diff --git a/Cargo.lock b/Cargo.lock
index b729521..1f71ab3 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -198,6 +198,12 @@ version = "1.13.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "60b1af1c220855b6ceac025d3f6ecdd2b7c4894bfe9cd9bda4fbb4bc7c0d4cf0"
 
+[[package]]
+name = "fnv"
+version = "1.0.7"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "3f9eec918d3f24069decb9af1554cad7c880e2da24a9afd88aca000531ab82c1"
+
 [[package]]
 name = "getrandom"
 version = "0.2.15"
@@ -309,6 +315,7 @@ dependencies = [
  "rand_distr",
  "sampling",
  "sprs",
+ "utils",
 ]
 
 [[package]]
@@ -733,6 +740,13 @@ version = "1.0.14"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "adb9e6ca4f869e1180728b7950e35922a7fc6397f7b641499e8f3ef06e50dc83"
 
+[[package]]
+name = "utils"
+version = "0.1.0"
+dependencies = [
+ "fnv",
+]
+
 [[package]]
 name = "walkdir"
 version = "2.5.0"
diff --git a/Cargo.toml b/Cargo.toml
index 634efaf..c13f7e4 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -1,2 +1,2 @@
 [workspace]
-members = ["math", "sampling"]
+members = ["math", "sampling", "utils"]
diff --git a/math/Cargo.toml b/math/Cargo.toml
index 7a90af4..90a5161 100644
--- a/math/Cargo.toml
+++ b/math/Cargo.toml
@@ -16,6 +16,7 @@ rand_distr = "0.4.3"
 sprs = "0.11.2"
 
 sampling = { path = "../sampling" }
+utils = { path = "../utils" }
 
 [[bench]]
 name = "ntt"
diff --git a/math/src/automorphism.rs b/math/src/automorphism.rs
index 6f985ff..417d46a 100644
--- a/math/src/automorphism.rs
+++ b/math/src/automorphism.rs
@@ -1,4 +1,29 @@
 use crate::modulus::WordOps;
+use crate::ring::Ring;
+use num::Unsigned;
+use utils::map::Map;
+
+pub struct AutoPermMap(Map<usize, AutoPerm>);
+
+impl AutoPermMap {
+    pub fn new() -> Self {
+        Self {
+            0: Map::<usize, AutoPerm>::new(),
+        }
+    }
+
+    pub fn insert(&mut self, perm: AutoPerm) {
+        self.0.insert(perm.gal_el, perm);
+    }
+
+    pub fn gen<O: Unsigned, const NTT: bool>(&mut self, ring: &Ring<O>, gen_1: usize, gen_2: bool) {
+        self.insert(AutoPerm::new::<O, NTT>(ring, gen_1, gen_2))
+    }
+
+    pub fn get(&self, gal_el: &usize) -> Option<&AutoPerm> {
+        self.0.get(gal_el)
+    }
+}
 
 pub struct AutoPerm {
     pub gal_el: usize,
@@ -9,29 +34,21 @@ impl AutoPerm {
     /// 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
-        );
+    pub fn new<O: Unsigned, const NTT: bool>(ring: &Ring<O>, gen_1: usize, gen_2: bool) -> Self {
+        let n = ring.n();
+        let cyclotomic_order = ring.cyclotomic_order();
+
+        let gal_el = ring.galois_element(gen_1, gen_2);
 
         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;
+            let mask = cyclotomic_order - 1;
+            let log_cyclotomic_order_half: u32 = cyclotomic_order.log2() as u32 - 1;
             for i in 0..n {
-                let i_rev: usize = 2 * i.reverse_bits_msb(log_nth_root_half) + 1;
+                let i_rev: usize = 2 * i.reverse_bits_msb(log_cyclotomic_order_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));
+                permutation.push(gal_el_i.reverse_bits_msb(log_cyclotomic_order_half));
             }
         } else {
             let log_n: usize = n.log2();
diff --git a/math/src/ring.rs b/math/src/ring.rs
index 62dfb01..1387c9a 100644
--- a/math/src/ring.rs
+++ b/math/src/ring.rs
@@ -1,15 +1,16 @@
 pub mod impl_u64;
-
 use crate::dft::DFT;
 use crate::modulus::prime::Prime;
 use crate::modulus::WordOps;
 use crate::poly::{Poly, PolyRNS};
+use crate::GALOISGENERATOR;
 use num::traits::Unsigned;
 use std::rc::Rc;
 
 pub struct Ring<O: Unsigned> {
     pub n: usize,
     pub modulus: Prime<O>,
+    pub cyclotomic_order: usize,
     pub dft: Box<dyn DFT<O>>,
 }
 
@@ -25,6 +26,33 @@ impl<O: Unsigned> Ring<O> {
     pub fn new_poly(&self) -> Poly<u64> {
         Poly::<u64>::new(self.n())
     }
+
+    pub fn cyclotomic_order(&self) -> usize {
+        self.cyclotomic_order
+    }
+
+    // Returns GALOISGENERATOR^gen_1 * (-1)^gen_2 mod 2^log_nth_root.
+    pub fn galois_element(&self, gen_1: usize, gen_2: bool) -> usize {
+        let mut gal_el: usize = 1;
+        let mut gen_1_pow: usize = GALOISGENERATOR;
+        let mut e: usize = gen_1;
+        while e > 0 {
+            if e & 1 == 1 {
+                gal_el = gal_el.wrapping_mul(gen_1_pow);
+            }
+
+            gen_1_pow = gen_1_pow.wrapping_mul(gen_1_pow);
+            e >>= 1;
+        }
+
+        let nth_root = 1 << self.cyclotomic_order;
+        gal_el &= nth_root - 1;
+
+        if gen_2 {
+            return nth_root - gal_el;
+        }
+        gal_el
+    }
 }
 
 pub struct RingRNS<O: Unsigned>(pub Vec<Rc<Ring<O>>>);
diff --git a/math/src/ring/impl_u64/ring.rs b/math/src/ring/impl_u64/ring.rs
index 2f090e3..adc55af 100644
--- a/math/src/ring/impl_u64/ring.rs
+++ b/math/src/ring/impl_u64/ring.rs
@@ -7,7 +7,6 @@ use crate::modulus::{BARRETT, REDUCEMOD};
 use crate::poly::Poly;
 use crate::ring::Ring;
 use crate::CHUNK;
-use crate::GALOISGENERATOR;
 use num_bigint::BigInt;
 use num_traits::ToPrimitive;
 
@@ -17,6 +16,7 @@ impl Ring<u64> {
         Self {
             n: n,
             modulus: prime.clone(),
+            cyclotomic_order: n << 1,
             dft: Box::new(Table::<u64>::new(prime, n << 1)),
         }
     }
@@ -40,29 +40,6 @@ impl Ring<u64> {
             .enumerate()
             .for_each(|(i, v)| *v = (&coeffs[i] % &q_big).to_u64().unwrap());
     }
-
-    // Returns GALOISGENERATOR^gen_1 * (-1)^gen_2 mod 2^log_nth_root.
-    pub fn galois_element(&self, gen_1: usize, gen_2: bool, log_nth_root: usize) -> usize {
-        let mut gal_el: usize = 1;
-        let mut gen_1_pow: usize = GALOISGENERATOR;
-        let mut e: usize = gen_1;
-        while e > 0 {
-            if e & 1 == 1 {
-                gal_el = gal_el.wrapping_mul(gen_1_pow);
-            }
-
-            gen_1_pow = gen_1_pow.wrapping_mul(gen_1_pow);
-            e >>= 1;
-        }
-
-        let nth_root = 1 << log_nth_root;
-        gal_el &= nth_root - 1;
-
-        if gen_2 {
-            return nth_root - gal_el;
-        }
-        gal_el
-    }
 }
 
 impl Ring<u64> {
diff --git a/math/tests/automorphism.rs b/math/tests/automorphism.rs
index 0adb306..ec7f7f2 100644
--- a/math/tests/automorphism.rs
+++ b/math/tests/automorphism.rs
@@ -77,9 +77,10 @@ fn test_automorphism_from_perm_u64<const NTT: bool>(ring: &Ring<u64>, nth_root:
         ring.ntt_inplace::<false>(&mut p0);
     }
 
-    let gal_el: usize = 2 * nth_root - 1;
+    let gen_1 = 0;
+    let gen_2 = true;
 
-    let auto_perm = AutoPerm::new::<NTT>(n, gal_el, nth_root);
+    let auto_perm = AutoPerm::new::<_, NTT>(&ring, gen_1, gen_2);
 
     ring.a_apply_automorphism_from_perm_into_b::<NTT>(&p0, &auto_perm, &mut p1);
 
diff --git a/utils/Cargo.toml b/utils/Cargo.toml
new file mode 100644
index 0000000..5f23bb6
--- /dev/null
+++ b/utils/Cargo.toml
@@ -0,0 +1,7 @@
+[package]
+name = "utils"
+version = "0.1.0"
+edition = "2021"
+
+[dependencies]
+fnv = "1.0.7"
\ No newline at end of file
diff --git a/utils/src/lib.rs b/utils/src/lib.rs
new file mode 100644
index 0000000..1d7f53b
--- /dev/null
+++ b/utils/src/lib.rs
@@ -0,0 +1 @@
+pub mod map;
diff --git a/utils/src/map.rs b/utils/src/map.rs
new file mode 100644
index 0000000..9709ada
--- /dev/null
+++ b/utils/src/map.rs
@@ -0,0 +1,20 @@
+use fnv::FnvHashMap;
+use std::hash::Hash;
+
+pub struct Map<K, V>(pub FnvHashMap<K, V>);
+
+impl<K: Eq + Hash, V> Map<K, V> {
+    pub fn new() -> Self {
+        Self {
+            0: FnvHashMap::<K, V>::default(),
+        }
+    }
+
+    pub fn insert(&mut self, k: K, data: V) -> Option<V> {
+        self.0.insert(k, data)
+    }
+
+    pub fn get(&self, k: &K) -> Option<&V> {
+        self.0.get(k)
+    }
+}