution, Normal};
+use sampling::source::Source;
+
+pub trait FillUniform {
+    /// Fills the first `size` size with uniform values in \[-2^{basek-1}, 2^{basek-1}\]
+    fn fill_uniform(&mut self, basek: usize, col_i: usize, size: usize, source: &mut Source);
+}
+
+pub trait FillDistF64 {
+    fn fill_dist_f64<D: Distribution<f64>>(
+        &mut self,
+        basek: usize,
+        col_i: usize,
+        k: usize,
+        source: &mut Source,
+        dist: D,
+        bound: f64,
+    );
+}
+
+pub trait AddDistF64 {
+    /// Adds vector sampled according to the provided distribution, scaled by 2^{-k} and bounded to \[-bound, bound\].
+    fn add_dist_f64<D: Distribution<f64>>(
+        &mut self,
+        basek: usize,
+        col_i: usize,
+        k: usize,
+        source: &mut Source,
+        dist: D,
+        bound: f64,
+    );
+}
+
+pub trait FillNormal {
+    fn fill_normal(&mut self, basek: usize, col_i: usize, k: usize, source: &mut Source, sigma: f64, bound: f64);
+}
+
+pub trait AddNormal {
+    /// Adds a discrete normal vector scaled by 2^{-k} with the provided standard deviation and bounded to \[-bound, bound\].
+    fn add_normal(&mut self, basek: usize, col_i: usize, k: usize, source: &mut Source, sigma: f64, bound: f64);
+}
+
+impl<T> FillUniform for VecZnx<T>
+where
+    VecZnx<T>: VecZnxToMut,
+{
+    fn fill_uniform(&mut self, basek: usize, col_i: usize, size: usize, source: &mut Source) {
+        let mut a: VecZnx<&mut [u8]> = self.to_mut();
+        let base2k: u64 = 1 << basek;
+        let mask: u64 = base2k - 1;
+        let base2k_half: i64 = (base2k >> 1) as i64;
+        (0..size).for_each(|j| {
+            a.at_mut(col_i, j)
+                .iter_mut()
+                .for_each(|x| *x = (source.next_u64n(base2k, mask) as i64) - base2k_half);
+        })
+    }
+}
+
+impl<T> FillDistF64 for VecZnx<T>
+where
+    VecZnx<T>: VecZnxToMut,
+{
+    fn fill_dist_f64<D: Distribution<f64>>(
+        &mut self,
+        basek: usize,
+        col_i: usize,
+        k: usize,
+        source: &mut Source,
+        dist: D,
+        bound: f64,
+    ) {
+        let mut a: VecZnx<&mut [u8]> = self.to_mut();
+        assert!(
+            (bound.log2().ceil() as i64) < 64,
+            "invalid bound: ceil(log2(bound))={} > 63",
+            (bound.log2().ceil() as i64)
+        );
+
+        let limb: usize = (k + basek - 1) / basek - 1;
+        let basek_rem: usize = (limb + 1) * basek - k;
+
+        if basek_rem != 0 {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a = (dist_f64.round() as i64) << basek_rem;
+            });
+        } else {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a = dist_f64.round() as i64
+            });
+        }
+    }
+}
+
+impl<T> AddDistF64 for VecZnx<T>
+where
+    VecZnx<T>: VecZnxToMut,
+{
+    fn add_dist_f64<D: Distribution<f64>>(
+        &mut self,
+        basek: usize,
+        col_i: usize,
+        k: usize,
+        source: &mut Source,
+        dist: D,
+        bound: f64,
+    ) {
+        let mut a: VecZnx<&mut [u8]> = self.to_mut();
+        assert!(
+            (bound.log2().ceil() as i64) < 64,
+            "invalid bound: ceil(log2(bound))={} > 63",
+            (bound.log2().ceil() as i64)
+        );
+
+        let limb: usize = (k + basek - 1) / basek - 1;
+        let basek_rem: usize = (limb + 1) * basek - k;
+
+        if basek_rem != 0 {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a += (dist_f64.round() as i64) << basek_rem;
+            });
+        } else {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a += dist_f64.round() as i64
+            });
+        }
+    }
+}
+
+impl<T> FillNormal for VecZnx<T>
+where
+    VecZnx<T>: VecZnxToMut,
+{
+    fn fill_normal(&mut self, basek: usize, col_i: usize, k: usize, source: &mut Source, sigma: f64, bound: f64) {
+        self.fill_dist_f64(
+            basek,
+            col_i,
+            k,
+            source,
+            Normal::new(0.0, sigma).unwrap(),
+            bound,
+        );
+    }
+}
+
+impl<T> AddNormal for VecZnx<T>
+where
+    VecZnx<T>: VecZnxToMut,
+{
+    fn add_normal(&mut self, basek: usize, col_i: usize, k: usize, source: &mut Source, sigma: f64, bound: f64) {
+        self.add_dist_f64(
+            basek,
+            col_i,
+            k,
+            source,
+            Normal::new(0.0, sigma).unwrap(),
+            bound,
+        );
+    }
+}
+
+impl<T> FillDistF64 for VecZnxBig<T, FFT64>
+where
+    VecZnxBig<T, FFT64>: VecZnxBigToMut<FFT64>,
+{
+    fn fill_dist_f64<D: Distribution<f64>>(
+        &mut self,
+        basek: usize,
+        col_i: usize,
+        k: usize,
+        source: &mut Source,
+        dist: D,
+        bound: f64,
+    ) {
+        let mut a: VecZnxBig<&mut [u8], FFT64> = self.to_mut();
+        assert!(
+            (bound.log2().ceil() as i64) < 64,
+            "invalid bound: ceil(log2(bound))={} > 63",
+            (bound.log2().ceil() as i64)
+        );
+
+        let limb: usize = (k + basek - 1) / basek - 1;
+        let basek_rem: usize = (limb + 1) * basek - k;
+
+        if basek_rem != 0 {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a = (dist_f64.round() as i64) << basek_rem;
+            });
+        } else {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a = dist_f64.round() as i64
+            });
+        }
+    }
+}
+
+impl<T> AddDistF64 for VecZnxBig<T, FFT64>
+where
+    VecZnxBig<T, FFT64>: VecZnxBigToMut<FFT64>,
+{
+    fn add_dist_f64<D: Distribution<f64>>(
+        &mut self,
+        basek: usize,
+        col_i: usize,
+        k: usize,
+        source: &mut Source,
+        dist: D,
+        bound: f64,
+    ) {
+        let mut a: VecZnxBig<&mut [u8], FFT64> = self.to_mut();
+        assert!(
+            (bound.log2().ceil() as i64) < 64,
+            "invalid bound: ceil(log2(bound))={} > 63",
+            (bound.log2().ceil() as i64)
+        );
+
+        let limb: usize = (k + basek - 1) / basek - 1;
+        let basek_rem: usize = (limb + 1) * basek - k;
+
+        if basek_rem != 0 {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a += (dist_f64.round() as i64) << basek_rem;
+            });
+        } else {
+            a.at_mut(col_i, limb).iter_mut().for_each(|a| {
+                let mut dist_f64: f64 = dist.sample(source);
+                while dist_f64.abs() > bound {
+                    dist_f64 = dist.sample(source)
+                }
+                *a += dist_f64.round() as i64
+            });
+        }
+    }
+}
+
+impl<T> FillNormal for VecZnxBig<T, FFT64>
+where
+    VecZnxBig<T, FFT64>: VecZnxBigToMut<FFT64>,
+{
+    fn fill_normal(&mut self, basek: usize, col_i: usize, k: usize, source: &mut Source, sigma: f64, bound: f64) {
+        self.fill_dist_f64(
+            basek,
+            col_i,
+            k,
+            source,
+            Normal::new(0.0, sigma).unwrap(),
+            bound,
+        );
+    }
+}
+
+impl<T> AddNormal for VecZnxBig<T, FFT64>
+where
+    VecZnxBig<T, FFT64>: VecZnxBigToMut<FFT64>,
+{
+    fn add_normal(&mut self, basek: usize, col_i: usize, k: usize, source: &mut Source, sigma: f64, bound: f64) {
+        self.add_dist_f64(
+            basek,
+            col_i,
+            k,
+            source,
+            Normal::new(0.0, sigma).unwrap(),
+            bound,
+        );
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::{AddNormal, FillUniform};
+    use crate::vec_znx_ops::*;
+    use crate::znx_base::*;
+    use crate::{FFT64, Module, Stats, VecZnx};
+    use sampling::source::Source;
+
+    #[test]
+    fn vec_znx_fill_uniform() {
+        let n: usize = 4096;
+        let module: Module<FFT64> = Module::<FFT64>::new(n);
+        let basek: usize = 17;
+        let size: usize = 5;
+        let mut source: Source = Source::new([0u8; 32]);
+        let cols: usize = 2;
+        let zero: Vec<i64> = vec![0; n];
+        let one_12_sqrt: f64 = 0.28867513459481287;
+        (0..cols).for_each(|col_i| {
+            let mut a: VecZnx<_> = module.new_vec_znx(cols, size);
+            a.fill_uniform(basek, col_i, size, &mut source);
+            (0..cols).for_each(|col_j| {
+                if col_j != col_i {
+                    (0..size).for_each(|limb_i| {
+                        assert_eq!(a.at(col_j, limb_i), zero);
+                    })
+                } else {
+                    let std: f64 = a.std(col_i, basek);
+                    assert!(
+                        (std - one_12_sqrt).abs() < 0.01,
+                        "std={} ~!= {}",
+                        std,
+                        one_12_sqrt
+                    );
+                }
+            })
+        });
+    }
+
+    #[test]
+    fn vec_znx_add_normal() {
+        let n: usize = 4096;
+        let module: Module<FFT64> = Module::<FFT64>::new(n);
+        let basek: usize = 17;
+        let k: usize = 2 * 17;
+        let size: usize = 5;
+        let sigma: f64 = 3.2;
+        let bound: f64 = 6.0 * sigma;
+        let mut source: Source = Source::new([0u8; 32]);
+        let cols: usize = 2;
+        let zero: Vec<i64> = vec![0; n];
+        let k_f64: f64 = (1u64 << k as u64) as f64;
+        (0..cols).for_each(|col_i| {
+            let mut a: VecZnx<_> = module.new_vec_znx(cols, size);
+            a.add_normal(basek, col_i, k, &mut source, sigma, bound);
+            (0..cols).for_each(|col_j| {
+                if col_j != col_i {
+                    (0..size).for_each(|limb_i| {
+                        assert_eq!(a.at(col_j, limb_i), zero);
+                    })
+                } else {
+                    let std: f64 = a.std(col_i, basek) * k_f64;
+                    assert!((std - sigma).abs() < 0.1, "std={} ~!= {}", std, sigma);
+                }
+            })
+        });
+    }
+}

backend/src/scalar_znx.rs

@@ -0,0 +1,306 @@
+use crate::ffi::vec_znx;
+use crate::znx_base::ZnxInfos;
+use crate::{
+    Backend, DataView, DataViewMut, Module, VecZnx, VecZnxToMut, VecZnxToRef, ZnxSliceSize, ZnxView, ZnxViewMut, alloc_aligned,
+};
+use rand::seq::SliceRandom;
+use rand_core::RngCore;
+use rand_distr::{Distribution, weighted::WeightedIndex};
+use sampling::source::Source;
+
+pub struct ScalarZnx<D> {
+    pub(crate) data: D,
+    pub(crate) n: usize,
+    pub(crate) cols: usize,
+}
+
+impl<D> ZnxInfos for ScalarZnx<D> {
+    fn cols(&self) -> usize {
+        self.cols
+    }
+
+    fn rows(&self) -> usize {
+        1
+    }
+
+    fn n(&self) -> usize {
+        self.n
+    }
+
+    fn size(&self) -> usize {
+        1
+    }
+}
+
+impl<D> ZnxSliceSize for ScalarZnx<D> {
+    fn sl(&self) -> usize {
+        self.n()
+    }
+}
+
+impl<D> DataView for ScalarZnx<D> {
+    type D = D;
+    fn data(&self) -> &Self::D {
+        &self.data
+    }
+}
+
+impl<D> DataViewMut for ScalarZnx<D> {
+    fn data_mut(&mut self) -> &mut Self::D {
+        &mut self.data
+    }
+}
+
+impl<D: AsRef<[u8]>> ZnxView for ScalarZnx<D> {
+    type Scalar = i64;
+}
+
+impl<D: AsMut<[u8]> + AsRef<[u8]>> ScalarZnx<D> {
+    pub fn fill_ternary_prob(&mut self, col: usize, prob: f64, source: &mut Source) {
+        let choices: [i64; 3] = [-1, 0, 1];
+        let weights: [f64; 3] = [prob / 2.0, 1.0 - prob, prob / 2.0];
+        let dist: WeightedIndex<f64> = WeightedIndex::new(&weights).unwrap();
+        self.at_mut(col, 0)
+            .iter_mut()
+            .for_each(|x: &mut i64| *x = choices[dist.sample(source)]);
+    }
+
+    pub fn fill_ternary_hw(&mut self, col: usize, hw: usize, source: &mut Source) {
+        assert!(hw <= self.n());
+        self.at_mut(col, 0)[..hw]
+            .iter_mut()
+            .for_each(|x: &mut i64| *x = (((source.next_u32() & 1) as i64) << 1) - 1);
+        self.at_mut(col, 0).shuffle(source);
+    }
+}
+
+impl<D: From<Vec<u8>>> ScalarZnx<D> {
+    pub(crate) fn bytes_of<S: Sized>(n: usize, cols: usize) -> usize {
+        n * cols * size_of::<S>()
+    }
+
+    pub(crate) fn new<S: Sized>(n: usize, cols: usize) -> Self {
+        let data = alloc_aligned::<u8>(Self::bytes_of::<S>(n, cols));
+        Self {
+            data: data.into(),
+            n,
+            cols,
+        }
+    }
+
+    pub(crate) fn new_from_bytes<S: Sized>(n: usize, cols: usize, bytes: impl Into<Vec<u8>>) -> Self {
+        let data: Vec<u8> = bytes.into();
+        assert!(data.len() == Self::bytes_of::<S>(n, cols));
+        Self {
+            data: data.into(),
+            n,
+            cols,
+        }
+    }
+}
+
+pub type ScalarZnxOwned = ScalarZnx<Vec<u8>>;
+
+pub(crate) fn bytes_of_scalar_znx<B: Backend>(module: &Module<B>, cols: usize) -> usize {
+    ScalarZnxOwned::bytes_of::<i64>(module.n(), cols)
+}
+
+pub trait ScalarZnxAlloc {
+    fn bytes_of_scalar_znx(&self, cols: usize) -> usize;
+    fn new_scalar_znx(&self, cols: usize) -> ScalarZnxOwned;
+    fn new_scalar_znx_from_bytes(&self, cols: usize, bytes: Vec<u8>) -> ScalarZnxOwned;
+}
+
+impl<B: Backend> ScalarZnxAlloc for Module<B> {
+    fn bytes_of_scalar_znx(&self, cols: usize) -> usize {
+        ScalarZnxOwned::bytes_of::<i64>(self.n(), cols)
+    }
+    fn new_scalar_znx(&self, cols: usize) -> ScalarZnxOwned {
+        ScalarZnxOwned::new::<i64>(self.n(), cols)
+    }
+    fn new_scalar_znx_from_bytes(&self, cols: usize, bytes: Vec<u8>) -> ScalarZnxOwned {
+        ScalarZnxOwned::new_from_bytes::<i64>(self.n(), cols, bytes)
+    }
+}
+
+pub trait ScalarZnxOps {
+    fn scalar_znx_automorphism<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: ScalarZnxToMut,
+        A: ScalarZnxToRef;
+
+    /// Applies the automorphism X^i -> X^ik on the selected column of `a`.
+    fn scalar_znx_automorphism_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: ScalarZnxToMut;
+}
+
+impl<B: Backend> ScalarZnxOps for Module<B> {
+    fn scalar_znx_automorphism<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: ScalarZnxToMut,
+        A: ScalarZnxToRef,
+    {
+        let a: ScalarZnx<&[u8]> = a.to_ref();
+        let mut res: ScalarZnx<&mut [u8]> = res.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_automorphism(
+                self.ptr,
+                k,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn scalar_znx_automorphism_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: ScalarZnxToMut,
+    {
+        let mut a: ScalarZnx<&mut [u8]> = a.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_automorphism(
+                self.ptr,
+                k,
+                a.at_mut_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+}
+
+impl<D> ScalarZnx<D> {
+    pub(crate) fn from_data(data: D, n: usize, cols: usize) -> Self {
+        Self { data, n, cols }
+    }
+}
+
+pub trait ScalarZnxToRef {
+    fn to_ref(&self) -> ScalarZnx<&[u8]>;
+}
+
+pub trait ScalarZnxToMut {
+    fn to_mut(&mut self) -> ScalarZnx<&mut [u8]>;
+}
+
+impl ScalarZnxToMut for ScalarZnx<Vec<u8>> {
+    fn to_mut(&mut self) -> ScalarZnx<&mut [u8]> {
+        ScalarZnx {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+        }
+    }
+}
+
+impl VecZnxToMut for ScalarZnx<Vec<u8>> {
+    fn to_mut(&mut self) -> VecZnx<&mut [u8]> {
+        VecZnx {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+        }
+    }
+}
+
+impl ScalarZnxToRef for ScalarZnx<Vec<u8>> {
+    fn to_ref(&self) -> ScalarZnx<&[u8]> {
+        ScalarZnx {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+        }
+    }
+}
+
+impl VecZnxToRef for ScalarZnx<Vec<u8>> {
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        VecZnx {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+        }
+    }
+}
+
+impl ScalarZnxToMut for ScalarZnx<&mut [u8]> {
+    fn to_mut(&mut self) -> ScalarZnx<&mut [u8]> {
+        ScalarZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+        }
+    }
+}
+
+impl VecZnxToMut for ScalarZnx<&mut [u8]> {
+    fn to_mut(&mut self) -> VecZnx<&mut [u8]> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+        }
+    }
+}
+
+impl ScalarZnxToRef for ScalarZnx<&mut [u8]> {
+    fn to_ref(&self) -> ScalarZnx<&[u8]> {
+        ScalarZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+        }
+    }
+}
+
+impl VecZnxToRef for ScalarZnx<&mut [u8]> {
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+        }
+    }
+}
+
+impl ScalarZnxToRef for ScalarZnx<&[u8]> {
+    fn to_ref(&self) -> ScalarZnx<&[u8]> {
+        ScalarZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+        }
+    }
+}
+
+impl VecZnxToRef for ScalarZnx<&[u8]> {
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+        }
+    }
+}

backend/src/scalar_znx_dft.rs

@@ -0,0 +1,233 @@
+use std::marker::PhantomData;
+
+use crate::ffi::svp;
+use crate::znx_base::ZnxInfos;
+use crate::{
+    Backend, DataView, DataViewMut, FFT64, Module, VecZnxDft, VecZnxDftToMut, VecZnxDftToRef, ZnxSliceSize, ZnxView,
+    alloc_aligned,
+};
+
+pub struct ScalarZnxDft<D, B: Backend> {
+    data: D,
+    n: usize,
+    cols: usize,
+    _phantom: PhantomData<B>,
+}
+
+impl<D, B: Backend> ZnxInfos for ScalarZnxDft<D, B> {
+    fn cols(&self) -> usize {
+        self.cols
+    }
+
+    fn rows(&self) -> usize {
+        1
+    }
+
+    fn n(&self) -> usize {
+        self.n
+    }
+
+    fn size(&self) -> usize {
+        1
+    }
+}
+
+impl<D> ZnxSliceSize for ScalarZnxDft<D, FFT64> {
+    fn sl(&self) -> usize {
+        self.n()
+    }
+}
+
+impl<D, B: Backend> DataView for ScalarZnxDft<D, B> {
+    type D = D;
+    fn data(&self) -> &Self::D {
+        &self.data
+    }
+}
+
+impl<D, B: Backend> DataViewMut for ScalarZnxDft<D, B> {
+    fn data_mut(&mut self) -> &mut Self::D {
+        &mut self.data
+    }
+}
+
+impl<D: AsRef<[u8]>> ZnxView for ScalarZnxDft<D, FFT64> {
+    type Scalar = f64;
+}
+
+pub(crate) fn bytes_of_scalar_znx_dft<B: Backend>(module: &Module<B>, cols: usize) -> usize {
+    ScalarZnxDftOwned::bytes_of(module, cols)
+}
+
+impl<D: From<Vec<u8>>, B: Backend> ScalarZnxDft<D, B> {
+    pub(crate) fn bytes_of(module: &Module<B>, cols: usize) -> usize {
+        unsafe { svp::bytes_of_svp_ppol(module.ptr) as usize * cols }
+    }
+
+    pub(crate) fn new(module: &Module<B>, cols: usize) -> Self {
+        let data = alloc_aligned::<u8>(Self::bytes_of(module, cols));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            cols,
+            _phantom: PhantomData,
+        }
+    }
+
+    pub(crate) fn new_from_bytes(module: &Module<B>, cols: usize, bytes: impl Into<Vec<u8>>) -> Self {
+        let data: Vec<u8> = bytes.into();
+        assert!(data.len() == Self::bytes_of(module, cols));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            cols,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D, B: Backend> ScalarZnxDft<D, B> {
+    pub(crate) fn from_data(data: D, n: usize, cols: usize) -> Self {
+        Self {
+            data,
+            n,
+            cols,
+            _phantom: PhantomData,
+        }
+    }
+
+    pub fn as_vec_znx_dft(self) -> VecZnxDft<D, B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+pub type ScalarZnxDftOwned<B> = ScalarZnxDft<Vec<u8>, B>;
+
+pub trait ScalarZnxDftToRef<B: Backend> {
+    fn to_ref(&self) -> ScalarZnxDft<&[u8], B>;
+}
+
+pub trait ScalarZnxDftToMut<B: Backend> {
+    fn to_mut(&mut self) -> ScalarZnxDft<&mut [u8], B>;
+}
+
+impl<B: Backend> ScalarZnxDftToMut<B> for ScalarZnxDft<Vec<u8>, B> {
+    fn to_mut(&mut self) -> ScalarZnxDft<&mut [u8], B> {
+        ScalarZnxDft {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> ScalarZnxDftToRef<B> for ScalarZnxDft<Vec<u8>, B> {
+    fn to_ref(&self) -> ScalarZnxDft<&[u8], B> {
+        ScalarZnxDft {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> ScalarZnxDftToMut<B> for ScalarZnxDft<&mut [u8], B> {
+    fn to_mut(&mut self) -> ScalarZnxDft<&mut [u8], B> {
+        ScalarZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> ScalarZnxDftToRef<B> for ScalarZnxDft<&mut [u8], B> {
+    fn to_ref(&self) -> ScalarZnxDft<&[u8], B> {
+        ScalarZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> ScalarZnxDftToRef<B> for ScalarZnxDft<&[u8], B> {
+    fn to_ref(&self) -> ScalarZnxDft<&[u8], B> {
+        ScalarZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToMut<B> for ScalarZnxDft<Vec<u8>, B> {
+    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B> {
+        VecZnxDft {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToRef<B> for ScalarZnxDft<Vec<u8>, B> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        VecZnxDft {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToMut<B> for ScalarZnxDft<&mut [u8], B> {
+    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToRef<B> for ScalarZnxDft<&mut [u8], B> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToRef<B> for ScalarZnxDft<&[u8], B> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: 1,
+            _phantom: PhantomData,
+        }
+    }
+}

backend/src/scalar_znx_dft_ops.rs

@@ -0,0 +1,103 @@
+use crate::ffi::svp;
+use crate::ffi::vec_znx_dft::vec_znx_dft_t;
+use crate::znx_base::{ZnxInfos, ZnxView, ZnxViewMut};
+use crate::{
+    Backend, FFT64, Module, ScalarZnxDft, ScalarZnxDftOwned, ScalarZnxDftToMut, ScalarZnxDftToRef, ScalarZnxToRef, VecZnxDft,
+    VecZnxDftToMut, VecZnxDftToRef,
+};
+
+pub trait ScalarZnxDftAlloc<B: Backend> {
+    fn new_scalar_znx_dft(&self, cols: usize) -> ScalarZnxDftOwned<B>;
+    fn bytes_of_scalar_znx_dft(&self, cols: usize) -> usize;
+    fn new_scalar_znx_dft_from_bytes(&self, cols: usize, bytes: Vec<u8>) -> ScalarZnxDftOwned<B>;
+}
+
+pub trait ScalarZnxDftOps<BACKEND: Backend> {
+    fn svp_prepare<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: ScalarZnxDftToMut<BACKEND>,
+        A: ScalarZnxToRef;
+    fn svp_apply<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxDftToMut<BACKEND>,
+        A: ScalarZnxDftToRef<BACKEND>,
+        B: VecZnxDftToRef<FFT64>;
+    fn svp_apply_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<BACKEND>,
+        A: ScalarZnxDftToRef<BACKEND>;
+}
+
+impl<B: Backend> ScalarZnxDftAlloc<B> for Module<B> {
+    fn new_scalar_znx_dft(&self, cols: usize) -> ScalarZnxDftOwned<B> {
+        ScalarZnxDftOwned::new(self, cols)
+    }
+
+    fn bytes_of_scalar_znx_dft(&self, cols: usize) -> usize {
+        ScalarZnxDftOwned::bytes_of(self, cols)
+    }
+
+    fn new_scalar_znx_dft_from_bytes(&self, cols: usize, bytes: Vec<u8>) -> ScalarZnxDftOwned<B> {
+        ScalarZnxDftOwned::new_from_bytes(self, cols, bytes)
+    }
+}
+
+impl ScalarZnxDftOps<FFT64> for Module<FFT64> {
+    fn svp_prepare<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: ScalarZnxDftToMut<FFT64>,
+        A: ScalarZnxToRef,
+    {
+        unsafe {
+            svp::svp_prepare(
+                self.ptr,
+                res.to_mut().at_mut_ptr(res_col, 0) as *mut svp::svp_ppol_t,
+                a.to_ref().at_ptr(a_col, 0),
+            )
+        }
+    }
+
+    fn svp_apply<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: ScalarZnxDftToRef<FFT64>,
+        B: VecZnxDftToRef<FFT64>,
+    {
+        let mut res: VecZnxDft<&mut [u8], FFT64> = res.to_mut();
+        let a: ScalarZnxDft<&[u8], FFT64> = a.to_ref();
+        let b: VecZnxDft<&[u8], FFT64> = b.to_ref();
+        unsafe {
+            svp::svp_apply_dft_to_dft(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0) as *mut vec_znx_dft_t,
+                res.size() as u64,
+                res.cols() as u64,
+                a.at_ptr(a_col, 0) as *const svp::svp_ppol_t,
+                b.at_ptr(b_col, 0) as *const vec_znx_dft_t,
+                b.size() as u64,
+                b.cols() as u64,
+            )
+        }
+    }
+
+    fn svp_apply_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: ScalarZnxDftToRef<FFT64>,
+    {
+        let mut res: VecZnxDft<&mut [u8], FFT64> = res.to_mut();
+        let a: ScalarZnxDft<&[u8], FFT64> = a.to_ref();
+        unsafe {
+            svp::svp_apply_dft_to_dft(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0) as *mut vec_znx_dft_t,
+                res.size() as u64,
+                res.cols() as u64,
+                a.at_ptr(a_col, 0) as *const svp::svp_ppol_t,
+                res.at_ptr(res_col, 0) as *const vec_znx_dft_t,
+                res.size() as u64,
+                res.cols() as u64,
+            )
+        }
+    }
+}

backend/src/stats.rs

@@ -0,0 +1,32 @@
+use crate::znx_base::ZnxInfos;
+use crate::{Decoding, VecZnx};
+use rug::Float;
+use rug::float::Round;
+use rug::ops::{AddAssignRound, DivAssignRound, SubAssignRound};
+
+pub trait Stats {
+    /// Returns the standard devaition of the i-th polynomial.
+    fn std(&self, col_i: usize, basek: usize) -> f64;
+}
+
+impl<D: AsRef<[u8]>> Stats for VecZnx<D> {
+    fn std(&self, col_i: usize, basek: usize) -> f64 {
+        let prec: u32 = (self.size() * basek) as u32;
+        let mut data: Vec<Float> = (0..self.n()).map(|_| Float::with_val(prec, 0)).collect();
+        self.decode_vec_float(col_i, basek, &mut data);
+        // std = sqrt(sum((xi - avg)^2) / n)
+        let mut avg: Float = Float::with_val(prec, 0);
+        data.iter().for_each(|x| {
+            avg.add_assign_round(x, Round::Nearest);
+        });
+        avg.div_assign_round(Float::with_val(prec, data.len()), Round::Nearest);
+        data.iter_mut().for_each(|x| {
+            x.sub_assign_round(&avg, Round::Nearest);
+        });
+        let mut std: Float = Float::with_val(prec, 0);
+        data.iter().for_each(|x| std += x * x);
+        std.div_assign_round(Float::with_val(prec, data.len()), Round::Nearest);
+        std = std.sqrt();
+        std.to_f64()
+    }
+}

backend/src/vec_znx.rs

@@ -0,0 +1,373 @@
+use itertools::izip;
+
+use crate::DataView;
+use crate::DataViewMut;
+use crate::ScalarZnx;
+use crate::Scratch;
+use crate::ZnxSliceSize;
+use crate::ZnxZero;
+use crate::alloc_aligned;
+use crate::assert_alignement;
+use crate::cast_mut;
+use crate::ffi::znx;
+use crate::znx_base::{ZnxInfos, ZnxView, ZnxViewMut};
+use std::{cmp::min, fmt};
+
+/// [VecZnx] represents collection of contiguously stacked vector of small norm polynomials of
+/// Zn\[X\] with [i64] coefficients.
+/// A [VecZnx] is composed of multiple Zn\[X\] polynomials stored in a single contiguous array
+/// in the memory.
+///
+/// # Example
+///
+/// Given 3 polynomials (a, b, c) of Zn\[X\], each with 4 columns, then the memory
+/// layout is: `[a0, b0, c0, a1, b1, c1, a2, b2, c2, a3, b3, c3]`, where ai, bi, ci
+/// are small polynomials of Zn\[X\].
+pub struct VecZnx<D> {
+    pub data: D,
+    pub n: usize,
+    pub cols: usize,
+    pub size: usize,
+}
+
+impl<D> ZnxInfos for VecZnx<D> {
+    fn cols(&self) -> usize {
+        self.cols
+    }
+
+    fn rows(&self) -> usize {
+        1
+    }
+
+    fn n(&self) -> usize {
+        self.n
+    }
+
+    fn size(&self) -> usize {
+        self.size
+    }
+}
+
+impl<D> ZnxSliceSize for VecZnx<D> {
+    fn sl(&self) -> usize {
+        self.n() * self.cols()
+    }
+}
+
+impl<D> DataView for VecZnx<D> {
+    type D = D;
+    fn data(&self) -> &Self::D {
+        &self.data
+    }
+}
+
+impl<D> DataViewMut for VecZnx<D> {
+    fn data_mut(&mut self) -> &mut Self::D {
+        &mut self.data
+    }
+}
+
+impl<D: AsRef<[u8]>> ZnxView for VecZnx<D> {
+    type Scalar = i64;
+}
+
+impl<D: AsRef<[u8]>> VecZnx<D> {
+    pub fn rsh_scratch_space(n: usize) -> usize {
+        n * std::mem::size_of::<i64>()
+    }
+}
+
+impl<D: AsMut<[u8]> + AsRef<[u8]>> VecZnx<D> {
+    /// Truncates the precision of the [VecZnx] by k bits.
+    ///
+    /// # Arguments
+    ///
+    /// * `basek`: the base two logarithm of the coefficients decomposition.
+    /// * `k`: the number of bits of precision to drop.
+    pub fn trunc_pow2(&mut self, basek: usize, k: usize, col: usize) {
+        if k == 0 {
+            return;
+        }
+
+        self.size -= k / basek;
+
+        let k_rem: usize = k % basek;
+
+        if k_rem != 0 {
+            let mask: i64 = ((1 << (basek - k_rem - 1)) - 1) << k_rem;
+            self.at_mut(col, self.size() - 1)
+                .iter_mut()
+                .for_each(|x: &mut i64| *x &= mask)
+        }
+    }
+
+    pub fn rsh(&mut self, basek: usize, k: usize, scratch: &mut Scratch) {
+        let n: usize = self.n();
+        let cols: usize = self.cols();
+        let size: usize = self.size();
+        let steps: usize = k / basek;
+
+        self.raw_mut().rotate_right(n * steps * cols);
+        (0..cols).for_each(|i| {
+            (0..steps).for_each(|j| {
+                self.zero_at(i, j);
+            })
+        });
+
+        let k_rem: usize = k % basek;
+
+        if k_rem != 0 {
+            let (carry, _) = scratch.tmp_slice::<i64>(n);
+            let shift = i64::BITS as usize - k_rem;
+            (0..cols).for_each(|i| {
+                carry.fill(0);
+                (steps..size).for_each(|j| {
+                    izip!(carry.iter_mut(), self.at_mut(i, j).iter_mut()).for_each(|(ci, xi)| {
+                        *xi += *ci << basek;
+                        *ci = (*xi << shift) >> shift;
+                        *xi = (*xi - *ci) >> k_rem;
+                    });
+                });
+            })
+        }
+    }
+}
+
+impl<D: From<Vec<u8>>> VecZnx<D> {
+    pub(crate) fn bytes_of<Scalar: Sized>(n: usize, cols: usize, size: usize) -> usize {
+        n * cols * size * size_of::<Scalar>()
+    }
+
+    pub(crate) fn new<Scalar: Sized>(n: usize, cols: usize, size: usize) -> Self {
+        let data = alloc_aligned::<u8>(Self::bytes_of::<Scalar>(n, cols, size));
+        Self {
+            data: data.into(),
+            n,
+            cols,
+            size,
+        }
+    }
+
+    pub(crate) fn new_from_bytes<Scalar: Sized>(n: usize, cols: usize, size: usize, bytes: impl Into<Vec<u8>>) -> Self {
+        let data: Vec<u8> = bytes.into();
+        assert!(data.len() == Self::bytes_of::<Scalar>(n, cols, size));
+        Self {
+            data: data.into(),
+            n,
+            cols,
+            size,
+        }
+    }
+}
+
+impl<D> VecZnx<D> {
+    pub(crate) fn from_data(data: D, n: usize, cols: usize, size: usize) -> Self {
+        Self {
+            data,
+            n,
+            cols,
+            size,
+        }
+    }
+
+    pub fn to_scalar_znx(self) -> ScalarZnx<D> {
+        debug_assert_eq!(
+            self.size, 1,
+            "cannot convert VecZnx to ScalarZnx if cols: {} != 1",
+            self.cols
+        );
+        ScalarZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+        }
+    }
+}
+
+/// Copies the coefficients of `a` on the receiver.
+/// Copy is done with the minimum size matching both backing arrays.
+/// Panics if the cols do not match.
+pub fn copy_vec_znx_from<DataMut, Data>(b: &mut VecZnx<DataMut>, a: &VecZnx<Data>)
+where
+    DataMut: AsMut<[u8]> + AsRef<[u8]>,
+    Data: AsRef<[u8]>,
+{
+    assert_eq!(b.cols(), a.cols());
+    let data_a: &[i64] = a.raw();
+    let data_b: &mut [i64] = b.raw_mut();
+    let size = min(data_b.len(), data_a.len());
+    data_b[..size].copy_from_slice(&data_a[..size])
+}
+
+#[allow(dead_code)]
+fn normalize_tmp_bytes(n: usize) -> usize {
+    n * std::mem::size_of::<i64>()
+}
+
+#[allow(dead_code)]
+fn normalize<D: AsMut<[u8]> + AsRef<[u8]>>(basek: usize, a: &mut VecZnx<D>, a_col: usize, tmp_bytes: &mut [u8]) {
+    let n: usize = a.n();
+
+    debug_assert!(
+        tmp_bytes.len() >= normalize_tmp_bytes(n),
+        "invalid tmp_bytes: tmp_bytes.len()={} < normalize_tmp_bytes({})",
+        tmp_bytes.len(),
+        n,
+    );
+
+    #[cfg(debug_assertions)]
+    {
+        assert_alignement(tmp_bytes.as_ptr())
+    }
+
+    let carry_i64: &mut [i64] = cast_mut(tmp_bytes);
+
+    unsafe {
+        znx::znx_zero_i64_ref(n as u64, carry_i64.as_mut_ptr());
+        (0..a.size()).rev().for_each(|i| {
+            znx::znx_normalize(
+                n as u64,
+                basek as u64,
+                a.at_mut_ptr(a_col, i),
+                carry_i64.as_mut_ptr(),
+                a.at_mut_ptr(a_col, i),
+                carry_i64.as_mut_ptr(),
+            )
+        });
+    }
+}
+
+impl<D> VecZnx<D>
+where
+    VecZnx<D>: VecZnxToMut + ZnxInfos,
+{
+    /// Extracts the a_col-th column of 'a' and stores it on the self_col-th column [Self].
+    pub fn extract_column<R>(&mut self, self_col: usize, a: &R, a_col: usize)
+    where
+        R: VecZnxToRef + ZnxInfos,
+    {
+        #[cfg(debug_assertions)]
+        {
+            assert!(self_col < self.cols());
+            assert!(a_col < a.cols());
+        }
+
+        let min_size: usize = self.size.min(a.size());
+        let max_size: usize = self.size;
+
+        let mut self_mut: VecZnx<&mut [u8]> = self.to_mut();
+        let a_ref: VecZnx<&[u8]> = a.to_ref();
+
+        (0..min_size).for_each(|i: usize| {
+            self_mut
+                .at_mut(self_col, i)
+                .copy_from_slice(a_ref.at(a_col, i));
+        });
+
+        (min_size..max_size).for_each(|i| {
+            self_mut.zero_at(self_col, i);
+        });
+    }
+}
+
+impl<D: AsRef<[u8]>> fmt::Display for VecZnx<D> {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        writeln!(
+            f,
+            "VecZnx(n={}, cols={}, size={})",
+            self.n, self.cols, self.size
+        )?;
+
+        for col in 0..self.cols {
+            writeln!(f, "Column {}:", col)?;
+            for size in 0..self.size {
+                let coeffs = self.at(col, size);
+                write!(f, "  Size {}: [", size)?;
+
+                let max_show = 100;
+                let show_count = coeffs.len().min(max_show);
+
+                for (i, &coeff) in coeffs.iter().take(show_count).enumerate() {
+                    if i > 0 {
+                        write!(f, ", ")?;
+                    }
+                    write!(f, "{}", coeff)?;
+                }
+
+                if coeffs.len() > max_show {
+                    write!(f, ", ... ({} more)", coeffs.len() - max_show)?;
+                }
+
+                writeln!(f, "]")?;
+            }
+        }
+        Ok(())
+    }
+}
+
+pub type VecZnxOwned = VecZnx<Vec<u8>>;
+pub type VecZnxMut<'a> = VecZnx<&'a mut [u8]>;
+pub type VecZnxRef<'a> = VecZnx<&'a [u8]>;
+
+pub trait VecZnxToRef {
+    fn to_ref(&self) -> VecZnx<&[u8]>;
+}
+
+pub trait VecZnxToMut {
+    fn to_mut(&mut self) -> VecZnx<&mut [u8]>;
+}
+
+impl VecZnxToMut for VecZnx<Vec<u8>> {
+    fn to_mut(&mut self) -> VecZnx<&mut [u8]> {
+        VecZnx {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+        }
+    }
+}
+
+impl VecZnxToRef for VecZnx<Vec<u8>> {
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        VecZnx {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+        }
+    }
+}
+
+impl VecZnxToMut for VecZnx<&mut [u8]> {
+    fn to_mut(&mut self) -> VecZnx<&mut [u8]> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+        }
+    }
+}
+
+impl VecZnxToRef for VecZnx<&mut [u8]> {
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+        }
+    }
+}
+
+impl VecZnxToRef for VecZnx<&[u8]> {
+    fn to_ref(&self) -> VecZnx<&[u8]> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+        }
+    }
+}

backend/src/vec_znx_big.rs

@@ -0,0 +1,244 @@
+use crate::ffi::vec_znx_big;
+use crate::znx_base::{ZnxInfos, ZnxView};
+use crate::{Backend, DataView, DataViewMut, FFT64, Module, VecZnx, ZnxSliceSize, ZnxViewMut, ZnxZero, alloc_aligned};
+use std::fmt;
+use std::marker::PhantomData;
+
+pub struct VecZnxBig<D, B: Backend> {
+    data: D,
+    n: usize,
+    cols: usize,
+    size: usize,
+    _phantom: PhantomData<B>,
+}
+
+impl<D, B: Backend> ZnxInfos for VecZnxBig<D, B> {
+    fn cols(&self) -> usize {
+        self.cols
+    }
+
+    fn rows(&self) -> usize {
+        1
+    }
+
+    fn n(&self) -> usize {
+        self.n
+    }
+
+    fn size(&self) -> usize {
+        self.size
+    }
+}
+
+impl<D> ZnxSliceSize for VecZnxBig<D, FFT64> {
+    fn sl(&self) -> usize {
+        self.n() * self.cols()
+    }
+}
+
+impl<D, B: Backend> DataView for VecZnxBig<D, B> {
+    type D = D;
+    fn data(&self) -> &Self::D {
+        &self.data
+    }
+}
+
+impl<D, B: Backend> DataViewMut for VecZnxBig<D, B> {
+    fn data_mut(&mut self) -> &mut Self::D {
+        &mut self.data
+    }
+}
+
+impl<D: AsRef<[u8]>> ZnxView for VecZnxBig<D, FFT64> {
+    type Scalar = i64;
+}
+
+pub(crate) fn bytes_of_vec_znx_big<B: Backend>(module: &Module<B>, cols: usize, size: usize) -> usize {
+    unsafe { vec_znx_big::bytes_of_vec_znx_big(module.ptr, size as u64) as usize * cols }
+}
+
+impl<D: From<Vec<u8>>, B: Backend> VecZnxBig<D, B> {
+    pub(crate) fn new(module: &Module<B>, cols: usize, size: usize) -> Self {
+        let data = alloc_aligned::<u8>(bytes_of_vec_znx_big(module, cols, size));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            cols,
+            size,
+            _phantom: PhantomData,
+        }
+    }
+
+    pub(crate) fn new_from_bytes(module: &Module<B>, cols: usize, size: usize, bytes: impl Into<Vec<u8>>) -> Self {
+        let data: Vec<u8> = bytes.into();
+        assert!(data.len() == bytes_of_vec_znx_big(module, cols, size));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            cols,
+            size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D, B: Backend> VecZnxBig<D, B> {
+    pub(crate) fn from_data(data: D, n: usize, cols: usize, size: usize) -> Self {
+        Self {
+            data,
+            n,
+            cols,
+            size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D> VecZnxBig<D, FFT64>
+where
+    VecZnxBig<D, FFT64>: VecZnxBigToMut<FFT64> + ZnxInfos,
+{
+    // Consumes the VecZnxBig to return a VecZnx.
+    // Useful when no normalization is needed.
+    pub fn to_vec_znx_small(self) -> VecZnx<D> {
+        VecZnx {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+        }
+    }
+
+    /// Extracts the a_col-th column of 'a' and stores it on the self_col-th column [Self].
+    pub fn extract_column<C>(&mut self, self_col: usize, a: &VecZnxBig<C, FFT64>, a_col: usize)
+    where
+        VecZnxBig<C, FFT64>: VecZnxBigToRef<FFT64> + ZnxInfos,
+    {
+        #[cfg(debug_assertions)]
+        {
+            assert!(self_col < self.cols());
+            assert!(a_col < a.cols());
+        }
+
+        let min_size: usize = self.size.min(a.size());
+        let max_size: usize = self.size;
+
+        let mut self_mut: VecZnxBig<&mut [u8], FFT64> = self.to_mut();
+        let a_ref: VecZnxBig<&[u8], FFT64> = a.to_ref();
+
+        (0..min_size).for_each(|i: usize| {
+            self_mut
+                .at_mut(self_col, i)
+                .copy_from_slice(a_ref.at(a_col, i));
+        });
+
+        (min_size..max_size).for_each(|i| {
+            self_mut.zero_at(self_col, i);
+        });
+    }
+}
+
+pub type VecZnxBigOwned<B> = VecZnxBig<Vec<u8>, B>;
+
+pub trait VecZnxBigToRef<B: Backend> {
+    fn to_ref(&self) -> VecZnxBig<&[u8], B>;
+}
+
+pub trait VecZnxBigToMut<B: Backend> {
+    fn to_mut(&mut self) -> VecZnxBig<&mut [u8], B>;
+}
+
+impl<B: Backend> VecZnxBigToMut<B> for VecZnxBig<Vec<u8>, B> {
+    fn to_mut(&mut self) -> VecZnxBig<&mut [u8], B> {
+        VecZnxBig {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxBigToRef<B> for VecZnxBig<Vec<u8>, B> {
+    fn to_ref(&self) -> VecZnxBig<&[u8], B> {
+        VecZnxBig {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxBigToMut<B> for VecZnxBig<&mut [u8], B> {
+    fn to_mut(&mut self) -> VecZnxBig<&mut [u8], B> {
+        VecZnxBig {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxBigToRef<B> for VecZnxBig<&mut [u8], B> {
+    fn to_ref(&self) -> VecZnxBig<&[u8], B> {
+        VecZnxBig {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxBigToRef<B> for VecZnxBig<&[u8], B> {
+    fn to_ref(&self) -> VecZnxBig<&[u8], B> {
+        VecZnxBig {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D: AsRef<[u8]>> fmt::Display for VecZnxBig<D, FFT64> {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        writeln!(
+            f,
+            "VecZnxBig(n={}, cols={}, size={})",
+            self.n, self.cols, self.size
+        )?;
+
+        for col in 0..self.cols {
+            writeln!(f, "Column {}:", col)?;
+            for size in 0..self.size {
+                let coeffs = self.at(col, size);
+                write!(f, "  Size {}: [", size)?;
+
+                let max_show = 100;
+                let show_count = coeffs.len().min(max_show);
+
+                for (i, &coeff) in coeffs.iter().take(show_count).enumerate() {
+                    if i > 0 {
+                        write!(f, ", ")?;
+                    }
+                    write!(f, "{}", coeff)?;
+                }
+
+                if coeffs.len() > max_show {
+                    write!(f, ", ... ({} more)", coeffs.len() - max_show)?;
+                }
+
+                writeln!(f, "]")?;
+            }
+        }
+        Ok(())
+    }
+}

backend/src/vec_znx_big_ops.rs

@@ -0,0 +1,632 @@
+use crate::ffi::vec_znx;
+use crate::znx_base::{ZnxInfos, ZnxView, ZnxViewMut};
+use crate::{
+    Backend, FFT64, Module, Scratch, VecZnx, VecZnxBig, VecZnxBigOwned, VecZnxBigToMut, VecZnxBigToRef, VecZnxScratch,
+    VecZnxToMut, VecZnxToRef, ZnxSliceSize, bytes_of_vec_znx_big,
+};
+
+pub trait VecZnxBigAlloc<B: Backend> {
+    /// Allocates a vector Z[X]/(X^N+1) that stores not normalized values.
+    fn new_vec_znx_big(&self, cols: usize, size: usize) -> VecZnxBigOwned<B>;
+
+    /// Returns a new [VecZnxBig] with the provided bytes array as backing array.
+    ///
+    /// Behavior: takes ownership of the backing array.
+    ///
+    /// # Arguments
+    ///
+    /// * `cols`: the number of polynomials..
+    /// * `size`: the number of polynomials per column.
+    /// * `bytes`: a byte array of size at least [Module::bytes_of_vec_znx_big].
+    ///
+    /// # Panics
+    /// If `bytes.len()` < [Module::bytes_of_vec_znx_big].
+    fn new_vec_znx_big_from_bytes(&self, cols: usize, size: usize, bytes: Vec<u8>) -> VecZnxBigOwned<B>;
+
+    // /// Returns a new [VecZnxBig] with the provided bytes array as backing array.
+    // ///
+    // /// Behavior: the backing array is only borrowed.
+    // ///
+    // /// # Arguments
+    // ///
+    // /// * `cols`: the number of polynomials..
+    // /// * `size`: the number of polynomials per column.
+    // /// * `bytes`: a byte array of size at least [Module::bytes_of_vec_znx_big].
+    // ///
+    // /// # Panics
+    // /// If `bytes.len()` < [Module::bytes_of_vec_znx_big].
+    // fn new_vec_znx_big_from_bytes_borrow(&self, cols: usize, size: usize, tmp_bytes: &mut [u8]) -> VecZnxBig<B>;
+
+    /// Returns the minimum number of bytes necessary to allocate
+    /// a new [VecZnxBig] through [VecZnxBig::from_bytes].
+    fn bytes_of_vec_znx_big(&self, cols: usize, size: usize) -> usize;
+}
+
+pub trait VecZnxBigOps<BACKEND: Backend> {
+    /// Adds `a` to `b` and stores the result on `c`.
+    fn vec_znx_big_add<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>,
+        B: VecZnxBigToRef<BACKEND>;
+
+    /// Adds `a` to `b` and stores the result on `b`.
+    fn vec_znx_big_add_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>;
+
+    /// Adds `a` to `b` and stores the result on `c`.
+    fn vec_znx_big_add_small<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>,
+        B: VecZnxToRef;
+
+    /// Adds `a` to `b` and stores the result on `b`.
+    fn vec_znx_big_add_small_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxToRef;
+
+    /// Subtracts `a` to `b` and stores the result on `c`.
+    fn vec_znx_big_sub<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>,
+        B: VecZnxBigToRef<BACKEND>;
+
+    /// Subtracts `a` from `b` and stores the result on `b`.
+    fn vec_znx_big_sub_ab_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>;
+
+    /// Subtracts `b` from `a` and stores the result on `b`.
+    fn vec_znx_big_sub_ba_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>;
+
+    /// Subtracts `b` from `a` and stores the result on `c`.
+    fn vec_znx_big_sub_small_a<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxToRef,
+        B: VecZnxBigToRef<BACKEND>;
+
+    /// Subtracts `a` from `res` and stores the result on `res`.
+    fn vec_znx_big_sub_small_a_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxToRef;
+
+    /// Subtracts `b` from `a` and stores the result on `c`.
+    fn vec_znx_big_sub_small_b<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>,
+        B: VecZnxToRef;
+
+    /// Subtracts `res` from `a` and stores the result on `res`.
+    fn vec_znx_big_sub_small_b_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxToRef;
+
+    /// Negates `a` inplace.
+    fn vec_znx_big_negate_inplace<A>(&self, a: &mut A, a_col: usize)
+    where
+        A: VecZnxBigToMut<BACKEND>;
+
+    /// Normalizes `a` and stores the result on `b`.
+    ///
+    /// # Arguments
+    ///
+    /// * `basek`: normalization basis.
+    /// * `tmp_bytes`: scratch space of size at least [VecZnxBigOps::vec_znx_big_normalize].
+    fn vec_znx_big_normalize<R, A>(
+        &self,
+        basek: usize,
+        res: &mut R,
+        res_col: usize,
+        a: &A,
+        a_col: usize,
+        scratch: &mut Scratch,
+    ) where
+        R: VecZnxToMut,
+        A: VecZnxBigToRef<BACKEND>;
+
+    /// Applies the automorphism X^i -> X^ik on `a` and stores the result on `b`.
+    fn vec_znx_big_automorphism<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<BACKEND>,
+        A: VecZnxBigToRef<BACKEND>;
+
+    /// Applies the automorphism X^i -> X^ik on `a` and stores the result on `a`.
+    fn vec_znx_big_automorphism_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: VecZnxBigToMut<BACKEND>;
+}
+
+pub trait VecZnxBigScratch {
+    /// Returns the minimum number of bytes to apply [VecZnxBigOps::vec_znx_big_normalize].
+    fn vec_znx_big_normalize_tmp_bytes(&self) -> usize;
+}
+
+impl<B: Backend> VecZnxBigAlloc<B> for Module<B> {
+    fn new_vec_znx_big(&self, cols: usize, size: usize) -> VecZnxBigOwned<B> {
+        VecZnxBig::new(self, cols, size)
+    }
+
+    fn new_vec_znx_big_from_bytes(&self, cols: usize, size: usize, bytes: Vec<u8>) -> VecZnxBigOwned<B> {
+        VecZnxBig::new_from_bytes(self, cols, size, bytes)
+    }
+
+    fn bytes_of_vec_znx_big(&self, cols: usize, size: usize) -> usize {
+        bytes_of_vec_znx_big(self, cols, size)
+    }
+}
+
+impl VecZnxBigOps<FFT64> for Module<FFT64> {
+    fn vec_znx_big_add<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+        B: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let b: VecZnxBig<&[u8], FFT64> = b.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_add_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+        B: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let b: VecZnxBig<&[u8], FFT64> = b.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub_ab_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub_ba_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub_small_b<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+        B: VecZnxToRef,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let b: VecZnx<&[u8]> = b.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub_small_b_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub_small_a<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxToRef,
+        B: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let b: VecZnxBig<&[u8], FFT64> = b.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_sub_small_a_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_add_small<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+        B: VecZnxToRef,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let b: VecZnx<&[u8]> = b.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_add_small_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_negate_inplace<A>(&self, a: &mut A, res_col: usize)
+    where
+        A: VecZnxBigToMut<FFT64>,
+    {
+        let mut a: VecZnxBig<&mut [u8], FFT64> = a.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_negate(
+                self.ptr,
+                a.at_mut_ptr(res_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(res_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_normalize<R, A>(
+        &self,
+        basek: usize,
+        res: &mut R,
+        res_col: usize,
+        a: &A,
+        a_col: usize,
+        scratch: &mut Scratch,
+    ) where
+        R: VecZnxToMut,
+        A: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            //(Jay)Note: This is calling VezZnxOps::vec_znx_normalize_tmp_bytes and not VecZnxBigOps::vec_znx_big_normalize_tmp_bytes.
+            // In the FFT backend the tmp sizes are same but will be different in the NTT backend
+            // assert!(tmp_bytes.len() >= <Self as VecZnxOps<&mut [u8], & [u8]>>::vec_znx_normalize_tmp_bytes(&self));
+            // assert_alignement(tmp_bytes.as_ptr());
+        }
+
+        let (tmp_bytes, _) = scratch.tmp_slice(<Self as VecZnxBigScratch>::vec_znx_big_normalize_tmp_bytes(
+            &self,
+        ));
+        unsafe {
+            vec_znx::vec_znx_normalize_base2k(
+                self.ptr,
+                basek as u64,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                tmp_bytes.as_mut_ptr(),
+            );
+        }
+    }
+
+    fn vec_znx_big_automorphism<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxBigToRef<FFT64>,
+    {
+        let a: VecZnxBig<&[u8], FFT64> = a.to_ref();
+        let mut res: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_automorphism(
+                self.ptr,
+                k,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_big_automorphism_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: VecZnxBigToMut<FFT64>,
+    {
+        let mut a: VecZnxBig<&mut [u8], FFT64> = a.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_automorphism(
+                self.ptr,
+                k,
+                a.at_mut_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+}
+
+impl<B: Backend> VecZnxBigScratch for Module<B> {
+    fn vec_znx_big_normalize_tmp_bytes(&self) -> usize {
+        <Self as VecZnxScratch>::vec_znx_normalize_tmp_bytes(self)
+    }
+}

backend/src/vec_znx_dft.rs

@@ -0,0 +1,242 @@
+use std::marker::PhantomData;
+
+use crate::ffi::vec_znx_dft;
+use crate::znx_base::ZnxInfos;
+use crate::{
+    Backend, DataView, DataViewMut, FFT64, Module, VecZnxBig, ZnxSliceSize, ZnxView, ZnxViewMut, ZnxZero, alloc_aligned,
+};
+use std::fmt;
+
+pub struct VecZnxDft<D, B: Backend> {
+    pub(crate) data: D,
+    pub(crate) n: usize,
+    pub(crate) cols: usize,
+    pub(crate) size: usize,
+    pub(crate) _phantom: PhantomData<B>,
+}
+
+impl<D, B: Backend> VecZnxDft<D, B> {
+    pub fn into_big(self) -> VecZnxBig<D, B> {
+        VecZnxBig::<D, B>::from_data(self.data, self.n, self.cols, self.size)
+    }
+}
+
+impl<D, B: Backend> ZnxInfos for VecZnxDft<D, B> {
+    fn cols(&self) -> usize {
+        self.cols
+    }
+
+    fn rows(&self) -> usize {
+        1
+    }
+
+    fn n(&self) -> usize {
+        self.n
+    }
+
+    fn size(&self) -> usize {
+        self.size
+    }
+}
+
+impl<D> ZnxSliceSize for VecZnxDft<D, FFT64> {
+    fn sl(&self) -> usize {
+        self.n() * self.cols()
+    }
+}
+
+impl<D, B: Backend> DataView for VecZnxDft<D, B> {
+    type D = D;
+    fn data(&self) -> &Self::D {
+        &self.data
+    }
+}
+
+impl<D, B: Backend> DataViewMut for VecZnxDft<D, B> {
+    fn data_mut(&mut self) -> &mut Self::D {
+        &mut self.data
+    }
+}
+
+impl<D: AsRef<[u8]>> ZnxView for VecZnxDft<D, FFT64> {
+    type Scalar = f64;
+}
+
+pub(crate) fn bytes_of_vec_znx_dft<B: Backend>(module: &Module<B>, cols: usize, size: usize) -> usize {
+    unsafe { vec_znx_dft::bytes_of_vec_znx_dft(module.ptr, size as u64) as usize * cols }
+}
+
+impl<D: From<Vec<u8>>, B: Backend> VecZnxDft<D, B> {
+    pub(crate) fn new(module: &Module<B>, cols: usize, size: usize) -> Self {
+        let data = alloc_aligned::<u8>(bytes_of_vec_znx_dft(module, cols, size));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            cols,
+            size,
+            _phantom: PhantomData,
+        }
+    }
+
+    pub(crate) fn new_from_bytes(module: &Module<B>, cols: usize, size: usize, bytes: impl Into<Vec<u8>>) -> Self {
+        let data: Vec<u8> = bytes.into();
+        assert!(data.len() == bytes_of_vec_znx_dft(module, cols, size));
+        Self {
+            data: data.into(),
+            n: module.n(),
+            cols,
+            size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D> VecZnxDft<D, FFT64>
+where
+    VecZnxDft<D, FFT64>: VecZnxDftToMut<FFT64> + ZnxInfos,
+{
+    /// Extracts the a_col-th column of 'a' and stores it on the self_col-th column [Self].
+    pub fn extract_column<C>(&mut self, self_col: usize, a: &VecZnxDft<C, FFT64>, a_col: usize)
+    where
+        VecZnxDft<C, FFT64>: VecZnxDftToRef<FFT64> + ZnxInfos,
+    {
+        #[cfg(debug_assertions)]
+        {
+            assert!(self_col < self.cols());
+            assert!(a_col < a.cols());
+        }
+
+        let min_size: usize = self.size.min(a.size());
+        let max_size: usize = self.size;
+
+        let mut self_mut: VecZnxDft<&mut [u8], FFT64> = self.to_mut();
+        let a_ref: VecZnxDft<&[u8], FFT64> = a.to_ref();
+
+        (0..min_size).for_each(|i: usize| {
+            self_mut
+                .at_mut(self_col, i)
+                .copy_from_slice(a_ref.at(a_col, i));
+        });
+
+        (min_size..max_size).for_each(|i| {
+            self_mut.zero_at(self_col, i);
+        });
+    }
+}
+
+pub type VecZnxDftOwned<B> = VecZnxDft<Vec<u8>, B>;
+
+impl<D, B: Backend> VecZnxDft<D, B> {
+    pub(crate) fn from_data(data: D, n: usize, cols: usize, size: usize) -> Self {
+        Self {
+            data,
+            n,
+            cols,
+            size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+pub trait VecZnxDftToRef<B: Backend> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B>;
+}
+
+pub trait VecZnxDftToMut<B: Backend> {
+    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B>;
+}
+
+impl<B: Backend> VecZnxDftToMut<B> for VecZnxDft<Vec<u8>, B> {
+    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B> {
+        VecZnxDft {
+            data: self.data.as_mut_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToRef<B> for VecZnxDft<Vec<u8>, B> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        VecZnxDft {
+            data: self.data.as_slice(),
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToMut<B> for VecZnxDft<&mut [u8], B> {
+    fn to_mut(&mut self) -> VecZnxDft<&mut [u8], B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToRef<B> for VecZnxDft<&mut [u8], B> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<B: Backend> VecZnxDftToRef<B> for VecZnxDft<&[u8], B> {
+    fn to_ref(&self) -> VecZnxDft<&[u8], B> {
+        VecZnxDft {
+            data: self.data,
+            n: self.n,
+            cols: self.cols,
+            size: self.size,
+            _phantom: PhantomData,
+        }
+    }
+}
+
+impl<D: AsRef<[u8]>> fmt::Display for VecZnxDft<D, FFT64> {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        writeln!(
+            f,
+            "VecZnxDft(n={}, cols={}, size={})",
+            self.n, self.cols, self.size
+        )?;
+
+        for col in 0..self.cols {
+            writeln!(f, "Column {}:", col)?;
+            for size in 0..self.size {
+                let coeffs = self.at(col, size);
+                write!(f, "  Size {}: [", size)?;
+
+                let max_show = 100;
+                let show_count = coeffs.len().min(max_show);
+
+                for (i, &coeff) in coeffs.iter().take(show_count).enumerate() {
+                    if i > 0 {
+                        write!(f, ", ")?;
+                    }
+                    write!(f, "{}", coeff)?;
+                }
+
+                if coeffs.len() > max_show {
+                    write!(f, ", ... ({} more)", coeffs.len() - max_show)?;
+                }
+
+                writeln!(f, "]")?;
+            }
+        }
+        Ok(())
+    }
+}

backend/src/vec_znx_dft_ops.rs

@@ -0,0 +1,287 @@
+use crate::ffi::{vec_znx_big, vec_znx_dft};
+use crate::vec_znx_dft::bytes_of_vec_znx_dft;
+use crate::znx_base::ZnxInfos;
+use crate::{
+    Backend, Scratch, VecZnxBig, VecZnxBigToMut, VecZnxDft, VecZnxDftOwned, VecZnxDftToMut, VecZnxDftToRef, VecZnxToRef,
+    ZnxSliceSize,
+};
+use crate::{FFT64, Module, ZnxView, ZnxViewMut, ZnxZero};
+use std::cmp::min;
+
+pub trait VecZnxDftAlloc<B: Backend> {
+    /// Allocates a vector Z[X]/(X^N+1) that stores normalized in the DFT space.
+    fn new_vec_znx_dft(&self, cols: usize, size: usize) -> VecZnxDftOwned<B>;
+
+    /// Returns a new [VecZnxDft] with the provided bytes array as backing array.
+    ///
+    /// Behavior: takes ownership of the backing array.
+    ///
+    /// # Arguments
+    ///
+    /// * `cols`: the number of cols of the [VecZnxDft].
+    /// * `bytes`: a byte array of size at least [Module::bytes_of_vec_znx_dft].
+    ///
+    /// # Panics
+    /// If `bytes.len()` < [Module::bytes_of_vec_znx_dft].
+    fn new_vec_znx_dft_from_bytes(&self, cols: usize, size: usize, bytes: Vec<u8>) -> VecZnxDftOwned<B>;
+
+    /// Returns a new [VecZnxDft] with the provided bytes array as backing array.
+    ///
+    /// # Arguments
+    ///
+    /// * `cols`: the number of cols of the [VecZnxDft].
+    /// * `bytes`: a byte array of size at least [Module::bytes_of_vec_znx_dft].
+    ///
+    /// # Panics
+    /// If `bytes.len()` < [Module::bytes_of_vec_znx_dft].
+    fn bytes_of_vec_znx_dft(&self, cols: usize, size: usize) -> usize;
+}
+
+pub trait VecZnxDftOps<B: Backend> {
+    /// Returns the minimum number of bytes necessary to allocate
+    /// a new [VecZnxDft] through [VecZnxDft::from_bytes].
+    fn vec_znx_idft_tmp_bytes(&self) -> usize;
+
+    fn vec_znx_dft_add<R, A, D>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &D, b_col: usize)
+    where
+        R: VecZnxDftToMut<B>,
+        A: VecZnxDftToRef<B>,
+        D: VecZnxDftToRef<B>;
+
+    fn vec_znx_dft_add_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<B>,
+        A: VecZnxDftToRef<B>;
+
+    fn vec_znx_dft_copy<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<B>,
+        A: VecZnxDftToRef<B>;
+
+    /// b <- IDFT(a), uses a as scratch space.
+    fn vec_znx_idft_tmp_a<R, A>(&self, res: &mut R, res_col: usize, a: &mut A, a_col: usize)
+    where
+        R: VecZnxBigToMut<B>,
+        A: VecZnxDftToMut<B>;
+
+    /// Consumes a to return IDFT(a) in big coeff space.
+    fn vec_znx_idft_consume<D>(&self, a: VecZnxDft<D, B>) -> VecZnxBig<D, FFT64>
+    where
+        VecZnxDft<D, FFT64>: VecZnxDftToMut<FFT64>;
+
+    fn vec_znx_idft<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, scratch: &mut Scratch)
+    where
+        R: VecZnxBigToMut<B>,
+        A: VecZnxDftToRef<B>;
+
+    fn vec_znx_dft<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<B>,
+        A: VecZnxToRef;
+}
+
+impl<B: Backend> VecZnxDftAlloc<B> for Module<B> {
+    fn new_vec_znx_dft(&self, cols: usize, size: usize) -> VecZnxDftOwned<B> {
+        VecZnxDftOwned::new(&self, cols, size)
+    }
+
+    fn new_vec_znx_dft_from_bytes(&self, cols: usize, size: usize, bytes: Vec<u8>) -> VecZnxDftOwned<B> {
+        VecZnxDftOwned::new_from_bytes(self, cols, size, bytes)
+    }
+
+    fn bytes_of_vec_znx_dft(&self, cols: usize, size: usize) -> usize {
+        bytes_of_vec_znx_dft(self, cols, size)
+    }
+}
+
+impl VecZnxDftOps<FFT64> for Module<FFT64> {
+    fn vec_znx_dft_add<R, A, D>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &D, b_col: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+        D: VecZnxDftToRef<FFT64>,
+    {
+        let mut res_mut: VecZnxDft<&mut [u8], FFT64> = res.to_mut();
+        let a_ref: VecZnxDft<&[u8], FFT64> = a.to_ref();
+        let b_ref: VecZnxDft<&[u8], FFT64> = b.to_ref();
+
+        let min_size: usize = res_mut.size().min(a_ref.size()).min(b_ref.size());
+
+        unsafe {
+            (0..min_size).for_each(|j| {
+                vec_znx_dft::vec_dft_add(
+                    self.ptr,
+                    res_mut.at_mut_ptr(res_col, j) as *mut vec_znx_dft::vec_znx_dft_t,
+                    1,
+                    a_ref.at_ptr(a_col, j) as *const vec_znx_dft::vec_znx_dft_t,
+                    1,
+                    b_ref.at_ptr(b_col, j) as *const vec_znx_dft::vec_znx_dft_t,
+                    1,
+                );
+            });
+        }
+        (min_size..res_mut.size()).for_each(|j| {
+            res_mut.zero_at(res_col, j);
+        })
+    }
+
+    fn vec_znx_dft_add_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+    {
+        let mut res_mut: VecZnxDft<&mut [u8], FFT64> = res.to_mut();
+        let a_ref: VecZnxDft<&[u8], FFT64> = a.to_ref();
+
+        let min_size: usize = res_mut.size().min(a_ref.size());
+
+        unsafe {
+            (0..min_size).for_each(|j| {
+                vec_znx_dft::vec_dft_add(
+                    self.ptr,
+                    res_mut.at_mut_ptr(res_col, j) as *mut vec_znx_dft::vec_znx_dft_t,
+                    1,
+                    res_mut.at_ptr(res_col, j) as *const vec_znx_dft::vec_znx_dft_t,
+                    1,
+                    a_ref.at_ptr(a_col, j) as *const vec_znx_dft::vec_znx_dft_t,
+                    1,
+                );
+            });
+        }
+    }
+
+    fn vec_znx_dft_copy<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+    {
+        let mut res_mut: VecZnxDft<&mut [u8], FFT64> = res.to_mut();
+        let a_ref: VecZnxDft<&[u8], FFT64> = a.to_ref();
+
+        let min_size: usize = min(res_mut.size(), a_ref.size());
+
+        (0..min_size).for_each(|j| {
+            res_mut
+                .at_mut(res_col, j)
+                .copy_from_slice(a_ref.at(a_col, j));
+        });
+        (min_size..res_mut.size()).for_each(|j| {
+            res_mut.zero_at(res_col, j);
+        })
+    }
+
+    fn vec_znx_idft_tmp_a<R, A>(&self, res: &mut R, res_col: usize, a: &mut A, a_col: usize)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxDftToMut<FFT64>,
+    {
+        let mut res_mut: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+        let mut a_mut: VecZnxDft<&mut [u8], FFT64> = a.to_mut();
+
+        let min_size: usize = min(res_mut.size(), a_mut.size());
+
+        unsafe {
+            (0..min_size).for_each(|j| {
+                vec_znx_dft::vec_znx_idft_tmp_a(
+                    self.ptr,
+                    res_mut.at_mut_ptr(res_col, j) as *mut vec_znx_big::vec_znx_big_t,
+                    1 as u64,
+                    a_mut.at_mut_ptr(a_col, j) as *mut vec_znx_dft::vec_znx_dft_t,
+                    1 as u64,
+                )
+            });
+            (min_size..res_mut.size()).for_each(|j| {
+                res_mut.zero_at(res_col, j);
+            })
+        }
+    }
+
+    fn vec_znx_idft_consume<D>(&self, mut a: VecZnxDft<D, FFT64>) -> VecZnxBig<D, FFT64>
+    where
+        VecZnxDft<D, FFT64>: VecZnxDftToMut<FFT64>,
+    {
+        let mut a_mut: VecZnxDft<&mut [u8], FFT64> = a.to_mut();
+
+        unsafe {
+            // Rev col and rows because ZnxDft.sl() >= ZnxBig.sl()
+            (0..a_mut.size()).for_each(|j| {
+                (0..a_mut.cols()).for_each(|i| {
+                    vec_znx_dft::vec_znx_idft_tmp_a(
+                        self.ptr,
+                        a_mut.at_mut_ptr(i, j) as *mut vec_znx_big::vec_znx_big_t,
+                        1 as u64,
+                        a_mut.at_mut_ptr(i, j) as *mut vec_znx_dft::vec_znx_dft_t,
+                        1 as u64,
+                    )
+                });
+            });
+        }
+
+        a.into_big()
+    }
+
+    fn vec_znx_idft_tmp_bytes(&self) -> usize {
+        unsafe { vec_znx_dft::vec_znx_idft_tmp_bytes(self.ptr) as usize }
+    }
+
+    /// b <- DFT(a)
+    ///
+    /// # Panics
+    /// If b.cols < a_col
+    fn vec_znx_dft<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxDftToMut<FFT64>,
+        A: VecZnxToRef,
+    {
+        let mut res_mut: VecZnxDft<&mut [u8], FFT64> = res.to_mut();
+        let a_ref: crate::VecZnx<&[u8]> = a.to_ref();
+
+        let min_size: usize = min(res_mut.size(), a_ref.size());
+
+        unsafe {
+            (0..min_size).for_each(|j| {
+                vec_znx_dft::vec_znx_dft(
+                    self.ptr,
+                    res_mut.at_mut_ptr(res_col, j) as *mut vec_znx_dft::vec_znx_dft_t,
+                    1 as u64,
+                    a_ref.at_ptr(a_col, j),
+                    1 as u64,
+                    a_ref.sl() as u64,
+                )
+            });
+            (min_size..res_mut.size()).for_each(|j| {
+                res_mut.zero_at(res_col, j);
+            });
+        }
+    }
+
+    // b <- IDFT(a), scratch space size obtained with [vec_znx_idft_tmp_bytes].
+    fn vec_znx_idft<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, scratch: &mut Scratch)
+    where
+        R: VecZnxBigToMut<FFT64>,
+        A: VecZnxDftToRef<FFT64>,
+    {
+        let mut res_mut: VecZnxBig<&mut [u8], FFT64> = res.to_mut();
+        let a_ref: VecZnxDft<&[u8], FFT64> = a.to_ref();
+
+        let (tmp_bytes, _) = scratch.tmp_slice(self.vec_znx_idft_tmp_bytes());
+
+        let min_size: usize = min(res_mut.size(), a_ref.size());
+
+        unsafe {
+            (0..min_size).for_each(|j| {
+                vec_znx_dft::vec_znx_idft(
+                    self.ptr,
+                    res_mut.at_mut_ptr(res_col, j) as *mut vec_znx_big::vec_znx_big_t,
+                    1 as u64,
+                    a_ref.at_ptr(a_col, j) as *const vec_znx_dft::vec_znx_dft_t,
+                    1 as u64,
+                    tmp_bytes.as_mut_ptr(),
+                )
+            });
+            (min_size..res_mut.size()).for_each(|j| {
+                res_mut.zero_at(res_col, j);
+            });
+        }
+    }
+}

backend/src/vec_znx_ops.rs

@@ -0,0 +1,694 @@
+use crate::ffi::vec_znx;
+use crate::{
+    Backend, Module, ScalarZnxToRef, Scratch, VecZnx, VecZnxOwned, VecZnxToMut, VecZnxToRef, ZnxInfos, ZnxSliceSize, ZnxView,
+    ZnxViewMut, ZnxZero,
+};
+use itertools::izip;
+use std::cmp::min;
+
+pub trait VecZnxAlloc {
+    /// Allocates a new [VecZnx].
+    ///
+    /// # Arguments
+    ///
+    /// * `cols`: the number of polynomials.
+    /// * `size`: the number small polynomials per column.
+    fn new_vec_znx(&self, cols: usize, size: usize) -> VecZnxOwned;
+
+    /// Instantiates a new [VecZnx] from a slice of bytes.
+    /// The returned [VecZnx] takes ownership of the slice of bytes.
+    ///
+    /// # Arguments
+    ///
+    /// * `cols`: the number of polynomials.
+    /// * `size`: the number small polynomials per column.
+    ///
+    /// # Panic
+    /// Requires the slice of bytes to be equal to [VecZnxOps::bytes_of_vec_znx].
+    fn new_vec_znx_from_bytes(&self, cols: usize, size: usize, bytes: Vec<u8>) -> VecZnxOwned;
+
+    /// Returns the number of bytes necessary to allocate
+    /// a new [VecZnx] through [VecZnxOps::new_vec_znx_from_bytes]
+    /// or [VecZnxOps::new_vec_znx_from_bytes_borrow].
+    fn bytes_of_vec_znx(&self, cols: usize, size: usize) -> usize;
+}
+
+pub trait VecZnxOps {
+    /// Normalizes the selected column of `a` and stores the result into the selected column of `res`.
+    fn vec_znx_normalize<R, A>(&self, basek: usize, res: &mut R, res_col: usize, a: &A, a_col: usize, scratch: &mut Scratch)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Normalizes the selected column of `a`.
+    fn vec_znx_normalize_inplace<A>(&self, basek: usize, a: &mut A, a_col: usize, scratch: &mut Scratch)
+    where
+        A: VecZnxToMut;
+
+    /// Adds the selected column of `a` to the selected column of `b` and writes the result on the selected column of `res`.
+    fn vec_znx_add<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+        B: VecZnxToRef;
+
+    /// Adds the selected column of `a` to the selected column of `res` and writes the result on the selected column of `res`.
+    fn vec_znx_add_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Adds the selected column of `a` on the selected column and limb of `res`.
+    fn vec_znx_add_scalar_inplace<R, A>(&self, res: &mut R, res_col: usize, res_limb: usize, a: &A, b_col: usize)
+    where
+        R: VecZnxToMut,
+        A: ScalarZnxToRef;
+
+    /// Subtracts the selected column of `b` from the selected column of `a` and writes the result on the selected column of `res`.
+    fn vec_znx_sub<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+        B: VecZnxToRef;
+
+    /// Subtracts the selected column of `a` from the selected column of `res` inplace.
+    ///
+    /// res[res_col] -= a[a_col]
+    fn vec_znx_sub_ab_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Subtracts the selected column of `res` from the selected column of `a` and inplace mutates `res`
+    ///
+    /// res[res_col] = a[a_col] - res[res_col]
+    fn vec_znx_sub_ba_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Subtracts the selected column of `a` on the selected column and limb of `res`.
+    fn vec_znx_sub_scalar_inplace<R, A>(&self, res: &mut R, res_col: usize, res_limb: usize, a: &A, b_col: usize)
+    where
+        R: VecZnxToMut,
+        A: ScalarZnxToRef;
+
+    // Negates the selected column of `a` and stores the result in `res_col` of `res`.
+    fn vec_znx_negate<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Negates the selected column of `a`.
+    fn vec_znx_negate_inplace<A>(&self, a: &mut A, a_col: usize)
+    where
+        A: VecZnxToMut;
+
+    /// Multiplies the selected column of `a` by X^k and stores the result in `res_col` of `res`.
+    fn vec_znx_rotate<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Multiplies the selected column of `a` by X^k.
+    fn vec_znx_rotate_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: VecZnxToMut;
+
+    /// Applies the automorphism X^i -> X^ik on the selected column of `a` and stores the result in `res_col` column of `res`.
+    fn vec_znx_automorphism<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Applies the automorphism X^i -> X^ik on the selected column of `a`.
+    fn vec_znx_automorphism_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: VecZnxToMut;
+
+    /// Splits the selected columns of `b` into subrings and copies them them into the selected column of `res`.
+    ///
+    /// # Panics
+    ///
+    /// This method requires that all [VecZnx] of b have the same ring degree
+    /// and that b.n() * b.len() <= a.n()
+    fn vec_znx_split<R, A>(&self, res: &mut Vec<R>, res_col: usize, a: &A, a_col: usize, scratch: &mut Scratch)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    /// Merges the subrings of the selected column of `a` into the selected column of `res`.
+    ///
+    /// # Panics
+    ///
+    /// This method requires that all [VecZnx] of a have the same ring degree
+    /// and that a.n() * a.len() <= b.n()
+    fn vec_znx_merge<R, A>(&self, res: &mut R, res_col: usize, a: Vec<A>, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+
+    fn switch_degree<R, A>(&self, r: &mut R, col_b: usize, a: &A, col_a: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef;
+}
+
+pub trait VecZnxScratch {
+    /// Returns the minimum number of bytes necessary for normalization.
+    fn vec_znx_normalize_tmp_bytes(&self) -> usize;
+}
+
+impl<B: Backend> VecZnxAlloc for Module<B> {
+    fn new_vec_znx(&self, cols: usize, size: usize) -> VecZnxOwned {
+        VecZnxOwned::new::<i64>(self.n(), cols, size)
+    }
+
+    fn bytes_of_vec_znx(&self, cols: usize, size: usize) -> usize {
+        VecZnxOwned::bytes_of::<i64>(self.n(), cols, size)
+    }
+
+    fn new_vec_znx_from_bytes(&self, cols: usize, size: usize, bytes: Vec<u8>) -> VecZnxOwned {
+        VecZnxOwned::new_from_bytes::<i64>(self.n(), cols, size, bytes)
+    }
+}
+
+impl<BACKEND: Backend> VecZnxOps for Module<BACKEND> {
+    fn vec_znx_normalize<R, A>(&self, basek: usize, res: &mut R, res_col: usize, a: &A, a_col: usize, scratch: &mut Scratch)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+
+        let (tmp_bytes, _) = scratch.tmp_slice(self.vec_znx_normalize_tmp_bytes());
+
+        unsafe {
+            vec_znx::vec_znx_normalize_base2k(
+                self.ptr,
+                basek as u64,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                tmp_bytes.as_mut_ptr(),
+            );
+        }
+    }
+
+    fn vec_znx_normalize_inplace<A>(&self, basek: usize, a: &mut A, a_col: usize, scratch: &mut Scratch)
+    where
+        A: VecZnxToMut,
+    {
+        let mut a: VecZnx<&mut [u8]> = a.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+        }
+
+        let (tmp_bytes, _) = scratch.tmp_slice(self.vec_znx_normalize_tmp_bytes());
+
+        unsafe {
+            vec_znx::vec_znx_normalize_base2k(
+                self.ptr,
+                basek as u64,
+                a.at_mut_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                tmp_bytes.as_mut_ptr(),
+            );
+        }
+    }
+
+    fn vec_znx_add<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+        B: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let b: VecZnx<&[u8]> = b.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_add_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_add_scalar_inplace<R, A>(&self, res: &mut R, res_col: usize, res_limb: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: ScalarZnxToRef,
+    {
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        let a: crate::ScalarZnx<&[u8]> = a.to_ref();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+
+        unsafe {
+            vec_znx::vec_znx_add(
+                self.ptr,
+                res.at_mut_ptr(res_col, res_limb),
+                1 as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, res_limb),
+                1 as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_sub<R, A, B>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize, b: &B, b_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+        B: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let b: VecZnx<&[u8]> = b.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(b.n(), self.n());
+            assert_eq!(res.n(), self.n());
+            assert_ne!(a.as_ptr(), b.as_ptr());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                b.at_ptr(b_col, 0),
+                b.size() as u64,
+                b.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_sub_scalar_inplace<R, A>(&self, res: &mut R, res_col: usize, res_limb: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: ScalarZnxToRef,
+    {
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        let a: crate::ScalarZnx<&[u8]> = a.to_ref();
+
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, res_limb),
+                1 as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, res_limb),
+                1 as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_sub_ab_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_sub_ba_inplace<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_sub(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                res.at_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_negate<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_negate(
+                self.ptr,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_negate_inplace<A>(&self, a: &mut A, a_col: usize)
+    where
+        A: VecZnxToMut,
+    {
+        let mut a: VecZnx<&mut [u8]> = a.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_negate(
+                self.ptr,
+                a.at_mut_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_rotate<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_rotate(
+                self.ptr,
+                k,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_rotate_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: VecZnxToMut,
+    {
+        let mut a: VecZnx<&mut [u8]> = a.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_rotate(
+                self.ptr,
+                k,
+                a.at_mut_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_automorphism<R, A>(&self, k: i64, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert_eq!(res.n(), self.n());
+        }
+        unsafe {
+            vec_znx::vec_znx_automorphism(
+                self.ptr,
+                k,
+                res.at_mut_ptr(res_col, 0),
+                res.size() as u64,
+                res.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_automorphism_inplace<A>(&self, k: i64, a: &mut A, a_col: usize)
+    where
+        A: VecZnxToMut,
+    {
+        let mut a: VecZnx<&mut [u8]> = a.to_mut();
+        #[cfg(debug_assertions)]
+        {
+            assert_eq!(a.n(), self.n());
+            assert!(
+                k & 1 != 0,
+                "invalid galois element: must be odd but is {}",
+                k
+            );
+        }
+        unsafe {
+            vec_znx::vec_znx_automorphism(
+                self.ptr,
+                k,
+                a.at_mut_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+                a.at_ptr(a_col, 0),
+                a.size() as u64,
+                a.sl() as u64,
+            )
+        }
+    }
+
+    fn vec_znx_split<R, A>(&self, res: &mut Vec<R>, res_col: usize, a: &A, a_col: usize, scratch: &mut Scratch)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+
+        let (n_in, n_out) = (a.n(), res[0].to_mut().n());
+
+        let (mut buf, _) = scratch.tmp_vec_znx(self, 1, a.size());
+
+        debug_assert!(
+            n_out < n_in,
+            "invalid a: output ring degree should be smaller"
+        );
+        res[1..].iter_mut().for_each(|bi| {
+            debug_assert_eq!(
+                bi.to_mut().n(),
+                n_out,
+                "invalid input a: all VecZnx must have the same degree"
+            )
+        });
+
+        res.iter_mut().enumerate().for_each(|(i, bi)| {
+            if i == 0 {
+                self.switch_degree(bi, res_col, &a, a_col);
+                self.vec_znx_rotate(-1, &mut buf, 0, &a, a_col);
+            } else {
+                self.switch_degree(bi, res_col, &mut buf, a_col);
+                self.vec_znx_rotate_inplace(-1, &mut buf, a_col);
+            }
+        })
+    }
+
+    fn vec_znx_merge<R, A>(&self, res: &mut R, res_col: usize, a: Vec<A>, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        let (n_in, n_out) = (res.n(), a[0].to_ref().n());
+
+        debug_assert!(
+            n_out < n_in,
+            "invalid a: output ring degree should be smaller"
+        );
+        a[1..].iter().for_each(|ai| {
+            debug_assert_eq!(
+                ai.to_ref().n(),
+                n_out,
+                "invalid input a: all VecZnx must have the same degree"
+            )
+        });
+
+        a.iter().enumerate().for_each(|(_, ai)| {
+            self.switch_degree(&mut res, res_col, ai, a_col);
+            self.vec_znx_rotate_inplace(-1, &mut res, res_col);
+        });
+
+        self.vec_znx_rotate_inplace(a.len() as i64, &mut res, res_col);
+    }
+
+    fn switch_degree<R, A>(&self, res: &mut R, res_col: usize, a: &A, a_col: usize)
+    where
+        R: VecZnxToMut,
+        A: VecZnxToRef,
+    {
+        let a: VecZnx<&[u8]> = a.to_ref();
+        let mut res: VecZnx<&mut [u8]> = res.to_mut();
+
+        let (n_in, n_out) = (a.n(), res.n());
+        let (gap_in, gap_out): (usize, usize);
+
+        if n_in > n_out {
+            (gap_in, gap_out) = (n_in / n_out, 1)
+        } else {
+            (gap_in, gap_out) = (1, n_out / n_in);
+            res.zero();
+        }
+
+        let size: usize = min(a.size(), res.size());
+
+        (0..size).for_each(|i| {
+            izip!(
+                a.at(a_col, i).iter().step_by(gap_in),
+                res.at_mut(res_col, i).iter_mut().step_by(gap_out)
+            )
+            .for_each(|(x_in, x_out)| *x_out = *x_in);
+        });
+    }
+}
+
+impl<B: Backend> VecZnxScratch for Module<B> {
+    fn vec_znx_normalize_tmp_bytes(&self) -> usize {
+        unsafe { vec_znx::vec_znx_normalize_base2k_tmp_bytes(self.ptr) as usize }
+    }
+}

backend/src/znx_base.rs

@@ -0,0 +1,199 @@
+use itertools::izip;
+use rand_distr::num_traits::Zero;
+
+pub trait ZnxInfos {
+    /// Returns the ring degree of the polynomials.
+    fn n(&self) -> usize;
+
+    /// Returns the base two logarithm of the ring dimension of the polynomials.
+    fn log_n(&self) -> usize {
+        (usize::BITS - (self.n() - 1).leading_zeros()) as _
+    }
+
+    /// Returns the number of rows.
+    fn rows(&self) -> usize;
+
+    /// Returns the number of polynomials in each row.
+    fn cols(&self) -> usize;
+
+    /// Returns the number of size per polynomial.
+    fn size(&self) -> usize;
+
+    /// Returns the total number of small polynomials.
+    fn poly_count(&self) -> usize {
+        self.rows() * self.cols() * self.size()
+    }
+}
+
+pub trait ZnxSliceSize {
+    /// Returns the slice size, which is the offset between
+    /// two size of the same column.
+    fn sl(&self) -> usize;
+}
+
+pub trait DataView {
+    type D;
+    fn data(&self) -> &Self::D;
+}
+
+pub trait DataViewMut: DataView {
+    fn data_mut(&mut self) -> &mut Self::D;
+}
+
+pub trait ZnxView: ZnxInfos + DataView<D: AsRef<[u8]>> {
+    type Scalar: Copy;
+
+    /// Returns a non-mutable pointer to the underlying coefficients array.
+    fn as_ptr(&self) -> *const Self::Scalar {
+        self.data().as_ref().as_ptr() as *const Self::Scalar
+    }
+
+    /// Returns a non-mutable reference to the entire underlying coefficient array.
+    fn raw(&self) -> &[Self::Scalar] {
+        unsafe { std::slice::from_raw_parts(self.as_ptr(), self.n() * self.poly_count()) }
+    }
+
+    /// Returns a non-mutable pointer starting at the j-th small polynomial of the i-th column.
+    fn at_ptr(&self, i: usize, j: usize) -> *const Self::Scalar {
+        #[cfg(debug_assertions)]
+        {
+            assert!(i < self.cols());
+            assert!(j < self.size());
+        }
+        let offset: usize = self.n() * (j * self.cols() + i);
+        unsafe { self.as_ptr().add(offset) }
+    }
+
+    /// Returns non-mutable reference to the (i, j)-th small polynomial.
+    fn at(&self, i: usize, j: usize) -> &[Self::Scalar] {
+        unsafe { std::slice::from_raw_parts(self.at_ptr(i, j), self.n()) }
+    }
+}
+
+pub trait ZnxViewMut: ZnxView + DataViewMut<D: AsMut<[u8]>> {
+    /// Returns a mutable pointer to the underlying coefficients array.
+    fn as_mut_ptr(&mut self) -> *mut Self::Scalar {
+        self.data_mut().as_mut().as_mut_ptr() as *mut Self::Scalar
+    }
+
+    /// Returns a mutable reference to the entire underlying coefficient array.
+    fn raw_mut(&mut self) -> &mut [Self::Scalar] {
+        unsafe { std::slice::from_raw_parts_mut(self.as_mut_ptr(), self.n() * self.poly_count()) }
+    }
+
+    /// Returns a mutable pointer starting at the j-th small polynomial of the i-th column.
+    fn at_mut_ptr(&mut self, i: usize, j: usize) -> *mut Self::Scalar {
+        #[cfg(debug_assertions)]
+        {
+            assert!(i < self.cols());
+            assert!(j < self.size());
+        }
+        let offset: usize = self.n() * (j * self.cols() + i);
+        unsafe { self.as_mut_ptr().add(offset) }
+    }
+
+    /// Returns mutable reference to the (i, j)-th small polynomial.
+    fn at_mut(&mut self, i: usize, j: usize) -> &mut [Self::Scalar] {
+        unsafe { std::slice::from_raw_parts_mut(self.at_mut_ptr(i, j), self.n()) }
+    }
+}
+
+//(Jay)Note: Can't provide blanket impl. of ZnxView because Scalar is not known
+impl<T> ZnxViewMut for T where T: ZnxView + DataViewMut<D: AsMut<[u8]>> {}
+
+pub trait ZnxZero: ZnxViewMut + ZnxSliceSize
+where
+    Self: Sized,
+{
+    fn zero(&mut self) {
+        unsafe {
+            std::ptr::write_bytes(self.as_mut_ptr(), 0, self.n() * self.poly_count());
+        }
+    }
+
+    fn zero_at(&mut self, i: usize, j: usize) {
+        unsafe {
+            std::ptr::write_bytes(self.at_mut_ptr(i, j), 0, self.n());
+        }
+    }
+}
+
+// Blanket implementations
+impl<T> ZnxZero for T where T: ZnxViewMut + ZnxSliceSize {} // WARNING should not work for mat_znx_dft but it does
+
+use std::ops::{Add, AddAssign, Div, Mul, Neg, Shl, Shr, Sub};
+
+use crate::Scratch;
+pub trait Integer:
+    Copy
+    + Default
+    + PartialEq
+    + PartialOrd
+    + Add<Output = Self>
+    + Sub<Output = Self>
+    + Mul<Output = Self>
+    + Div<Output = Self>
+    + Neg<Output = Self>
+    + Shl<Output = Self>
+    + Shr<Output = Self>
+    + AddAssign
+{
+    const BITS: u32;
+}
+
+impl Integer for i64 {
+    const BITS: u32 = 64;
+}
+
+impl Integer for i128 {
+    const BITS: u32 = 128;
+}
+
+//(Jay)Note: `rsh` impl. ignores the column
+pub fn rsh<V: ZnxZero>(k: usize, basek: usize, a: &mut V, _a_col: usize, scratch: &mut Scratch)
+where
+    V::Scalar: From<usize> + Integer + Zero,
+{
+    let n: usize = a.n();
+    let _size: usize = a.size();
+    let cols: usize = a.cols();
+
+    let size: usize = a.size();
+    let steps: usize = k / basek;
+
+    a.raw_mut().rotate_right(n * steps * cols);
+    (0..cols).for_each(|i| {
+        (0..steps).for_each(|j| {
+            a.zero_at(i, j);
+        })
+    });
+
+    let k_rem: usize = k % basek;
+
+    if k_rem != 0 {
+        let (carry, _) = scratch.tmp_slice::<V::Scalar>(rsh_tmp_bytes::<V::Scalar>(n));
+
+        unsafe {
+            std::ptr::write_bytes(carry.as_mut_ptr(), 0, n * size_of::<V::Scalar>());
+        }
+
+        let basek_t = V::Scalar::from(basek);
+        let shift = V::Scalar::from(V::Scalar::BITS as usize - k_rem);
+        let k_rem_t = V::Scalar::from(k_rem);
+
+        (0..cols).for_each(|i| {
+            (steps..size).for_each(|j| {
+                izip!(carry.iter_mut(), a.at_mut(i, j).iter_mut()).for_each(|(ci, xi)| {
+                    *xi += *ci << basek_t;
+                    *ci = (*xi << shift) >> shift;
+                    *xi = (*xi - *ci) >> k_rem_t;
+                });
+            });
+            carry.iter_mut().for_each(|r| *r = V::Scalar::zero());
+        })
+    }
+}
+
+pub fn rsh_tmp_bytes<T>(n: usize) -> usize {
+    n * std::mem::size_of::<T>()
+}