@@ -3,25 +3,25 @@ use math::{modulus::prime::Prime,dft::ntt::Table};
 use math::dft::DFT;
 
 fn forward_inplace(c: &mut Criterion) {
-    fn runner<T: DFT<u64> + 'static>(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
+    fn runner(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
         let ntt_table: Table<u64> = Table::<u64>::new(prime_instance, nth_root);
         let mut a: Vec<u64> = vec![0; (nth_root >> 1) as usize];
         for i in 0..a.len(){
             a[i] = i as u64;
         }
         Box::new(move || {
-            ntt_table.forward_inplace(&mut a)
+            ntt_table.forward_inplace::<false>(&mut a)
         })
     }
 
     let mut b: criterion::BenchmarkGroup<'_, criterion::measurement::WallTime> = c.benchmark_group("forward_inplace");
     for log_nth_root in 11..18 {
 
-        let mut prime_instance: Prime<u64> = Prime::<u64>::new(0x1fffffffffe00001, 1);
+        let prime_instance: Prime<u64> = Prime::<u64>::new(0x1fffffffffe00001, 1);
 
         let runners = [
             ("prime", {
-                runner::<Table<u64>>(prime_instance, 1<<log_nth_root)
+                runner(prime_instance, 1<<log_nth_root)
             }),
         ];
         for (name, mut runner) in runners {
@@ -32,7 +32,7 @@ }
 }
 
 fn forward_inplace_lazy(c: &mut Criterion) {
-    fn runner<T: DFT<u64> + 'static>(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
+    fn runner(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
         let ntt_table: Table<u64> = Table::<u64>::new(prime_instance, nth_root);
         let mut a: Vec<u64> = vec![0; (nth_root >> 1) as usize];
         for i in 0..a.len(){
@@ -50,7 +50,7 @@ let prime_instance: Prime<u64> = Prime::<u64>::new(0x1fffffffffe00001, 1);
 
         let runners = [
             ("prime", {
-                runner::<Table<u64>>(prime_instance, 1<<log_nth_root)
+                runner(prime_instance, 1<<log_nth_root)
             }),
         ];
         for (name, mut runner) in runners {
@@ -61,14 +61,14 @@ }
 }
 
 fn backward_inplace(c: &mut Criterion) {
-    fn runner<T: DFT<u64> + 'static>(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
+    fn runner(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
         let ntt_table: Table<u64> = Table::<u64>::new(prime_instance, nth_root);
         let mut a: Vec<u64> = vec![0; (nth_root >> 1) as usize];
         for i in 0..a.len(){
             a[i] = i as u64;
         }
         Box::new(move || {
-            ntt_table.backward_inplace(&mut a)
+            ntt_table.backward_inplace::<false>(&mut a)
         })
     }
 
@@ -79,7 +79,7 @@ let prime_instance: Prime<u64> = Prime::<u64>::new(0x1fffffffffe00001, 1);
 
         let runners = [
             ("prime", {
-                runner::<Table<u64>>(prime_instance, 1<<log_nth_root)
+                runner(prime_instance, 1<<log_nth_root)
             }),
         ];
         for (name, mut runner) in runners {
@@ -90,14 +90,14 @@ }
 }
 
 fn backward_inplace_lazy(c: &mut Criterion) {
-    fn runner<T: DFT<u64> + 'static>(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
+    fn runner(prime_instance: Prime<u64>, nth_root: u64) -> Box<dyn FnMut()> {
         let ntt_table: Table<u64> = Table::<u64>::new(prime_instance, nth_root);
         let mut a: Vec<u64> = vec![0; (nth_root >> 1) as usize];
         for i in 0..a.len(){
             a[i] = i as u64;
         }
         Box::new(move || {
-            ntt_table.backward_inplace_lazy(&mut a)
+            ntt_table.backward_inplace::<true>(&mut a)
         })
     }
 
@@ -108,7 +108,7 @@ let prime_instance: Prime<u64> = Prime::<u64>::new(0x1fffffffffe00001, 1);
 
         let runners = [
             ("prime", {
-                runner::<Table<u64>>(prime_instance, 1<<log_nth_root)
+                runner(prime_instance, 1<<log_nth_root)
             }),
         ];
         for (name, mut runner) in runners {

@@ -1,12 +1,12 @@
 use criterion::{criterion_group, criterion_main, BenchmarkId, Criterion};
 use math::ring::Ring;
-use math::modulus::Operations;
+use math::modulus::VecOperations;
 use math::modulus::montgomery::Montgomery;
-use math::modulus::{NONE, ONCE};
+use math::modulus::{BARRETT, ONCE};
 
 const CHUNK: usize= 8;
 
-fn add_vec_unary(c: &mut Criterion) {
+fn vec_add_unary(c: &mut Criterion) {
     fn runner(r: Ring<u64>) -> Box<dyn FnMut()> {
         
         let mut p0: math::poly::Poly<u64> = r.new_poly();
@@ -16,7 +16,7 @@ p0.0[i] = i as u64;
             p1.0[i] = i as u64;
         }
         Box::new(move || {
-            r.modulus.add_vec_unary_assign::<CHUNK, ONCE>(&p0.0, &mut p1.0);
+            r.modulus.vec_add_unary_assign::<CHUNK, ONCE>(&p0.0, &mut p1.0);
         })
     }
 
@@ -39,17 +39,17 @@ }
     }
 }
 
-fn mul_vec_montgomery_external_unary_assign(c: &mut Criterion) {
+fn vec_mul_montgomery_external_unary_assign(c: &mut Criterion) {
     fn runner(r: Ring<u64>) -> Box<dyn FnMut()> {
         
-        let mut p0: math::poly::Poly<Montgomery<u64>> = r.new_poly_montgomery();
+        let mut p0: math::poly::Poly<Montgomery<u64>> = r.new_poly();
         let mut p1: math::poly::Poly<u64> = r.new_poly();
         for i in 0..p0.n(){
             p0.0[i] = r.modulus.montgomery.prepare::<ONCE>(i as u64);
             p1.0[i] = i as u64;
         }
         Box::new(move || {
-            r.modulus.mul_vec_montgomery_external_unary_assign::<CHUNK, NONE>(&p0.0, &mut p1.0);
+            r.modulus.vec_mul_montgomery_external_unary_assign::<CHUNK, ONCE>(&p0.0, &mut p1.0);
         })
     }
 
@@ -72,5 +72,39 @@ }
     }
 }
 
-criterion_group!(benches, add_vec_unary, mul_vec_montgomery_external_unary_assign);
+fn vec_mul_montgomery_external_binary_assign(c: &mut Criterion) {
+    fn runner(r: Ring<u64>) -> Box<dyn FnMut()> {
+        
+        let mut p0: math::poly::Poly<Montgomery<u64>> = r.new_poly();
+        let mut p1: math::poly::Poly<u64> = r.new_poly();
+        let mut p2: math::poly::Poly<u64> = r.new_poly();
+        for i in 0..p0.n(){
+            p0.0[i] = r.modulus.montgomery.prepare::<ONCE>(i as u64);
+            p1.0[i] = i as u64;
+        }
+        Box::new(move || {
+            r.modulus.vec_mul_montgomery_external_binary_assign::<CHUNK,ONCE>(&p0.0, & p1.0, &mut p2.0);
+        })
+    }
+
+    let mut b: criterion::BenchmarkGroup<'_, criterion::measurement::WallTime> = c.benchmark_group("mul_vec_montgomery_external_binary_assign");
+    for log_n in 11..17 {
+
+        let n: usize = 1<<log_n as usize;
+        let q_base: u64 = 0x1fffffffffe00001u64;
+        let q_power: usize = 1usize;
+        let r: Ring<u64> = Ring::<u64>::new(n, q_base, q_power);
+        let runners = [
+            ("prime", {
+                runner(r)
+            }),
+        ];
+        for (name, mut runner) in runners {
+            let id = BenchmarkId::new(name, n);
+            b.bench_with_input(id, &(), |b, _| b.iter(&mut runner));
+        }
+    }
+}
+
+criterion_group!(benches, vec_add_unary, vec_mul_montgomery_external_unary_assign, vec_mul_montgomery_external_binary_assign);
 criterion_main!(benches);

@@ -28,11 +28,11 @@ }
 
     println!("{:?}", a);
 
-    ntt_table.forward_inplace(&mut a);
+    ntt_table.forward_inplace::<false>(&mut a);
 
     println!("{:?}", a);
 
-    ntt_table.backward_inplace(&mut a);
+    ntt_table.backward_inplace::<false>(&mut a);
 
     println!("{:?}", a);
 

@@ -1,5 +1,5 @@
 use crate::modulus::montgomery::Montgomery;
-use crate::modulus::shoup::Shoup;
+use crate::modulus::barrett::Barrett;
 use crate::modulus::prime::Prime;
 use crate::modulus::ReduceOnce;
 use crate::modulus::WordOps;
@@ -9,8 +9,9 @@ use itertools::izip;
 
 pub struct Table<O>{
     prime:Prime<O>,
-    psi_forward_rev:Vec<Shoup<u64>>,
-    psi_backward_rev: Vec<Shoup<u64>>,
+    psi: O,
+    psi_forward_rev:Vec<Barrett<u64>>,
+    psi_backward_rev: Vec<Barrett<u64>>,
     q:O,
     two_q:O,
     four_q:O,
@@ -26,11 +27,11 @@
         let psi_mont: Montgomery<u64> = prime.montgomery.prepare::<ONCE>(psi);
         let psi_inv_mont: Montgomery<u64> = prime.montgomery.pow(psi_mont, prime.phi-1);
         
-        let mut psi_forward_rev: Vec<Shoup<u64>> = vec![Shoup(0, 0); (nth_root >> 1) as usize];
-        let mut psi_backward_rev: Vec<Shoup<u64>> = vec![Shoup(0, 0); (nth_root >> 1) as usize];
+        let mut psi_forward_rev: Vec<Barrett<u64>> = vec![Barrett(0, 0); (nth_root >> 1) as usize];
+        let mut psi_backward_rev: Vec<Barrett<u64>> = vec![Barrett(0, 0); (nth_root >> 1) as usize];
 
-        psi_forward_rev[0] = prime.shoup.prepare(1);
-        psi_backward_rev[0] = prime.shoup.prepare(1);
+        psi_forward_rev[0] = prime.barrett.prepare(1);
+        psi_backward_rev[0] = prime.barrett.prepare(1);
 
         let log_nth_root_half: u32 = (nth_root>>1).log2() as _;
 
@@ -44,14 +45,15 @@
             prime.montgomery.mul_external_assign::<ONCE>(psi_mont, &mut powers_forward);
             prime.montgomery.mul_external_assign::<ONCE>(psi_inv_mont, &mut powers_backward);
 
-            psi_forward_rev[i_rev] = prime.shoup.prepare(powers_forward);
-            psi_backward_rev[i_rev] = prime.shoup.prepare(powers_backward); 
+            psi_forward_rev[i_rev] = prime.barrett.prepare(powers_forward);
+            psi_backward_rev[i_rev] = prime.barrett.prepare(powers_backward); 
         }
 
         let q: u64 = prime.q();
 
         Self{ 
             prime: prime, 
+            psi:psi,
             psi_forward_rev: psi_forward_rev, 
             psi_backward_rev: psi_backward_rev,
             q:q,
@@ -59,65 +61,60 @@ two_q:q<<1,
             four_q:q<<2,
         }
     }
-
-    // Returns n^-1 mod q in Montgomery.
-    fn inv(&self, n:u64) -> Montgomery<u64>{
-        self.prime.montgomery.pow(self.prime.montgomery.prepare::<ONCE>(n), self.prime.phi-1)
-    }
 }
 
 
 impl DFT<u64> for Table<u64>{
+
     fn forward_inplace(&self, a: &mut [u64]){
-        self.forward_inplace(a)
+        self.forward_inplace::<false>(a)
     }
 
     fn forward_inplace_lazy(&self, a: &mut [u64]){
-        self.forward_inplace_lazy(a)
+        self.forward_inplace::<true>(a)
     }
 
     fn backward_inplace(&self, a: &mut [u64]){
-        self.backward_inplace(a)
+        self.backward_inplace::<false>(a)
     }
 
     fn backward_inplace_lazy(&self, a: &mut [u64]){
-        self.backward_inplace_lazy(a)
+        self.backward_inplace::<true>(a)
     }
 }
 
 impl Table<u64>{
 
-    pub fn forward_inplace_lazy(&self, a: &mut [u64]){
-        self.forward_inplace_core::<true>(a);
+    pub fn forward_inplace<const LAZY:bool>(&self, a: &mut [u64]){
+        self.forward_inplace_core::<LAZY, 0, 0>(a);
     }
 
-    pub fn forward_inplace(&self, a: &mut [u64]){
-        self.forward_inplace_core::<false>(a);
-    }
-
-    pub fn forward_inplace_core<const LAZY: bool>(&self, a: &mut [u64]) {
+    pub fn forward_inplace_core<const LAZY: bool, const SKIPSTART: u8, const SKIPEND: u8>(&self, a: &mut [u64]) {
 
         let n: usize = a.len();
         assert!(n & n-1 == 0, "invalid x.len()= {} must be a power of two", n);
         let log_n: u32 = usize::BITS - ((n as usize)-1).leading_zeros();
 
-        for layer in 0..log_n {
+        let start: u32 = SKIPSTART as u32;
+        let end: u32 = log_n - (SKIPEND as u32);
+
+        for layer in start..end {
             let (m, size) = (1 << layer, 1 << (log_n - layer - 1));
             let t: usize = 2*size;
             if layer == log_n - 1 {
                 if LAZY{
                     izip!(a.chunks_exact_mut(t), &self.psi_forward_rev[m..]).for_each(|(a, psi)| {
                         let (a, b) = a.split_at_mut(size);
-                        self.dit::<false>(&mut a[0], &mut b[0], *psi);
+                        self.dit_inplace::<false>(&mut a[0], &mut b[0], *psi);
                         debug_assert!(a[0] < self.two_q, "forward_inplace_core::<LAZY=true> output {} > {} (2q-1)", a[0], self.two_q-1);
                         debug_assert!(b[0] < self.two_q, "forward_inplace_core::<LAZY=true> output {} > {} (2q-1)", b[0], self.two_q-1);
                     });
                 }else{
                     izip!(a.chunks_exact_mut(t), &self.psi_forward_rev[m..]).for_each(|(a, psi)| {
                         let (a, b) = a.split_at_mut(size);
-                        self.dit::<true>(&mut a[0], &mut b[0], *psi);
-                        self.prime.shoup.reduce_assign(&mut a[0]);
-                        self.prime.shoup.reduce_assign(&mut b[0]);
+                        self.dit_inplace::<true>(&mut a[0], &mut b[0], *psi);
+                        self.prime.barrett.reduce_assign(&mut a[0]);
+                        self.prime.barrett.reduce_assign(&mut b[0]);
                         debug_assert!(a[0] < self.q, "forward_inplace_core::<LAZY=false> output {} > {} (q-1)", a[0], self.q-1);
                         debug_assert!(b[0] < self.q, "forward_inplace_core::<LAZY=false> output {} > {} (q-1)", b[0], self.q-1);
                     });
@@ -127,31 +124,31 @@ } else if t >= 16{
                 izip!(a.chunks_exact_mut(t), &self.psi_forward_rev[m..]).for_each(|(a, psi)| {
                     let (a, b) = a.split_at_mut(size);
                     izip!(a.chunks_exact_mut(8), b.chunks_exact_mut(8)).for_each(|(a, b)| {
-                        self.dit::<true>(&mut a[0], &mut b[0], *psi);
-                        self.dit::<true>(&mut a[1], &mut b[1], *psi);
-                        self.dit::<true>(&mut a[2], &mut b[2], *psi);
-                        self.dit::<true>(&mut a[3], &mut b[3], *psi);
-                        self.dit::<true>(&mut a[4], &mut b[4], *psi);
-                        self.dit::<true>(&mut a[5], &mut b[5], *psi);
-                        self.dit::<true>(&mut a[6], &mut b[6], *psi);
-                        self.dit::<true>(&mut a[7], &mut b[7], *psi);
+                        self.dit_inplace::<true>(&mut a[0], &mut b[0], *psi);
+                        self.dit_inplace::<true>(&mut a[1], &mut b[1], *psi);
+                        self.dit_inplace::<true>(&mut a[2], &mut b[2], *psi);
+                        self.dit_inplace::<true>(&mut a[3], &mut b[3], *psi);
+                        self.dit_inplace::<true>(&mut a[4], &mut b[4], *psi);
+                        self.dit_inplace::<true>(&mut a[5], &mut b[5], *psi);
+                        self.dit_inplace::<true>(&mut a[6], &mut b[6], *psi);
+                        self.dit_inplace::<true>(&mut a[7], &mut b[7], *psi);
                     });
                 });
             }else{
                 izip!(a.chunks_exact_mut(t), &self.psi_forward_rev[m..]).for_each(|(a, psi)| {
                     let (a, b) = a.split_at_mut(size);
-                    izip!(a, b).for_each(|(a, b)| self.dit::<true>(a, b, *psi));
+                    izip!(a, b).for_each(|(a, b)| self.dit_inplace::<true>(a, b, *psi));
                 });
             }
         }
     }
 
     #[inline(always)]
-    fn dit<const LAZY: bool>(&self, a: &mut u64, b: &mut u64, t: Shoup<u64>) {
+    fn dit_inplace<const LAZY: bool>(&self, a: &mut u64, b: &mut u64, t: Barrett<u64>) {
         debug_assert!(*a < self.four_q, "a:{} q:{}", a, self.four_q);
         debug_assert!(*b < self.four_q, "b:{} q:{}", b, self.four_q);
         a.reduce_once_assign(self.two_q);
-        let bt: u64 = self.prime.shoup.mul_external_lazy(t, *b);
+        let bt: u64 = self.prime.barrett.mul_external_lazy(t, *b);
         *b = a.wrapping_add(self.two_q-bt);
         *a = a.wrapping_add(bt);
         if !LAZY {
@@ -160,39 +157,38 @@ b.reduce_once_assign(self.two_q);
         }
     }
 
-    pub fn backward_inplace_lazy(&self, a: &mut [u64]){
-        self.backward_inplace_core::<true>(a);
-    }
-
-    pub fn backward_inplace(&self, a: &mut [u64]){
-        self.backward_inplace_core::<false>(a);
+    pub fn backward_inplace<const LAZY:bool>(&self, a: &mut [u64]){
+        self.backward_inplace_core::<LAZY, 0, 0>(a);
     }
 
-    pub fn backward_inplace_core<const LAZY:bool>(&self, a: &mut [u64]) {
+    pub fn backward_inplace_core<const LAZY:bool, const SKIPSTART: u8, const SKIPEND: u8>(&self, a: &mut [u64]) {
         let n: usize = a.len();
         assert!(n & n-1 == 0, "invalid x.len()= {} must be a power of two", n);
         let log_n = usize::BITS - ((n as usize)-1).leading_zeros();
 
-        for layer in (0..log_n).rev() {
+        let start: u32 = SKIPEND as u32;
+        let end: u32 = log_n - (SKIPSTART as u32);
+
+        for layer in (start..end).rev() {
             let (m, size) = (1 << layer, 1 << (log_n - layer - 1));
             let t: usize = 2*size;
             if layer == 0 {
 
-                let n_inv: Shoup<u64> = self.prime.shoup.prepare(self.prime.inv(n as u64));
-                let psi: Shoup<u64> = self.prime.shoup.prepare(self.prime.shoup.mul_external(n_inv, self.psi_backward_rev[1].0));
+                let n_inv: Barrett<u64> = self.prime.barrett.prepare(self.prime.inv(n as u64));
+                let psi: Barrett<u64> = self.prime.barrett.prepare(self.prime.barrett.mul_external(n_inv, self.psi_backward_rev[1].0));
 
                 izip!(a.chunks_exact_mut(2 * size)).for_each(
                     |a| {
                         let (a, b) = a.split_at_mut(size);
                         izip!(a.chunks_exact_mut(8), b.chunks_exact_mut(8)).for_each(|(a, b)| {
-                            self.dif_last::<LAZY>(&mut a[0], &mut b[0], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[1], &mut b[1], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[2], &mut b[2], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[3], &mut b[3], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[4], &mut b[4], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[5], &mut b[5], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[6], &mut b[6], psi, n_inv);
-                            self.dif_last::<LAZY>(&mut a[7], &mut b[7], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[0], &mut b[0], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[1], &mut b[1], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[2], &mut b[2], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[3], &mut b[3], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[4], &mut b[4], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[5], &mut b[5], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[6], &mut b[6], psi, n_inv);
+                            self.dif_last_inplace::<LAZY>(&mut a[7], &mut b[7], psi, n_inv);
                         });
                     },
                 );
@@ -202,14 +198,14 @@ izip!(a.chunks_exact_mut(t), &self.psi_backward_rev[m..]).for_each(
                     |(a, psi)| {
                         let (a, b) = a.split_at_mut(size);
                         izip!(a.chunks_exact_mut(8), b.chunks_exact_mut(8)).for_each(|(a, b)| {
-                            self.dif::<true>(&mut a[0], &mut b[0], *psi);
-                            self.dif::<true>(&mut a[1], &mut b[1], *psi);
-                            self.dif::<true>(&mut a[2], &mut b[2], *psi);
-                            self.dif::<true>(&mut a[3], &mut b[3], *psi);
-                            self.dif::<true>(&mut a[4], &mut b[4], *psi);
-                            self.dif::<true>(&mut a[5], &mut b[5], *psi);
-                            self.dif::<true>(&mut a[6], &mut b[6], *psi);
-                            self.dif::<true>(&mut a[7], &mut b[7], *psi);
+                            self.dif_inplace::<true>(&mut a[0], &mut b[0], *psi);
+                            self.dif_inplace::<true>(&mut a[1], &mut b[1], *psi);
+                            self.dif_inplace::<true>(&mut a[2], &mut b[2], *psi);
+                            self.dif_inplace::<true>(&mut a[3], &mut b[3], *psi);
+                            self.dif_inplace::<true>(&mut a[4], &mut b[4], *psi);
+                            self.dif_inplace::<true>(&mut a[5], &mut b[5], *psi);
+                            self.dif_inplace::<true>(&mut a[6], &mut b[6], *psi);
+                            self.dif_inplace::<true>(&mut a[7], &mut b[7], *psi);
                         });
                     },
                 );
@@ -217,7 +213,7 @@ } else {
                 izip!(a.chunks_exact_mut(2 * size), &self.psi_backward_rev[m..]).for_each(
                     |(a, psi)| {
                         let (a, b) = a.split_at_mut(size);
-                        izip!(a, b).for_each(|(a, b)| self.dif::<true>(a, b, *psi));
+                        izip!(a, b).for_each(|(a, b)| self.dif_inplace::<true>(a, b, *psi));
                     },
                 );
             }
@@ -225,10 +221,10 @@ }
     }
 
     #[inline(always)]
-    fn dif<const LAZY: bool>(&self, a: &mut u64, b: &mut u64, t: Shoup<u64>) {
-        debug_assert!(*a < self.two_q);
-        debug_assert!(*b < self.two_q);
-        let d: u64 = self.prime.shoup.mul_external_lazy(t, *a + self.two_q - *b);
+    fn dif_inplace<const LAZY: bool>(&self, a: &mut u64, b: &mut u64, t: Barrett<u64>) {
+        debug_assert!(*a < self.two_q, "a:{} q:{}", a, self.four_q);
+        debug_assert!(*b < self.two_q, "b:{} q:{}", b, self.four_q);
+        let d: u64 = self.prime.barrett.mul_external_lazy(t, *a + self.two_q - *b);
         *a = a.wrapping_add(*b);
         a.reduce_once_assign(self.two_q);
         *b = d;
@@ -238,17 +234,41 @@ b.reduce_once_assign(self.q);
         }
     }
 
-    fn dif_last<const LAZY:bool>(&self, a: &mut u64, b: &mut u64, psi: Shoup<u64>, n_inv: Shoup<u64>){
+    fn dif_last_inplace<const LAZY:bool>(&self, a: &mut u64, b: &mut u64, psi: Barrett<u64>, n_inv: Barrett<u64>){
         debug_assert!(*a < self.two_q);
         debug_assert!(*b < self.two_q);
         if LAZY{
-            let d: u64 = self.prime.shoup.mul_external_lazy(psi, *a + self.two_q - *b);
-            *a = self.prime.shoup.mul_external_lazy(n_inv, *a + *b);
+            let d: u64 = self.prime.barrett.mul_external_lazy(psi, *a + self.two_q - *b);
+            *a = self.prime.barrett.mul_external_lazy(n_inv, *a + *b);
             *b = d;
         }else{
-            let d: u64 = self.prime.shoup.mul_external(psi, *a + self.two_q - *b);
-            *a = self.prime.shoup.mul_external(n_inv, *a + *b);
+            let d: u64 = self.prime.barrett.mul_external(psi, *a + self.two_q - *b);
+            *a = self.prime.barrett.mul_external(n_inv, *a + *b);
             *b = d;
         }
     }
 }
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn test_ntt() {
+        let q_base: u64 = 0x800000000004001;
+        let q_power: usize = 1;
+        let prime_instance: Prime<u64> = Prime::<u64>::new(q_base, q_power);
+        let n: u64 = 32;
+        let two_nth_root: u64 = n<<1;
+        let ntt_table: Table<u64> = Table::<u64>::new(prime_instance, two_nth_root);
+        let mut a: Vec<u64> = vec![0; n as usize];
+        for i in 0..a.len(){
+            a[i] = i as u64;
+        }
+
+        let b: Vec<u64> = a.clone();
+        ntt_table.forward_inplace::<false>(&mut a);
+        ntt_table.backward_inplace::<false>(&mut a);
+        assert!(a == b);
+    }
+}

@@ -1,7 +1,6 @@
 pub mod prime;
 pub mod barrett;
 pub mod montgomery;
-pub mod shoup;
 pub mod impl_u64;
 
 pub type REDUCEMOD = u8;
@@ -55,75 +54,64 @@ /// /// User must ensure that 2q fits in O.
     fn reduce_once(&self, q:O) -> O;
 }
 
-impl ReduceOnce<u64> for u64{
-    #[inline(always)]
-    fn reduce_once_constant_time_assign(&mut self, q: u64){
-        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
-        *self -= (q.wrapping_sub(*self)>>63)*q;
-    }
-    
-    #[inline(always)]
-    fn reduce_once_constant_time(&self, q:u64) -> u64{
-        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
-        self - (q.wrapping_sub(*self)>>63)*q
-    }
+pub trait WordOperations<O>{
 
-    #[inline(always)]
-    fn reduce_once_assign(&mut self, q: u64){
-        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
-        *self = *self.min(&mut self.wrapping_sub(q))
-    }
+    // Applies a parameterized modular reduction.
+    fn word_reduce_assign<const REDUCE:REDUCEMOD>(&self, x: &mut O);
 
-    #[inline(always)]
-    fn reduce_once(&self, q:u64) -> u64{
-        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
-        *self.min(&mut self.wrapping_sub(q))
-    }
-}
-
-
-pub trait Operations<O>{
     // Assigns a + b to c.
-    fn add_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b:&O, c: &mut O);
+    fn word_add_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b:&O, c: &mut O);
 
     // Assigns a + b to b.
-    fn add_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b: &mut O);
-
-    // Assigns a[i] + b[i] to c[i]
-    fn add_vec_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b:&[O], c: &mut [O]);
-
-    // Assigns a[i] + b[i] to b[i]
-    fn add_vec_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b: &mut [O]);
+    fn word_add_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b: &mut O);
 
     // Assigns a - b to c.
-    fn sub_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b:&O, c: &mut O);
+    fn word_sub_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b:&O, c: &mut O);
 
     // Assigns b - a to b.
-    fn sub_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b: &mut O);
-
-    // Assigns a[i] - b[i] to c[i]
-    fn sub_vec_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b:&[O], c: &mut [O]);
-
-    // Assigns a[i] - b[i] to b[i]
-    fn sub_vec_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b: &mut [O]);
+    fn word_sub_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b: &mut O);
 
     // Assigns -a to a.
-    fn neg_assign<const REDUCE:REDUCEMOD>(&self, a:&mut O);
+    fn word_neg_assign<const REDUCE:REDUCEMOD>(&self, a:&mut O);
 
-    // Assigns -a[i] to a[i].
-    fn neg_vec_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &mut [O]);
+    // Assigns a * 2^64 to b.
+    fn word_prepare_montgomery_assign<const REDUCE:REDUCEMOD>(&self, a: &O, b: &mut montgomery::Montgomery<O>);
 
     // Assigns a * b to c.
-    fn mul_montgomery_external_binary_assign<const REDUCE:REDUCEMOD>(&self, a:&montgomery::Montgomery<O>, b:&O, c: &mut O);
+    fn word_mul_montgomery_external_binary_assign<const REDUCE:REDUCEMOD>(&self, a:&montgomery::Montgomery<O>, b:&O, c: &mut O);
 
     // Assigns a * b to b.
-    fn mul_montgomery_external_unary_assign<const REDUCE:REDUCEMOD>(&self, a:&montgomery::Montgomery<O>, b:&mut O);
+    fn word_mul_montgomery_external_unary_assign<const REDUCE:REDUCEMOD>(&self, a:&montgomery::Montgomery<O>, b:&mut O);
+}
+
+pub trait VecOperations<O>{
+
+    // Applies a parameterized modular reduction.
+    fn vec_reduce_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, x: &mut [O]);
+
+    // Assigns a[i] + b[i] to c[i]
+    fn vec_add_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b:&[O], c: &mut [O]);
+
+    // Assigns a[i] + b[i] to b[i]
+    fn vec_add_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b: &mut [O]);
+
+    // Assigns a[i] - b[i] to c[i]
+    fn vec_sub_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b:&[O], c: &mut [O]);
+
+    // Assigns a[i] - b[i] to b[i]
+    fn vec_sub_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b: &mut [O]);
+
+    // Assigns -a[i] to a[i].
+    fn vec_neg_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &mut [O]);
+
+    // Assigns a * 2^64 to b.
+    fn vec_prepare_montgomery_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[O], b: &mut [montgomery::Montgomery<O>]);
 
     // Assigns a[i] * b[i] to c[i].
-    fn mul_vec_montgomery_external_binary_assign<const CHUNK:usize,const REDUCE:REDUCEMOD>(&self, a:&[montgomery::Montgomery<O>], b:&[O], c: &mut [O]);
+    fn vec_mul_montgomery_external_binary_assign<const CHUNK:usize,const REDUCE:REDUCEMOD>(&self, a:&[montgomery::Montgomery<O>], b:&[O], c: &mut [O]);
 
     // Assigns a[i] * b[i] to b[i].
-    fn mul_vec_montgomery_external_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a:&[montgomery::Montgomery<O>], b:&mut [O]);
+    fn vec_mul_montgomery_external_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a:&[montgomery::Montgomery<O>], b:&mut [O]);
 }
 
 

@@ -1,8 +1,25 @@
+#[derive(Clone, Copy, Debug, PartialEq, Eq)]
+pub struct Barrett<O>(pub O, pub O);
+
+impl<O> Barrett<O> {
+    
+    #[inline(always)]
+    pub fn value(&self) -> &O {
+        &self.0
+    }
+
+    #[inline(always)]
+    pub fn quotient(&self) -> &O {
+        &self.1
+    }
+}
+
 #[derive(Clone, Copy, Debug, PartialEq, Eq)]
 pub struct BarrettPrecomp<O>{
     pub q: O,
     pub lo:O,
     pub hi:O,
+    pub one: Barrett<O>,
 }
 
 impl<O> BarrettPrecomp<O>{

@@ -1,15 +1,30 @@
-use crate::modulus::barrett::BarrettPrecomp;
+use crate::modulus::barrett::{Barrett, BarrettPrecomp};
 use crate::modulus::ReduceOnce;
 
 use num_bigint::BigUint;
 use num_traits::cast::ToPrimitive;
 
 impl BarrettPrecomp<u64>{
+    
     pub fn new(q: u64) -> BarrettPrecomp<u64> {
         let big_r: BigUint = (BigUint::from(1 as usize)<<((u64::BITS<<1) as usize)) / BigUint::from(q);
         let lo: u64 = (&big_r & BigUint::from(u64::MAX)).to_u64().unwrap();
         let hi: u64 = (big_r >> u64::BITS).to_u64().unwrap();
-        Self{q, lo, hi}
+        let mut precomp: BarrettPrecomp<u64> = Self{q, lo, hi, one:Barrett(0,0)};
+        precomp.one = precomp.prepare(1);
+        precomp
+    }
+
+    #[inline(always)]
+    pub fn one(&self) -> Barrett<u64> {
+        self.one
+    }
+
+    #[inline(always)]
+    pub fn prepare(&self, v: u64) -> Barrett<u64> {
+        debug_assert!(v < self.q);
+        let quotient: u64 = (((v as u128) << 64) / self.q as u128) as _;
+        Barrett(v, quotient)
     }
 
     /// Returns lhs mod q.
@@ -39,5 +54,31 @@ #[inline(always)]
     pub fn reduce_lazy_assign(&self, lhs: &mut u64){
         let (_, mhi) = lhs.widening_mul(self.hi);
         *lhs = *lhs - mhi.wrapping_mul(self.q)
+    }
+
+    #[inline(always)]
+    pub fn mul_external(&self, lhs: Barrett<u64>, rhs: u64) -> u64 {
+        let mut r: u64 = self.mul_external_lazy(lhs, rhs);
+        r.reduce_once_assign(self.q);
+        r
+    }
+
+    #[inline(always)]
+    pub fn mul_external_assign(&self, lhs: Barrett<u64>, rhs: &mut u64){
+        self.mul_external_lazy_assign(lhs, rhs);
+        rhs.reduce_once_assign(self.q);
+    }
+
+    #[inline(always)]
+    pub fn mul_external_lazy(&self, lhs: Barrett<u64>, rhs: u64) -> u64 {
+        let mut r: u64 = rhs;
+        self.mul_external_lazy_assign(lhs, &mut r);
+        r
+    }
+
+    #[inline(always)]
+    pub fn mul_external_lazy_assign(&self, lhs: Barrett<u64>, rhs: &mut u64){
+        let t: u64 = ((*lhs.quotient() as u128 * *rhs as u128) >> 64) as _;
+        *rhs = (rhs.wrapping_mul(*lhs.value())).wrapping_sub(self.q.wrapping_mul(t));
     }
 }

@@ -1,5 +1,32 @@
 pub mod prime;
 pub mod barrett;
 pub mod montgomery;
-pub mod shoup;
-pub mod operations;+pub mod operations;
+
+use crate::modulus::ReduceOnce;
+
+impl ReduceOnce<u64> for u64{
+    #[inline(always)]
+    fn reduce_once_constant_time_assign(&mut self, q: u64){
+        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
+        *self -= (q.wrapping_sub(*self)>>63)*q;
+    }
+    
+    #[inline(always)]
+    fn reduce_once_constant_time(&self, q:u64) -> u64{
+        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
+        self - (q.wrapping_sub(*self)>>63)*q
+    }
+
+    #[inline(always)]
+    fn reduce_once_assign(&mut self, q: u64){
+        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
+        *self = *self.min(&mut self.wrapping_sub(q))
+    }
+
+    #[inline(always)]
+    fn reduce_once(&self, q:u64) -> u64{
+        debug_assert!(q < 0x8000000000000000, "2q >= 2^64");
+        *self.min(&mut self.wrapping_sub(q))
+    }
+}

@@ -26,12 +26,12 @@ two_q: q<<1,
             four_q: q<<2,
             barrett: BarrettPrecomp::new(q), 
             q_inv: q_inv,
-            one: Montgomery(0),
-            minus_one: Montgomery(0),
+            one: 0,
+            minus_one:0,
         };
 
         precomp.one = precomp.prepare::<ONCE>(1);
-        precomp.minus_one = Montgomery(q-precomp.one.value());
+        precomp.minus_one = q-precomp.one;
 
         precomp
     }
@@ -71,7 +71,7 @@ ONCE =>{x.reduce_once_assign(self.q)},
             TWICE=>{x.reduce_once_assign(self.two_q)},
             FOURTIMES =>{x.reduce_once_assign(self.four_q)},
             BARRETT =>{self.barrett.reduce_assign(x)},
-            BARRETTLAZY =>{self.barrett.reduce_assign(x)},
+            BARRETTLAZY =>{self.barrett.reduce_lazy_assign(x)},
             _ => unreachable!("invalid REDUCE argument")
         }
     }
@@ -79,7 +79,7 @@
     /// Returns lhs * 2^64 mod q as a Montgomery<u64>.
     #[inline(always)]
     pub fn prepare<const REDUCE:REDUCEMOD>(&self, lhs: u64) -> Montgomery<u64>{
-        let mut rhs = Montgomery(0);
+        let mut rhs: u64 = 0;
         self.prepare_assign::<REDUCE>(lhs, &mut rhs);
         rhs
     }
@@ -88,8 +88,8 @@ /// Assigns lhs * 2^64 mod q to rhs.
     #[inline(always)]
     pub fn prepare_assign<const REDUCE:REDUCEMOD>(&self, lhs: u64, rhs: &mut Montgomery<u64>){
         let (_, mhi) = lhs.widening_mul(*self.barrett.value_lo());
-        *rhs = Montgomery((lhs.wrapping_mul(*self.barrett.value_hi()).wrapping_add(mhi)).wrapping_mul(self.q).wrapping_neg());
-        self.reduce_assign::<REDUCE>(rhs.value_mut()); 
+        *rhs = (lhs.wrapping_mul(*self.barrett.value_hi()).wrapping_add(mhi)).wrapping_mul(self.q).wrapping_neg();
+        self.reduce_assign::<REDUCE>(rhs); 
     }
 
     /// Returns lhs * (2^64)^-1 mod q as a u64.
@@ -103,7 +103,7 @@
     /// Assigns lhs * (2^64)^-1 mod q to rhs.
     #[inline(always)]
     pub fn unprepare_assign<const REDUCE:REDUCEMOD>(&self, lhs: Montgomery<u64>, rhs: &mut u64){
-        let (_, r) = self.q.widening_mul(lhs.value().wrapping_mul(self.q_inv));
+        let (_, r) = self.q.widening_mul(lhs.wrapping_mul(self.q_inv));
         *rhs = self.reduce::<REDUCE>(self.q.wrapping_sub(r));        
     }
 
@@ -118,7 +118,7 @@
     /// Assigns lhs * rhs * (2^{64})^-1 mod q to rhs.
     #[inline(always)]
     pub fn mul_external_assign<const REDUCE:REDUCEMOD>(&self, lhs: Montgomery<u64>, rhs: &mut u64){
-        let (mlo, mhi) = lhs.value().widening_mul(*rhs);
+        let (mlo, mhi) = lhs.widening_mul(*rhs);
         let (_, hhi) = self.q.widening_mul(mlo.wrapping_mul(self.q_inv));
         *rhs = self.reduce::<REDUCE>(mhi.wrapping_sub(hhi).wrapping_add(self.q));
     }
@@ -126,31 +126,31 @@
     /// Returns lhs * rhs * (2^{64})^-1 mod q in range [0, 2q-1].
     #[inline(always)]
     pub fn mul_internal<const REDUCE:REDUCEMOD>(&self, lhs: Montgomery<u64>, rhs: Montgomery<u64>) -> Montgomery<u64>{
-        Montgomery(self.mul_external::<REDUCE>(lhs, *rhs.value()))
+        self.mul_external::<REDUCE>(lhs, rhs)
     }
 
     /// Assigns lhs * rhs * (2^{64})^-1 mod q to rhs.
     #[inline(always)]
     pub fn mul_internal_assign<const REDUCE:REDUCEMOD>(&self, lhs: Montgomery<u64>, rhs: &mut Montgomery<u64>){
-        self.mul_external_assign::<REDUCE>(lhs, rhs.value_mut());
+        self.mul_external_assign::<REDUCE>(lhs, rhs);
     }
 
     #[inline(always)]
     pub fn add_internal(&self, lhs: Montgomery<u64>, rhs: Montgomery<u64>) -> Montgomery<u64>{
-        Montgomery(self.barrett.reduce(rhs.value() + lhs.value()))
+        self.barrett.reduce(rhs + lhs)
     }
 
     /// Assigns lhs + rhs to rhs.
     #[inline(always)]
     pub fn add_internal_lazy_assign(&self, lhs: Montgomery<u64>, rhs: &mut Montgomery<u64>){
-        *rhs.value_mut() += lhs.value()
+        *rhs += lhs
     }
 
     /// Assigns lhs + rhs - q if (lhs + rhs) >= q to rhs.
     #[inline(always)]
     pub fn add_internal_reduce_once_assign<const LAZY:bool>(&self, lhs: Montgomery<u64>, rhs: &mut Montgomery<u64>){
         self.add_internal_lazy_assign(lhs, rhs);
-        rhs.value_mut().reduce_once_assign(self.q);
+        rhs.reduce_once_assign(self.q);
     }
 
     /// Returns lhs mod q in range [0, 2q-1].
@@ -173,28 +173,9 @@ self.mul_internal_assign::<ONCE>(x_mut, &mut x_mut);
             i >>= 1;
         }
 
-        y.value_mut().reduce_once_assign(self.q);
+        y.reduce_once_assign(self.q);
         y
     }
-}
-
-/// Returns x^exponent mod q.
-/// This function internally instantiate a new MontgomeryPrecomp<u64>
-/// To be used when called only a few times and if there 
-/// is no Prime instantiated with q.
-fn pow(x:u64, exponent:u64, q:u64) -> u64{
-    let montgomery: MontgomeryPrecomp<u64> = MontgomeryPrecomp::<u64>::new(q);
-    let mut y_mont: Montgomery<u64> = montgomery.one();
-    let mut x_mont: Montgomery<u64> = montgomery.prepare::<ONCE>(x);
-    while exponent > 0{
-        if exponent & 1 == 1{
-            montgomery.mul_internal_assign::<ONCE>(x_mont, &mut y_mont);
-        }
-
-        montgomery.mul_internal_assign::<ONCE>(x_mont, &mut x_mont);
-    }
-    
-    montgomery.unprepare::<ONCE>(y_mont)
 }
 
 #[cfg(test)]

@@ -1,84 +1,121 @@
 
-use crate::modulus::Operations;
+use crate::modulus::{WordOperations, VecOperations};
 use crate::modulus::prime::Prime;
 use crate::modulus::ReduceOnce;
 use crate::modulus::montgomery::Montgomery;
-use crate::modulus::{REDUCEMOD, NONE, ONCE, BARRETT, BARRETTLAZY};
+use crate::modulus::REDUCEMOD;
 use crate::{apply_unary, apply_binary, apply_ternary};
 use itertools::izip;
 
-impl Operations<u64> for Prime<u64>{
+impl WordOperations<u64> for Prime<u64>{
 
+    /// Applies a modular reduction on x based on REDUCE:
+    /// - LAZY: no modular reduction.
+    /// - ONCE: subtracts q if x >= q.
+    /// - TWO: subtracts 2q if x >= 2q.
+    /// - FOUR: subtracts 4q if x >= 4q.
+    /// - BARRETT: maps x to x mod q using Barrett reduction.
+    /// - BARRETTLAZY: maps x to x mod q using Barrett reduction with values in [0, 2q-1].
     #[inline(always)]
-    fn add_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &u64, c: &mut u64){
+    fn word_reduce_assign<const REDUCE:REDUCEMOD>(&self, x: &mut u64){
+        self.montgomery.reduce_assign::<REDUCE>(x);
+    }
+
+    #[inline(always)]
+    fn word_add_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &u64, c: &mut u64){
         *c = a.wrapping_add(*b);
-        self.montgomery.reduce_assign::<REDUCE>(c);
+        self.word_reduce_assign::<REDUCE>(c);
     }
 
     #[inline(always)]
-    fn add_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &mut u64){
+    fn word_add_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &mut u64){
         *b = a.wrapping_add(*b);
-        self.montgomery.reduce_assign::<REDUCE>(b);
+        self.word_reduce_assign::<REDUCE>(b);
+    }
+
+    #[inline(always)]
+    fn word_sub_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &u64, c: &mut u64){
+        *c = a.wrapping_add(self.q.wrapping_sub(*b)).reduce_once(self.q);
+    }
+
+    #[inline(always)]
+    fn word_sub_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &mut u64){
+        *b = a.wrapping_add(self.q.wrapping_sub(*b)).reduce_once(self.q);
+    }
+
+    #[inline(always)]
+    fn word_neg_assign<const REDUCE:REDUCEMOD>(&self, a: &mut u64){
+        *a = self.q.wrapping_sub(*a);
+        self.word_reduce_assign::<REDUCE>(a)
     }
 
     #[inline(always)]
-    fn add_vec_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&[u64], c:&mut [u64]){
-        apply_ternary!(self, Self::add_binary_assign::<REDUCE>, a, b, c, CHUNK);
+    fn word_prepare_montgomery_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &mut Montgomery<u64>){
+        self.montgomery.prepare_assign::<REDUCE>(*a, b);
     }
 
     #[inline(always)]
-    fn add_vec_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&mut [u64]){
-        apply_binary!(self, Self::add_unary_assign::<REDUCE>, a, b, CHUNK);
+    fn word_mul_montgomery_external_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &Montgomery<u64>, b:&u64, c: &mut u64){
+        *c = self.montgomery.mul_external::<REDUCE>(*a, *b);
     }
 
     #[inline(always)]
-    fn sub_binary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &u64, c: &mut u64){
-        *c = a.wrapping_add(self.q.wrapping_sub(*b)).reduce_once(self.q);
+    fn word_mul_montgomery_external_unary_assign<const REDUCE:REDUCEMOD>(&self, lhs:&Montgomery<u64>, rhs:&mut u64){
+        self.montgomery.mul_external_assign::<REDUCE>(*lhs, rhs);
     }
+}
 
+impl VecOperations<u64> for Prime<u64>{
+
+    /// Applies a modular reduction on x based on REDUCE:
+    /// - LAZY: no modular reduction.
+    /// - ONCE: subtracts q if x >= q.
+    /// - TWO: subtracts 2q if x >= 2q.
+    /// - FOUR: subtracts 4q if x >= 4q.
+    /// - BARRETT: maps x to x mod q using Barrett reduction.
+    /// - BARRETTLAZY: maps x to x mod q using Barrett reduction with values in [0, 2q-1].
     #[inline(always)]
-    fn sub_unary_assign<const REDUCE:REDUCEMOD>(&self, a: &u64, b: &mut u64){
-        *b = a.wrapping_add(self.q.wrapping_sub(*b)).reduce_once(self.q);
+    fn vec_reduce_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, x: &mut [u64]){
+        apply_unary!(self, Self::word_reduce_assign::<REDUCE>, x, CHUNK);
     }
 
     #[inline(always)]
-    fn sub_vec_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&[u64], c:&mut [u64]){
-        apply_ternary!(self, Self::sub_binary_assign::<REDUCE>, a, b, c, CHUNK);
+    fn vec_add_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&[u64], c:&mut [u64]){
+        apply_ternary!(self, Self::word_add_binary_assign::<REDUCE>, a, b, c, CHUNK);
     }
 
     #[inline(always)]
-    fn sub_vec_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&mut [u64]){
-        apply_binary!(self, Self::sub_unary_assign::<REDUCE>, a, b, CHUNK);
+    fn vec_add_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&mut [u64]){
+        apply_binary!(self, Self::word_add_unary_assign::<REDUCE>, a, b, CHUNK);
     }
 
     #[inline(always)]
-    fn neg_assign<const REDUCE:REDUCEMOD>(&self, a: &mut u64){
-        *a = self.q.wrapping_sub(*a);
-        self.montgomery.reduce_assign::<REDUCE>(a)
+    fn vec_sub_binary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&[u64], c:&mut [u64]){
+        apply_ternary!(self, Self::word_sub_binary_assign::<REDUCE>, a, b, c, CHUNK);
     }
 
     #[inline(always)]
-    fn neg_vec_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &mut [u64]){
-        apply_unary!(self, Self::neg_assign::<REDUCE>, a, CHUNK);
+    fn vec_sub_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b:&mut [u64]){
+        apply_binary!(self, Self::word_sub_unary_assign::<REDUCE>, a, b, CHUNK);
     }
 
     #[inline(always)]
-    fn mul_montgomery_external_binary_assign<const REDUCE:REDUCEMOD>(&self, a:& Montgomery<u64>, b:&u64, c: &mut u64){
-        *c = self.montgomery.mul_external::<REDUCE>(*a, *b);
+    fn vec_neg_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &mut [u64]){
+        apply_unary!(self, Self::word_neg_assign::<REDUCE>, a, CHUNK);
     }
 
     #[inline(always)]
-    fn mul_montgomery_external_unary_assign<const REDUCE:REDUCEMOD>(&self, lhs:&Montgomery<u64>, rhs:&mut u64){
-        *rhs = self.montgomery.mul_external::<REDUCE>(*lhs, *rhs);
+    fn vec_prepare_montgomery_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a: &[u64], b: &mut [Montgomery<u64>]){
+        apply_binary!(self, Self::word_prepare_montgomery_assign::<REDUCE>, a, b, CHUNK);
     }
 
     #[inline(always)]
-    fn mul_vec_montgomery_external_binary_assign<const CHUNK:usize,const REDUCE:REDUCEMOD>(&self, a:& [Montgomery<u64>], b:&[u64], c: &mut [u64]){
-        apply_ternary!(self, Self::mul_montgomery_external_binary_assign::<REDUCE>, a, b, c, CHUNK);
+    fn vec_mul_montgomery_external_binary_assign<const CHUNK:usize,const REDUCE:REDUCEMOD>(&self, a:& [Montgomery<u64>], b:&[u64], c: &mut [u64]){
+        apply_ternary!(self, Self::word_mul_montgomery_external_binary_assign::<REDUCE>, a, b, c, CHUNK);
     }
 
     #[inline(always)]
-    fn mul_vec_montgomery_external_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a:&[Montgomery<u64>], b:&mut [u64]){
-        apply_binary!(self, Self::mul_montgomery_external_unary_assign::<REDUCE>, a, b, CHUNK);
+    fn vec_mul_montgomery_external_unary_assign<const CHUNK:usize, const REDUCE:REDUCEMOD>(&self, a:& [Montgomery<u64>], b:&mut [u64]){
+        apply_binary!(self, Self::word_mul_montgomery_external_unary_assign::<REDUCE>, a, b, CHUNK);
     }
 }

@@ -1,6 +1,6 @@
 use crate::modulus::prime::Prime;
 use crate::modulus::montgomery::{Montgomery, MontgomeryPrecomp};
-use crate::modulus::shoup::{ShoupPrecomp};
+use crate::modulus::barrett::BarrettPrecomp;
 use crate::modulus::ONCE;
 use primality_test::is_prime;
 use prime_factorization::Factorization;
@@ -34,11 +34,13 @@ }
 
         let mut prime: Prime<u64> = Self {
             q:q,
+            two_q:q<<1,
+            four_q:q<<2,
             q_base:q_base,
             q_power:q_power,
             factors: Vec::new(),
             montgomery:MontgomeryPrecomp::new(q),
-            shoup:ShoupPrecomp::new(q),
+            barrett:BarrettPrecomp::new(q),
             phi:phi,
         };
 
@@ -101,7 +103,7 @@ candidate += 1;
 
             for &factor in &self.factors{
 
-                if  Pow(candidate, (self.q_base-1)/factor, self.q_base) == 1{
+                if  pow(candidate, (self.q_base-1)/factor, self.q_base) == 1{
                     not_found = true;
                     break
                 }
@@ -124,7 +126,7 @@
         let psi: u64 = self.primitive_root();
 
         // nth primitive root mod q_base: psi_nth^(prime.q_base-1)/nth_root mod q_base
-        let psi_nth_q_base: u64 = Pow(psi, (self.q_base-1)/nth_root, self.q_base);
+        let psi_nth_q_base: u64 = pow(psi, (self.q_base-1)/nth_root, self.q_base);
 
         // lifts nth primitive root mod q_base to q = q_base^q_power
         let psi_nth_q: u64 = self.hensel_lift(psi_nth_q_base, nth_root);
@@ -171,7 +173,7 @@
     /// Returns (psi + a * q_base)^{nth_root} = 1 mod q = q_base^q_power given psi^{nth_root} = 1 mod q_base.
     /// Panics if psi^{nth_root} != 1 mod q_base.
     fn hensel_lift(&self, psi: u64, nth_root: u64) -> u64{
-        assert!(Pow(psi, nth_root, self.q_base)==1, "invalid argument psi: psi^nth_root = {} != 1", Pow(psi, nth_root, self.q_base));
+        assert!(pow(psi, nth_root, self.q_base)==1, "invalid argument psi: psi^nth_root = {} != 1", pow(psi, nth_root, self.q_base));
 
         let mut psi_mont: Montgomery<u64> = self.montgomery.prepare::<ONCE>(psi);
         let nth_root_mont: Montgomery<u64> = self.montgomery.prepare::<ONCE>(nth_root);
@@ -180,7 +182,7 @@ for _i in 1..self.q_power{
 
             let psi_pow: Montgomery<u64> = self.montgomery.pow(psi_mont, nth_root-1);
 
-            let num: Montgomery<u64> = Montgomery(self.montgomery.one().value() + self.q - self.montgomery.mul_internal::<ONCE>(psi_pow, psi_mont).value());
+            let num: Montgomery<u64> = self.montgomery.one() + self.q - self.montgomery.mul_internal::<ONCE>(psi_pow, psi_mont);
 
             let mut den: Montgomery<u64> = self.montgomery.mul_internal::<ONCE>(nth_root_mont, psi_pow);
 
@@ -197,7 +199,7 @@ /// Returns x^exponent mod q.
 /// This function internally instantiate a new MontgomeryPrecomp<u64>
 /// To be used when called only a few times and if there 
 /// is no Prime instantiated with q.
-pub fn Pow(x:u64, exponent:u64, q:u64) -> u64{
+pub fn pow(x:u64, exponent:u64, q:u64) -> u64{
     let montgomery: MontgomeryPrecomp<u64> = MontgomeryPrecomp::<u64>::new(q);
     let mut y_mont: Montgomery<u64> = montgomery.one();
     let mut x_mont: Montgomery<u64> = montgomery.prepare::<ONCE>(x);

@@ -1,60 +0,0 @@
-use crate::modulus::ReduceOnce;
-use crate::modulus::shoup::{ShoupPrecomp, Shoup};
-
-impl ShoupPrecomp<u64>{
-
-    pub fn new(q: u64) -> Self {
-        let mut precomp: ShoupPrecomp<u64> = Self{q:q, one:Shoup(0,0)};
-        precomp.one = precomp.prepare(1);
-        precomp
-    }
-
-    #[inline(always)]
-    pub fn one(&self) -> Shoup<u64> {
-        self.one
-    }
-
-    #[inline(always)]
-    pub fn prepare(&self, v: u64) -> Shoup<u64> {
-        debug_assert!(v < self.q);
-        let quotient: u64 = (((v as u128) << 64) / self.q as u128) as _;
-        Shoup(v, quotient)
-    }
-
-    #[inline(always)]
-    pub fn mul_external(&self, lhs: Shoup<u64>, rhs: u64) -> u64 {
-        let mut r: u64 = self.mul_external_lazy(lhs, rhs);
-        r.reduce_once_assign(self.q);
-        r
-    }
-
-    #[inline(always)]
-    pub fn mul_external_assign(&self, lhs: Shoup<u64>, rhs: &mut u64){
-        self.mul_external_lazy_assign(lhs, rhs);
-        rhs.reduce_once_assign(self.q);
-    }
-
-    #[inline(always)]
-    pub fn mul_external_lazy(&self, lhs: Shoup<u64>, rhs: u64) -> u64 {
-        let mut r: u64 = rhs;
-        self.mul_external_lazy_assign(lhs, &mut r);
-        r
-    }
-
-    #[inline(always)]
-    pub fn mul_external_lazy_assign(&self, lhs: Shoup<u64>, rhs: &mut u64){
-        let t: u64 = ((*lhs.quotient() as u128 * *rhs as u128) >> 64) as _;
-        *rhs = (rhs.wrapping_mul(*lhs.value())).wrapping_sub(self.q.wrapping_mul(t));
-    }
-
-    #[inline(always)]
-    pub fn reduce_assign(&self, rhs: &mut u64){
-        self.reduce_assign_lazy(rhs);
-        rhs.reduce_once_assign(self.q);
-    }
-
-    #[inline(always)]
-    pub fn reduce_assign_lazy(&self, rhs: &mut u64){
-        *rhs = rhs.wrapping_sub(self.q.wrapping_mul(((self.one.1 as u128 * *rhs as u128) >> 64) as _))
-    }
-}

@@ -2,35 +2,7 @@ use crate::modulus::barrett::BarrettPrecomp;
 
 /// Montgomery is a generic struct storing
 /// an element in the Montgomery domain.
-#[derive(Clone, Copy, Debug, PartialEq, Eq)]
-pub struct Montgomery<O>(pub O);
-
-/// Implements helper methods on the struct Montgomery<O>.
-impl<O> Montgomery<O>{
-
-    #[inline(always)]
-    pub fn new(lhs: O) -> Self{
-        Self(lhs)
-    }
-
-    #[inline(always)]
-    pub fn value(&self) -> &O{
-        &self.0
-    }
-
-    pub fn value_mut(&mut self) -> &mut O{
-        &mut self.0
-    }
-}
-
-/// Default instantiation.
-impl<O> Default for  Montgomery<O> where O:Default {
-    fn default() -> Self {
-        Self {
-            0: O::default(),
-        }
-    }
-}
+pub type Montgomery<O> = O;
 
 /// MontgomeryPrecomp is a generic struct storing 
 /// precomputations for Montgomery arithmetic.

@@ -1,14 +1,16 @@
 use crate::modulus::montgomery::MontgomeryPrecomp;
-use crate::modulus::shoup::ShoupPrecomp;
+use crate::modulus::barrett::BarrettPrecomp;
 
 #[derive(Clone, Debug, PartialEq, Eq)]
 pub struct Prime<O> {
 	pub q: O, /// q_base^q_powers
+    pub two_q: O,
+    pub four_q: O,
     pub q_base: O,
     pub q_power: usize,
     pub factors: Vec<O>, /// distinct factors of q-1
     pub montgomery: MontgomeryPrecomp<O>,
-    pub shoup:ShoupPrecomp<O>,
+    pub barrett:BarrettPrecomp<O>,
     pub phi: O,
 }
 

@@ -1,22 +0,0 @@
-#[derive(Clone, Copy, Debug, PartialEq, Eq)]
-pub struct Shoup<O>(pub O, pub O);
-
-impl<O> Shoup<O> {
-    
-    #[inline(always)]
-    pub fn value(&self) -> &O {
-        &self.0
-    }
-
-    #[inline(always)]
-    pub fn quotient(&self) -> &O {
-        &self.1
-    }
-}
-
-#[derive(Clone, Copy, Debug, PartialEq, Eq)]
-pub struct ShoupPrecomp<O>{
-    pub q: O,
-    pub one: Shoup<O>,
-}
-

@@ -5,7 +5,7 @@
 /// Returns a lookup table for the automorphism X^{i} -> X^{i * k mod nth_root}.
 /// Method will panic if n or nth_root are not power-of-two.
 /// Method will panic if gal_el is not coprime with nth_root.
-pub fn automorphism_index_ntt(n: usize, nth_root:u64, gal_el: u64) -> (Vec<u64>){
+pub fn automorphism_index_ntt(n: usize, nth_root:u64, gal_el: u64) -> Vec<u64>{
     assert!(n&(n-1) != 0, "invalid n={}: not a power-of-two", n);
     assert!(nth_root&(nth_root-1) != 0, "invalid nth_root={}: not a power-of-two", n);
     assert!(gal_el & 1 == 1, "invalid gal_el={}: not coprime with nth_root={}", gal_el, nth_root);
@@ -39,6 +39,5 @@ let sign: u64 = (gal_el_i>>log_n) & 1;
             let i_out: u64 = gal_el_i & mask;
             b_vec[i_out as usize] = ai * (sign^1) | (q - ai) * sign
         });
-
     }
 }

@@ -1,7 +1,6 @@
 use crate::ring::Ring;
 use crate::dft::ntt::Table;
 use crate::modulus::prime::Prime;
-use crate::modulus::montgomery::Montgomery;
 use crate::poly::Poly;
 
 impl Ring<u64>{
@@ -18,15 +17,37 @@ pub fn n(&self) -> usize{
         return self.n
     }
 
-    fn new_poly_core<O>(&self) -> Poly<O> where O: Default + Clone {
-        Poly::<O>::new(self.n())
+    pub fn new_poly(&self) -> Poly<u64>{
+        Poly::<u64>::new(self.n())
+    }
+
+    pub fn ntt_inplace<const LAZY:bool>(&self, poly: &mut Poly<u64>){
+        match LAZY{
+            true => self.dft.forward_inplace_lazy(&mut poly.0),
+            false => self.dft.forward_inplace(&mut poly.0)
+        }
+    }
+
+    pub fn intt_inplace<const LAZY:bool>(&self, poly: &mut Poly<u64>){
+        match LAZY{
+            true => self.dft.forward_inplace_lazy(&mut poly.0),
+            false => self.dft.forward_inplace(&mut poly.0)
+        }
     }
 
-    pub fn new_poly(&self) -> Poly<u64>{
-        self.new_poly_core::<u64>()
+    pub fn ntt<const LAZY:bool>(&self, poly_in: &Poly<u64>, poly_out: &mut Poly<u64>){
+        poly_out.0.copy_from_slice(&poly_in.0);
+        match LAZY{
+            true => self.dft.backward_inplace_lazy(&mut poly_out.0),
+            false => self.dft.backward_inplace(&mut poly_out.0)
+        }
     }
 
-    pub fn new_poly_montgomery(&self) -> Poly<Montgomery<u64>>{
-        self.new_poly_core::<Montgomery<u64>>()
+    pub fn intt<const LAZY:bool>(&self, poly_in: &Poly<u64>, poly_out: &mut Poly<u64>){
+        poly_out.0.copy_from_slice(&poly_in.0);
+        match LAZY{
+            true => self.dft.backward_inplace_lazy(&mut poly_out.0),
+            false => self.dft.backward_inplace(&mut poly_out.0)
+        }
     }
 }