@@ -44,7 +44,7 @@ for i in 0..p0.n() {
         p0.0[i] = i as u64
     }
 
-    r.a_apply_automorphism_into_b::<false>(&p0, 2 * r.n - 1, nth_root, &mut p1);
+    r.a_apply_automorphism_native_into_b::<false>(&p0, 2 * r.n - 1, nth_root, &mut p1);
 
     println!("{:?}", p1);
 }

@@ -0,0 +1,52 @@
+use crate::modulus::WordOps;
+
+pub struct AutomorphismPermutation {
+    pub gal_el: usize,
+    pub permutation: Vec<usize>,
+}
+
+impl AutomorphismPermutation {
+    /// 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 new<const NTT: bool>(n: usize, gal_el: usize, nth_root: usize) -> Self {
+        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
+        );
+
+        let mut permutation: Vec<usize> = Vec::with_capacity(n);
+
+        if NTT {
+            let mask = nth_root - 1;
+            let log_nth_root_half: u32 = nth_root.log2() as u32 - 1;
+            for i in 0..n {
+                let i_rev: usize = 2 * i.reverse_bits_msb(log_nth_root_half) + 1;
+                let gal_el_i: usize = ((gal_el * i_rev) & mask) >> 1;
+                permutation.push(gal_el_i.reverse_bits_msb(log_nth_root_half));
+            }
+        } else {
+            let log_n: usize = n.log2();
+            let mask: usize = (n - 1) as usize;
+            for i in 0..n {
+                let gal_el_i: usize = i as usize * gal_el;
+                let sign: usize = (gal_el_i >> log_n) & 1;
+                let i_out: usize = (gal_el_i & mask) | (sign << (usize::BITS - 1));
+                permutation.push(i_out)
+            }
+        }
+
+        Self {
+            gal_el: gal_el,
+            permutation: permutation,
+        }
+    }
+}

@@ -1,6 +1,7 @@
 #![feature(bigint_helper_methods)]
 #![feature(test)]
 
+pub mod automorphism;
 pub mod dft;
 pub mod modulus;
 pub mod num_bigint;

@@ -1,84 +1,44 @@
+use crate::automorphism::AutomorphismPermutation;
 use crate::modulus::{ScalarOperations, ONCE};
 use crate::modulus::{WordOps, REDUCEMOD};
 use crate::poly::Poly;
 use crate::ring::Ring;
 
-/// 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<const NTT: bool>(n: usize, nth_root: usize, gal_el: usize) -> Vec<usize> {
-    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
-    );
-
-    let mut index: Vec<usize> = Vec::with_capacity(n);
-
-    if NTT {
-        let mask = nth_root - 1;
-        let log_nth_root_half: u32 = nth_root.log2() as u32 - 1;
-        for i in 0..n {
-            let i_rev: usize = 2 * i.reverse_bits_msb(log_nth_root_half) + 1;
-            let gal_el_i: usize = ((gal_el * i_rev) & mask) >> 1;
-            index.push(gal_el_i.reverse_bits_msb(log_nth_root_half));
-        }
-    } else {
-        let log_n: usize = n.log2();
-        let mask: usize = (n - 1) as usize;
-        for i in 0..n {
-            let gal_el_i: usize = i as usize * gal_el;
-            let sign: usize = (gal_el_i >> log_n) & 1;
-            let i_out: usize = (gal_el_i & mask) | (sign << (usize::BITS - 1));
-            index.push(i_out)
-        }
-    }
-
-    index
-}
-
 impl Ring<u64> {
     // b <- auto(a)
-    pub fn a_apply_automorphism_into_b<const NTT: bool>(
+    pub fn a_apply_automorphism_native_into_b<const NTT: bool>(
         &self,
         a: &Poly<u64>,
         gal_el: usize,
         nth_root: usize,
         b: &mut Poly<u64>,
     ) {
-        self.apply_automorphism_core::<0, ONCE, NTT>(a, gal_el, nth_root, b)
+        self.apply_automorphism_native_core::<0, ONCE, NTT>(a, gal_el, nth_root, b)
     }
 
     // b <- REDUCEMOD(b + auto(a))
-    pub fn a_apply_automorphism_add_b_into_b<const REDUCE: REDUCEMOD, const NTT: bool>(
+    pub fn a_apply_automorphism_native_add_b_into_b<const REDUCE: REDUCEMOD, const NTT: bool>(
         &self,
         a: &Poly<u64>,
         gal_el: usize,
         nth_root: usize,
         b: &mut Poly<u64>,
     ) {
-        self.apply_automorphism_core::<1, REDUCE, NTT>(a, gal_el, nth_root, b)
+        self.apply_automorphism_native_core::<1, REDUCE, NTT>(a, gal_el, nth_root, b)
     }
 
     // b <- REDUCEMOD(b - auto(a))
-    pub fn a_apply_automorphism_sub_b_into_b<const REDUCE: REDUCEMOD, const NTT: bool>(
+    pub fn a_apply_automorphism_native_sub_b_into_b<const REDUCE: REDUCEMOD, const NTT: bool>(
         &self,
         a: &Poly<u64>,
         gal_el: usize,
         nth_root: usize,
         b: &mut Poly<u64>,
     ) {
-        self.apply_automorphism_core::<2, REDUCE, NTT>(a, gal_el, nth_root, b)
+        self.apply_automorphism_native_core::<2, REDUCE, NTT>(a, gal_el, nth_root, b)
     }
 
-    fn apply_automorphism_core<const MOD: u8, const REDUCE: REDUCEMOD, const NTT: bool>(
+    fn apply_automorphism_native_core<const MOD: u8, const REDUCE: REDUCEMOD, const NTT: bool>(
         &self,
         a: &Poly<u64>,
         gal_el: usize,
@@ -155,46 +115,40 @@ }
     }
 
     // b <- auto(a)
-    pub fn a_apply_automorphism_from_index_into_b<const NTT: bool>(
+    pub fn a_apply_automorphism_from_perm_into_b<const NTT: bool>(
         &self,
         a: &Poly<u64>,
-        idx: &[usize],
+        auto_perm: &AutomorphismPermutation,
         b: &mut Poly<u64>,
     ) {
-        self.automorphism_from_index_core::<0, ONCE, NTT>(a, idx, b)
+        self.automorphism_from_perm_core::<0, ONCE, NTT>(a, auto_perm, b)
     }
 
     // b <- REDUCEMOD(b + auto(a))
-    pub fn a_apply_automorphism_from_index_add_b_into_b<
-        const REDUCE: REDUCEMOD,
-        const NTT: bool,
-    >(
+    pub fn a_apply_automorphism_from_perm_add_b_into_b<const REDUCE: REDUCEMOD, const NTT: bool>(
         &self,
         a: &Poly<u64>,
-        idx: &[usize],
+        auto_perm: &AutomorphismPermutation,
         b: &mut Poly<u64>,
     ) {
-        self.automorphism_from_index_core::<1, REDUCE, NTT>(a, idx, b)
+        self.automorphism_from_perm_core::<1, REDUCE, NTT>(a, auto_perm, b)
     }
 
     // b <- REDUCEMOD(b - auto(a))
-    pub fn a_apply_automorphism_from_index_sub_b_into_b<
-        const REDUCE: REDUCEMOD,
-        const NTT: bool,
-    >(
+    pub fn a_apply_automorphism_from_perm_sub_b_into_b<const REDUCE: REDUCEMOD, const NTT: bool>(
         &self,
         a: &Poly<u64>,
-        idx: &[usize],
+        auto_perm: &AutomorphismPermutation,
         b: &mut Poly<u64>,
     ) {
-        self.automorphism_from_index_core::<2, REDUCE, NTT>(a, idx, b)
+        self.automorphism_from_perm_core::<2, REDUCE, NTT>(a, auto_perm, b)
     }
 
     // b <- auto(a) if OVERWRITE else b <- REDUCEMOD(b + auto(a))
-    fn automorphism_from_index_core<const MOD: u8, const REDUCE: REDUCEMOD, const NTT: bool>(
+    fn automorphism_from_perm_core<const MOD: u8, const REDUCE: REDUCEMOD, const NTT: bool>(
         &self,
         a: &Poly<u64>,
-        idx: &[usize],
+        auto_perm: &AutomorphismPermutation,
         b: &mut Poly<u64>,
     ) {
         debug_assert!(
@@ -206,6 +160,8 @@ );
 
         let b_vec: &mut Vec<u64> = &mut b.0;
         let a_vec: &Vec<u64> = &a.0;
+
+        let idx: &Vec<usize> = &auto_perm.permutation;
 
         if NTT {
             a_vec.iter().enumerate().for_each(|(i, ai)| match MOD {

@@ -1,4 +1,5 @@
 use itertools::izip;
+use math::automorphism::AutomorphismPermutation;
 use math::poly::Poly;
 use math::ring::Ring;
 
@@ -10,11 +11,18 @@ let q_base: u64 = 65537u64;
     let q_power: usize = 1usize;
     let ring: Ring<u64> = Ring::new(n, q_base, q_power);
 
-    sub_test("test_automorphism_u64::<NTT:false>", || {
-        test_automorphism_u64::<false>(&ring, nth_root)
+    sub_test("test_automorphism_native_u64::<NTT:false>", || {
+        test_automorphism_native_u64::<false>(&ring, nth_root)
     });
-    sub_test("test_automorphism_u64::<NTT:true>", || {
-        test_automorphism_u64::<true>(&ring, nth_root)
+    sub_test("test_automorphism_native_u64::<NTT:true>", || {
+        test_automorphism_native_u64::<true>(&ring, nth_root)
+    });
+
+    sub_test("test_automorphism_from_perm_u64::<NTT:false>", || {
+        test_automorphism_from_perm_u64::<false>(&ring, nth_root)
+    });
+    sub_test("test_automorphism_from_perm_u64::<NTT:true>", || {
+        test_automorphism_from_perm_u64::<true>(&ring, nth_root)
     });
 }
 
@@ -23,7 +31,38 @@ println!("Running {}", name);
     f();
 }
 
-fn test_automorphism_u64<const NTT: bool>(ring: &Ring<u64>, nth_root: usize) {
+fn test_automorphism_native_u64<const NTT: bool>(ring: &Ring<u64>, nth_root: usize) {
+    let n: usize = ring.n();
+    let q: u64 = ring.modulus.q;
+
+    let mut p0: Poly<u64> = ring.new_poly();
+    let mut p1: Poly<u64> = ring.new_poly();
+
+    for i in 0..p0.n() {
+        p0.0[i] = i as u64
+    }
+
+    if NTT {
+        ring.ntt_inplace::<false>(&mut p0);
+    }
+
+    let gal_el: usize = 2 * nth_root - 1;
+
+    ring.a_apply_automorphism_native_into_b::<NTT>(&p0, gal_el, nth_root, &mut p1);
+
+    if NTT {
+        ring.intt_inplace::<false>(&mut p1);
+    }
+
+    p0.0[0] = 0;
+    for i in 1..p0.n() {
+        p0.0[i] = q - (n - i) as u64
+    }
+
+    izip!(p0.0, p1.0).for_each(|(a, b)| assert_eq!(a, b));
+}
+
+fn test_automorphism_from_perm_u64<const NTT: bool>(ring: &Ring<u64>, nth_root: usize) {
     let n: usize = ring.n();
     let q: u64 = ring.modulus.q;
 
@@ -38,7 +77,11 @@ if NTT {
         ring.ntt_inplace::<false>(&mut p0);
     }
 
-    ring.a_apply_automorphism_into_b::<NTT>(&p0, 2 * n - 1, nth_root, &mut p1);
+    let gal_el: usize = 2 * nth_root - 1;
+
+    let auto_perm = AutomorphismPermutation::new::<NTT>(n, gal_el, nth_root);
+
+    ring.a_apply_automorphism_from_perm_into_b::<NTT>(&p0, &auto_perm, &mut p1);
 
     if NTT {
         ring.intt_inplace::<false>(&mut p1);