diff --git a/src/backend/mod.rs b/src/backend/mod.rs
new file mode 100644
index 0000000..d3cc655
--- /dev/null
+++ b/src/backend/mod.rs
@@ -0,0 +1,135 @@
+use num_traits::ToPrimitive;
+
+use crate::{Matrix, RowMut};
+
+mod modulus_u64;
+mod word_size;
+
+pub use modulus_u64::ModularOpsU64;
+pub use word_size::WordSizeModulus;
+
+pub trait Modulus {
+    type Element;
+    /// Modulus value if it fits in Element
+    fn q(&self) -> Option<Self::Element>;
+    /// Modulus value as f64 if it fits in f64
+    fn q_as_f64(&self) -> Option<f64>;
+    /// Is modulus native?
+    fn is_native(&self) -> bool;
+    /// -1 in signed representaiton
+    fn neg_one(&self) -> Self::Element;
+    /// Largest unsigned value that fits in the modulus. That is, q - 1.
+    fn largest_unsigned_value(&self) -> Self::Element;
+    /// Smallest unsigned value that fits in the modulus
+    /// Always assmed to be 0.
+    fn smallest_unsigned_value(&self) -> Self::Element;
+    /// Convert unsigned value in signed represetation to i64
+    fn map_element_to_i64(&self, v: &Self::Element) -> i64;
+    /// Convert f64 to signed represented in modulus
+    fn map_element_from_f64(&self, v: f64) -> Self::Element;
+    /// Convert i64 to signed represented in modulus
+    fn map_element_from_i64(&self, v: i64) -> Self::Element;
+}
+
+impl Modulus for u64 {
+    type Element = u64;
+    fn is_native(&self) -> bool {
+        // q of size u64 can never be a naitve modulus
+        false
+    }
+    fn largest_unsigned_value(&self) -> Self::Element {
+        self - 1
+    }
+    fn neg_one(&self) -> Self::Element {
+        self - 1
+    }
+    fn smallest_unsigned_value(&self) -> Self::Element {
+        0
+    }
+    fn map_element_to_i64(&self, v: &Self::Element) -> i64 {
+        assert!(v <= self, "{v} must be <= {self}");
+        if *v >= (self >> 1) {
+            -ToPrimitive::to_i64(&(self - v)).unwrap()
+        } else {
+            ToPrimitive::to_i64(v).unwrap()
+        }
+    }
+    fn map_element_from_f64(&self, v: f64) -> Self::Element {
+        //FIXME (Jay): Before I check whether v is smaller than 0 with `let is_neg =
+        // o.is_sign_negative() && o != 0.0; I'm ocnfused why didn't I simply check <
+        // 0.0?
+        let v = v.round();
+        if v < 0.0 {
+            self - v.abs().to_u64().unwrap()
+        } else {
+            v.to_u64().unwrap()
+        }
+    }
+    fn map_element_from_i64(&self, v: i64) -> Self::Element {
+        if v < 0 {
+            self - v.abs().to_u64().unwrap()
+        } else {
+            v.to_u64().unwrap()
+        }
+    }
+    fn q(&self) -> Option<Self::Element> {
+        Some(*self)
+    }
+    fn q_as_f64(&self) -> Option<f64> {
+        self.to_f64()
+    }
+}
+
+pub trait ModInit {
+    type M;
+    fn new(modulus: Self::M) -> Self;
+}
+
+pub trait GetModulus {
+    type Element;
+    type M: Modulus<Element = Self::Element>;
+    fn modulus(&self) -> &Self::M;
+}
+
+pub trait VectorOps {
+    type Element;
+
+    fn elwise_scalar_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &Self::Element);
+    fn elwise_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &[Self::Element]);
+
+    fn elwise_add_mut(&self, a: &mut [Self::Element], b: &[Self::Element]);
+    fn elwise_sub_mut(&self, a: &mut [Self::Element], b: &[Self::Element]);
+    fn elwise_mul_mut(&self, a: &mut [Self::Element], b: &[Self::Element]);
+    fn elwise_scalar_mul_mut(&self, a: &mut [Self::Element], b: &Self::Element);
+    fn elwise_neg_mut(&self, a: &mut [Self::Element]);
+    /// inplace mutates `a`: a = a + b*c
+    fn elwise_fma_mut(&self, a: &mut [Self::Element], b: &[Self::Element], c: &[Self::Element]);
+    fn elwise_fma_scalar_mut(
+        &self,
+        a: &mut [Self::Element],
+        b: &[Self::Element],
+        c: &Self::Element,
+    );
+}
+
+pub trait ArithmeticOps {
+    type Element;
+    fn mul(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
+    fn add(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
+    fn sub(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
+    fn neg(&self, a: &Self::Element) -> Self::Element;
+}
+
+pub trait ArithmeticLazyOps {
+    type Element;
+    fn mul_lazy(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
+    fn add_lazy(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
+}
+
+pub trait ShoupMatrixFMA<M: Matrix>
+where
+    M::R: RowMut,
+{
+    /// Returns summation of row-wise product of matrix a and b.
+    fn shoup_matrix_fma(&self, out: &mut M::R, a: &M, a_shoup: &M, b: &M);
+}
diff --git a/src/backend.rs b/src/backend/modulus_u64.rs
similarity index 55%
rename from src/backend.rs
rename to src/backend/modulus_u64.rs
index 70f849c..16c0e85 100644
--- a/src/backend.rs
+++ b/src/backend/modulus_u64.rs
@@ -3,134 +3,11 @@ use std::marker::PhantomData;
 use itertools::izip;
 use num_traits::{PrimInt, Signed, ToPrimitive, WrappingAdd, WrappingMul, WrappingSub, Zero};
 
+use super::{
+    ArithmeticLazyOps, ArithmeticOps, GetModulus, ModInit, Modulus, ShoupMatrixFMA, VectorOps,
+};
 use crate::{utils::ShoupMul, Matrix, RowMut};
 
-pub trait Modulus {
-    type Element;
-    /// Modulus value if it fits in Element
-    fn q(&self) -> Option<Self::Element>;
-    /// Modulus value as f64 if it fits in f64
-    fn q_as_f64(&self) -> Option<f64>;
-    /// Is modulus native?
-    fn is_native(&self) -> bool;
-    /// -1 in signed representaiton
-    fn neg_one(&self) -> Self::Element;
-    /// Largest unsigned value that fits in the modulus. That is, q - 1.
-    fn largest_unsigned_value(&self) -> Self::Element;
-    /// Smallest unsigned value that fits in the modulus
-    /// Always assmed to be 0.
-    fn smallest_unsigned_value(&self) -> Self::Element;
-    /// Convert unsigned value in signed represetation to i64
-    fn map_element_to_i64(&self, v: &Self::Element) -> i64;
-    /// Convert f64 to signed represented in modulus
-    fn map_element_from_f64(&self, v: f64) -> Self::Element;
-    /// Convert i64 to signed represented in modulus
-    fn map_element_from_i64(&self, v: i64) -> Self::Element;
-}
-
-impl Modulus for u64 {
-    type Element = u64;
-    fn is_native(&self) -> bool {
-        // q of size u64 can never be a naitve modulus
-        false
-    }
-    fn largest_unsigned_value(&self) -> Self::Element {
-        self - 1
-    }
-    fn neg_one(&self) -> Self::Element {
-        self - 1
-    }
-    fn smallest_unsigned_value(&self) -> Self::Element {
-        0
-    }
-    fn map_element_to_i64(&self, v: &Self::Element) -> i64 {
-        assert!(v <= self, "{v} must be <= {self}");
-        if *v >= (self >> 1) {
-            -ToPrimitive::to_i64(&(self - v)).unwrap()
-        } else {
-            ToPrimitive::to_i64(v).unwrap()
-        }
-    }
-    fn map_element_from_f64(&self, v: f64) -> Self::Element {
-        //FIXME (Jay): Before I check whether v is smaller than 0 with `let is_neg =
-        // o.is_sign_negative() && o != 0.0; I'm ocnfused why didn't I simply check <
-        // 0.0?
-        let v = v.round();
-        if v < 0.0 {
-            self - v.abs().to_u64().unwrap()
-        } else {
-            v.to_u64().unwrap()
-        }
-    }
-    fn map_element_from_i64(&self, v: i64) -> Self::Element {
-        if v < 0 {
-            self - v.abs().to_u64().unwrap()
-        } else {
-            v.to_u64().unwrap()
-        }
-    }
-    fn q(&self) -> Option<Self::Element> {
-        Some(*self)
-    }
-    fn q_as_f64(&self) -> Option<f64> {
-        self.to_f64()
-    }
-}
-
-pub trait ModInit {
-    type M;
-    fn new(modulus: Self::M) -> Self;
-}
-
-pub trait GetModulus {
-    type Element;
-    type M: Modulus<Element = Self::Element>;
-    fn modulus(&self) -> &Self::M;
-}
-
-pub trait VectorOps {
-    type Element;
-
-    fn elwise_scalar_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &Self::Element);
-    fn elwise_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &[Self::Element]);
-
-    fn elwise_add_mut(&self, a: &mut [Self::Element], b: &[Self::Element]);
-    fn elwise_sub_mut(&self, a: &mut [Self::Element], b: &[Self::Element]);
-    fn elwise_mul_mut(&self, a: &mut [Self::Element], b: &[Self::Element]);
-    fn elwise_scalar_mul_mut(&self, a: &mut [Self::Element], b: &Self::Element);
-    fn elwise_neg_mut(&self, a: &mut [Self::Element]);
-    /// inplace mutates `a`: a = a + b*c
-    fn elwise_fma_mut(&self, a: &mut [Self::Element], b: &[Self::Element], c: &[Self::Element]);
-    fn elwise_fma_scalar_mut(
-        &self,
-        a: &mut [Self::Element],
-        b: &[Self::Element],
-        c: &Self::Element,
-    );
-}
-
-pub trait ArithmeticOps {
-    type Element;
-    fn mul(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
-    fn add(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
-    fn sub(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
-    fn neg(&self, a: &Self::Element) -> Self::Element;
-}
-
-pub trait ArithmeticLazyOps {
-    type Element;
-    fn mul_lazy(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
-    fn add_lazy(&self, a: &Self::Element, b: &Self::Element) -> Self::Element;
-}
-
-pub trait ShoupMatrixFMA<M: Matrix>
-where
-    M::R: RowMut,
-{
-    /// Returns summation of row-wise product of matrix a and b.
-    fn shoup_matrix_fma(&self, out: &mut M::R, a: &M, a_shoup: &M, b: &M);
-}
-
 pub struct ModularOpsU64<T> {
     q: u64,
     q_twice: u64,
@@ -413,126 +290,6 @@ where
     }
 }
 
-pub struct WordSizeModulus<T> {
-    modulus: T,
-}
-
-impl<T> ModInit for WordSizeModulus<T>
-where
-    T: Modulus,
-{
-    type M = T;
-    fn new(modulus: T) -> Self {
-        assert!(modulus.is_native());
-        // For now assume ModulusOpsU64 is only used for u64
-        Self { modulus: modulus }
-    }
-}
-
-impl<T> ArithmeticOps for WordSizeModulus<T>
-where
-    T: Modulus,
-    T::Element: WrappingAdd + WrappingSub + WrappingMul + Zero,
-{
-    type Element = T::Element;
-    fn add(&self, a: &Self::Element, b: &Self::Element) -> Self::Element {
-        T::Element::wrapping_add(a, b)
-    }
-
-    fn mul(&self, a: &Self::Element, b: &Self::Element) -> Self::Element {
-        T::Element::wrapping_mul(a, b)
-    }
-
-    fn neg(&self, a: &Self::Element) -> Self::Element {
-        T::Element::wrapping_sub(&T::Element::zero(), a)
-    }
-
-    fn sub(&self, a: &Self::Element, b: &Self::Element) -> Self::Element {
-        T::Element::wrapping_sub(a, b)
-    }
-}
-
-impl<T> VectorOps for WordSizeModulus<T>
-where
-    T: Modulus,
-    T::Element: WrappingAdd + WrappingSub + WrappingMul + Zero,
-{
-    type Element = T::Element;
-
-    fn elwise_add_mut(&self, a: &mut [Self::Element], b: &[Self::Element]) {
-        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
-            *ai = T::Element::wrapping_add(ai, bi);
-        });
-    }
-
-    fn elwise_sub_mut(&self, a: &mut [Self::Element], b: &[Self::Element]) {
-        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
-            *ai = T::Element::wrapping_sub(ai, bi);
-        });
-    }
-
-    fn elwise_mul_mut(&self, a: &mut [Self::Element], b: &[Self::Element]) {
-        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
-            *ai = T::Element::wrapping_mul(ai, bi);
-        });
-    }
-
-    fn elwise_neg_mut(&self, a: &mut [Self::Element]) {
-        a.iter_mut()
-            .for_each(|ai| *ai = T::Element::wrapping_sub(&T::Element::zero(), ai));
-    }
-
-    fn elwise_scalar_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &Self::Element) {
-        izip!(out.iter_mut(), a.iter()).for_each(|(oi, ai)| {
-            *oi = T::Element::wrapping_mul(ai, b);
-        });
-    }
-
-    fn elwise_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &[Self::Element]) {
-        izip!(out.iter_mut(), a.iter(), b.iter()).for_each(|(oi, ai, bi)| {
-            *oi = T::Element::wrapping_mul(ai, bi);
-        });
-    }
-
-    fn elwise_scalar_mul_mut(&self, a: &mut [Self::Element], b: &Self::Element) {
-        a.iter_mut().for_each(|ai| {
-            *ai = T::Element::wrapping_mul(ai, b);
-        });
-    }
-
-    fn elwise_fma_mut(&self, a: &mut [Self::Element], b: &[Self::Element], c: &[Self::Element]) {
-        izip!(a.iter_mut(), b.iter(), c.iter()).for_each(|(ai, bi, ci)| {
-            *ai = T::Element::wrapping_add(ai, &T::Element::wrapping_mul(bi, ci));
-        });
-    }
-
-    fn elwise_fma_scalar_mut(
-        &self,
-        a: &mut [Self::Element],
-        b: &[Self::Element],
-        c: &Self::Element,
-    ) {
-        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
-            *ai = T::Element::wrapping_add(ai, &T::Element::wrapping_mul(bi, c));
-        });
-    }
-
-    // fn modulus(&self) -> &T {
-    //     &self.modulus
-    // }
-}
-
-impl<T> GetModulus for WordSizeModulus<T>
-where
-    T: Modulus,
-{
-    type Element = T::Element;
-    type M = T;
-    fn modulus(&self) -> &Self::M {
-        &self.modulus
-    }
-}
-
 #[cfg(test)]
 mod tests {
     use super::*;
diff --git a/src/backend/word_size.rs b/src/backend/word_size.rs
new file mode 100644
index 0000000..0c2fbac
--- /dev/null
+++ b/src/backend/word_size.rs
@@ -0,0 +1,127 @@
+use itertools::izip;
+use num_traits::{PrimInt, Signed, ToPrimitive, WrappingAdd, WrappingMul, WrappingSub, Zero};
+
+use super::{
+    ArithmeticLazyOps, ArithmeticOps, GetModulus, ModInit, Modulus, ShoupMatrixFMA, VectorOps,
+};
+use crate::{utils::ShoupMul, Matrix, RowMut};
+
+pub struct WordSizeModulus<T> {
+    modulus: T,
+}
+
+impl<T> ModInit for WordSizeModulus<T>
+where
+    T: Modulus,
+{
+    type M = T;
+    fn new(modulus: T) -> Self {
+        assert!(modulus.is_native());
+        // For now assume ModulusOpsU64 is only used for u64
+        Self { modulus: modulus }
+    }
+}
+
+impl<T> ArithmeticOps for WordSizeModulus<T>
+where
+    T: Modulus,
+    T::Element: WrappingAdd + WrappingSub + WrappingMul + Zero,
+{
+    type Element = T::Element;
+    fn add(&self, a: &Self::Element, b: &Self::Element) -> Self::Element {
+        T::Element::wrapping_add(a, b)
+    }
+
+    fn mul(&self, a: &Self::Element, b: &Self::Element) -> Self::Element {
+        T::Element::wrapping_mul(a, b)
+    }
+
+    fn neg(&self, a: &Self::Element) -> Self::Element {
+        T::Element::wrapping_sub(&T::Element::zero(), a)
+    }
+
+    fn sub(&self, a: &Self::Element, b: &Self::Element) -> Self::Element {
+        T::Element::wrapping_sub(a, b)
+    }
+}
+
+impl<T> VectorOps for WordSizeModulus<T>
+where
+    T: Modulus,
+    T::Element: WrappingAdd + WrappingSub + WrappingMul + Zero,
+{
+    type Element = T::Element;
+
+    fn elwise_add_mut(&self, a: &mut [Self::Element], b: &[Self::Element]) {
+        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
+            *ai = T::Element::wrapping_add(ai, bi);
+        });
+    }
+
+    fn elwise_sub_mut(&self, a: &mut [Self::Element], b: &[Self::Element]) {
+        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
+            *ai = T::Element::wrapping_sub(ai, bi);
+        });
+    }
+
+    fn elwise_mul_mut(&self, a: &mut [Self::Element], b: &[Self::Element]) {
+        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
+            *ai = T::Element::wrapping_mul(ai, bi);
+        });
+    }
+
+    fn elwise_neg_mut(&self, a: &mut [Self::Element]) {
+        a.iter_mut()
+            .for_each(|ai| *ai = T::Element::wrapping_sub(&T::Element::zero(), ai));
+    }
+
+    fn elwise_scalar_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &Self::Element) {
+        izip!(out.iter_mut(), a.iter()).for_each(|(oi, ai)| {
+            *oi = T::Element::wrapping_mul(ai, b);
+        });
+    }
+
+    fn elwise_mul(&self, out: &mut [Self::Element], a: &[Self::Element], b: &[Self::Element]) {
+        izip!(out.iter_mut(), a.iter(), b.iter()).for_each(|(oi, ai, bi)| {
+            *oi = T::Element::wrapping_mul(ai, bi);
+        });
+    }
+
+    fn elwise_scalar_mul_mut(&self, a: &mut [Self::Element], b: &Self::Element) {
+        a.iter_mut().for_each(|ai| {
+            *ai = T::Element::wrapping_mul(ai, b);
+        });
+    }
+
+    fn elwise_fma_mut(&self, a: &mut [Self::Element], b: &[Self::Element], c: &[Self::Element]) {
+        izip!(a.iter_mut(), b.iter(), c.iter()).for_each(|(ai, bi, ci)| {
+            *ai = T::Element::wrapping_add(ai, &T::Element::wrapping_mul(bi, ci));
+        });
+    }
+
+    fn elwise_fma_scalar_mut(
+        &self,
+        a: &mut [Self::Element],
+        b: &[Self::Element],
+        c: &Self::Element,
+    ) {
+        izip!(a.iter_mut(), b.iter()).for_each(|(ai, bi)| {
+            *ai = T::Element::wrapping_add(ai, &T::Element::wrapping_mul(bi, c));
+        });
+    }
+
+    // fn modulus(&self) -> &T {
+    //     &self.modulus
+    // }
+}
+
+impl<T> GetModulus for WordSizeModulus<T>
+where
+    T: Modulus,
+{
+    type Element = T::Element;
+    type M = T;
+    fn modulus(&self) -> &Self::M {
+        &self.modulus
+    }
+}