diff --git a/src/agcd.rs b/src/agcd.rs
index 6a67977..59dde41 100644
--- a/src/agcd.rs
+++ b/src/agcd.rs
@@ -1,5 +1,6 @@
 use crate::bkz::bkz_reduce;
 use crate::deep_lll::deep_lll;
+// use crate::lll::lattice_reduce;
 use crate::matrix::Matrix;
 use crate::utils::abs;
 use lll_rs::l2::bigl2;
@@ -26,6 +27,10 @@ pub fn agcd(ciphertexts: Vec<Integer>, noise_bits: usize, algorithm: u8) -> Inte
             let reduced = deep_lll(basis_matrix.clone(), Rational::from((51, 100))).unwrap();
             reduced.columns[0][0].clone()
         }
+        // 3u8 => {
+        //     lattice_reduce(&mut basis_matrix, 0.51, 0.75);
+        //     lll_matrix[0][0].clone()
+        // }
         _ => panic!("Unknown algorithm value: {}", algorithm),
     };
 
diff --git a/src/deep_lll.rs b/src/deep_lll.rs
index e85a14b..da8eb29 100644
--- a/src/deep_lll.rs
+++ b/src/deep_lll.rs
@@ -1,9 +1,9 @@
 use crate::matrix::Matrix;
-use rug::Rational;
+use rug::{Rational, Integer};
 
 /// Perform DeepLLL reduction on a given lattice basis represented by Matrix.
 /// 1/4 < delta < 1.
-pub fn deep_lll(mut mat: Matrix, delta: Rational) -> Option<Matrix> {
+pub fn deep_lll(mut mat: Matrix<Integer>, delta: Rational) -> Option<Matrix<Integer>> {
     let n = mat.n;
     let (mut mu, mut b_star_sq) = gramm_schmidt(&mat);
     let mut k = 2;
@@ -12,7 +12,10 @@ pub fn deep_lll(mut mat: Matrix, delta: Rational) -> Option<Matrix> {
 
     while k <= n {
         if iterations >= MAX_ITERATIONS {
-            eprintln!("Warning: DeepLLL did not converge after {} iterations", MAX_ITERATIONS);
+            eprintln!(
+                "Warning: DeepLLL did not converge after {} iterations",
+                MAX_ITERATIONS
+            );
             return Some(mat);
         }
         iterations += 1;
@@ -40,7 +43,7 @@ pub fn deep_lll(mut mat: Matrix, delta: Rational) -> Option<Matrix> {
 }
 
 /// Compute Gram–Schmidt coefficients and squared norms of orthogonal vectors b*_i as Rationals.
-fn gramm_schmidt(mat: &Matrix) -> (Vec<Vec<Rational>>, Vec<Rational>) {
+fn gramm_schmidt(mat: &Matrix<Integer>) -> (Vec<Vec<Rational>>, Vec<Rational>) {
     let n = mat.n;
     let m = mat.m;
     let mut mu = vec![vec![Rational::from((0, 1)); n]; n];
@@ -75,7 +78,7 @@ fn gramm_schmidt(mat: &Matrix) -> (Vec<Vec<Rational>>, Vec<Rational>) {
 }
 
 /// Size-reduce column k in-place
-fn size_reduce(mat: &mut Matrix, mu: &mut [Vec<Rational>], b_star_sq: &mut [Rational], k: usize) {
+fn size_reduce(mat: &mut Matrix<Integer>, mu: &mut [Vec<Rational>], b_star_sq: &mut [Rational], k: usize) {
     let mut updated = true;
     while updated {
         updated = false;
@@ -100,13 +103,13 @@ fn size_reduce(mat: &mut Matrix, mu: &mut [Vec<Rational>], b_star_sq: &mut [Rati
 }
 
 /// Deep insertion: move column k into position i (1-based), shifting intermediate columns right.
-fn deep_insert(mat: &mut Matrix, i: usize, k: usize) {
+fn deep_insert(mat: &mut Matrix<Integer>, i: usize, k: usize) {
     let col = mat.columns.remove(k - 1);
     mat.columns.insert(i - 1, col);
 }
 
 /// Compute squared Euclidean norm of column k as a Rational.
-fn norm_sq(mat: &Matrix, k: usize) -> Rational {
+fn norm_sq(mat: &Matrix<Integer>, k: usize) -> Rational {
     let mut sum = Rational::from((0, 1));
     for row in 0..mat.n {
         let v = mat[(k - 1, row)].clone();
diff --git a/src/lll.rs b/src/lll.rs
index 3ee5513..a77dade 100644
--- a/src/lll.rs
+++ b/src/lll.rs
@@ -1,8 +1,10 @@
-use lll_rs::{matrix::Matrix as LLLMatrix, vector::BigVector};
-
 use crate::matrix::Matrix;
+use lll_rs::{matrix::Matrix as LLLMatrix, vector::BigVector};
+use rug::{Integer, Rational};
+use std::cmp::max;
+use std::ops::Sub;
 
-impl Matrix {
+impl Matrix<Integer> {
     pub fn to_lll_matrix(&self) -> LLLMatrix<BigVector> {
         let n = self.n;
         let mut lll_mat = LLLMatrix::init(n, n);
diff --git a/src/matrix.rs b/src/matrix.rs
index f1183c2..67e14a3 100644
--- a/src/matrix.rs
+++ b/src/matrix.rs
@@ -1,27 +1,27 @@
-use crate::vector::IntVector;
-use rug::ops::Pow;
-use rug::Integer;
-use std::ops::{Index, IndexMut};
-
-macro_rules! int {
-    ($x:expr) => {
-        rug::Integer::from($x)
-    };
-}
+use crate::vector::Vector;
+use rug::{ops::Pow, Integer};
+use std::{
+    fmt,
+    ops::{Index, IndexMut},
+};
 
 #[derive(Debug, PartialEq, Clone)]
-pub struct Matrix {
+pub struct Matrix<T> {
     pub n: usize, // number of columns
     pub m: usize, // number of rows
-    pub columns: Vec<IntVector>,
+    pub columns: Vec<Vector<T>>,
 }
 
-impl Matrix {
-    pub fn new(n: usize, m: usize, values: Vec<Integer>) -> Option<Self> {
+impl<T> Matrix<T> {
+    pub fn new(n: usize, m: usize, values: Vec<T>) -> Option<Self> {
         if n * m != values.len() {
             return None;
         }
-        let mut columns = vec![Vec::with_capacity(m); n];
+        // avoid requiring Vec<T>: Clone by building manually
+        let mut columns = Vec::with_capacity(n);
+        for _ in 0..n {
+            columns.push(Vec::with_capacity(m));
+        }
         for (i, value) in values.into_iter().enumerate() {
             let col = i % n;
             columns[col].push(value);
@@ -29,39 +29,47 @@ impl Matrix {
         Some(Matrix {
             n,
             m,
-            columns: columns.into_iter().map(IntVector::from_vec).collect(),
+            columns: columns.into_iter().map(Vector::from_vec).collect(),
         })
     }
+}
 
-    pub fn new_lattice(noise_bits: usize, ciphertexts: Vec<Integer>) -> Option<Self> {
+impl<T> Matrix<T>
+where
+    T: Clone + Default + From<Integer>,
+{
+    pub fn new_lattice(noise_bits: usize, ciphertexts: Vec<T>) -> Option<Self> {
         let n = ciphertexts.len();
         let mut columns = vec![Vec::with_capacity(n); n];
 
-        columns[0].push(int!(2u64).pow((noise_bits + 1) as u32));
+        // First row: [2^(noise+1), ciphertexts[1], ciphertexts[2], ...]
+        let two_pow = Integer::from(2u64).pow((noise_bits + 1) as u32);
+        columns[0].push(T::from(two_pow));
         for i in 1..n {
             columns[i].push(ciphertexts[i].clone());
         }
 
+        // Subsequent rows form identity matrix with ciphertexts[0]
         for i in 1..n {
             for (j, column) in columns.iter_mut().enumerate().take(n) {
-                if i == j {
-                    column.push(ciphertexts[0].clone());
+                column.push(if i == j {
+                    ciphertexts[0].clone()
                 } else {
-                    column.push(int!(0));
-                }
+                    T::default()
+                });
             }
         }
 
         Some(Matrix {
             n,
             m: n,
-            columns: columns.into_iter().map(IntVector::from_vec).collect(),
+            columns: columns.into_iter().map(Vector::from_vec).collect(),
         })
     }
 }
 
-impl Index<(usize, usize)> for Matrix {
-    type Output = Integer;
+impl<T> Index<(usize, usize)> for Matrix<T> {
+    type Output = T;
     fn index(&self, index: (usize, usize)) -> &Self::Output {
         let (col, row) = index;
         if row >= self.n || col >= self.m {
@@ -71,7 +79,7 @@ impl Index<(usize, usize)> for Matrix {
     }
 }
 
-impl IndexMut<(usize, usize)> for Matrix {
+impl<T> IndexMut<(usize, usize)> for Matrix<T> {
     fn index_mut(&mut self, index: (usize, usize)) -> &mut Self::Output {
         let (col, row) = index;
         if row >= self.n || col >= self.m {
@@ -84,16 +92,23 @@ impl IndexMut<(usize, usize)> for Matrix {
 #[cfg(test)]
 mod tests {
     use super::*;
-    use std::panic;
+    use rug::Rational;
+
+    macro_rules! int {
+        ($x:expr) => {
+            Integer::from($x)
+        };
+    }
+
+    macro_rules! rational {
+        ($x:expr) => {
+            Rational::from(Integer::from($x))
+        };
+    }
 
     #[test]
-    fn simple_dimensions_and_index() {
+    fn test_integer_matrix() {
         let m = Matrix::new(2, 2, vec![int!(1), int!(2), int!(3), int!(4)]).unwrap();
-        assert_eq!(m.n, 2);
-        assert_eq!(m.m, 2);
-
-        // values: [1,2,3,4]
-        // columns: [[1,3], [2,4]]
         assert_eq!(m[(0, 0)], int!(1));
         assert_eq!(m[(0, 1)], int!(2));
         assert_eq!(m[(1, 0)], int!(3));
@@ -101,47 +116,26 @@ mod tests {
     }
 
     #[test]
-    fn indexes_and_mutation() {
-        let mut m = Matrix::new(2, 2, vec![int!(1), int!(2), int!(3), int!(4)]).unwrap();
-        assert_eq!(m[(1, 0)], int!(3));
-
-        m[(1, 0)] = int!(5);
-        assert_eq!(m[(1, 0)], int!(5));
-
-        let m2 = Matrix::new(
-            3,
+    fn test_rational_matrix() {
+        let m = Matrix::new(
             2,
-            vec![int!(1), int!(2), int!(3), int!(4), int!(5), int!(6)],
+            2,
+            vec![rational!(1), rational!(2), rational!(3), rational!(4)],
         )
         .unwrap();
-
-        assert_eq!(m2[(0, 2)], int!(3));
-        assert_eq!(m2[(1, 0)], int!(4));
-
-        let result = panic::catch_unwind(|| {
-            let _ = m2[(2, 0)];
-        });
-        assert!(result.is_err(), "Expected panic on m2[(2, 0)]");
-
-        let result2 = panic::catch_unwind(|| {
-            let _ = m2[(0, 3)];
-        });
-        assert!(result2.is_err(), "Expected panic on m2[(0, 3)]");
+        assert_eq!(m[(0, 0)], rational!(1));
+        assert_eq!(m[(0, 1)], rational!(2));
+        assert_eq!(m[(1, 0)], rational!(3));
+        assert_eq!(m[(1, 1)], rational!(4));
     }
 
     #[test]
-    fn test_new_lattice_layout() {
+    fn test_lattice_matrix() {
         let ciphertexts = vec![int!(5), int!(8), int!(12)];
         let noise_bits = 2;
-        let lattice = Matrix::new_lattice(noise_bits, ciphertexts.clone()).unwrap();
+        let lattice = Matrix::new_lattice(noise_bits, ciphertexts).unwrap();
 
-        assert_eq!(lattice.n, 3);
-        assert_eq!(lattice.m, 3);
-
-        // 1st column = [2^(noise+1), 0, 0] = [8,0,0]
-        // 2nd column = [ciphertexts[1], ciphertexts[0], 0] = [8,5,0]
-        // 3rd column = [ciphertexts[2], 0, ciphertexts[0]] = [12,0,5]
-        let expected_flat = vec![
+        let expected = vec![
             int!(8),
             int!(8),
             int!(12),
@@ -152,14 +146,24 @@ mod tests {
             int!(0),
             int!(5),
         ];
-
-        let mut actual_flat = Vec::with_capacity(9);
-        for col in 0..lattice.m {
-            for row in 0..lattice.n {
-                actual_flat.push(lattice[(col, row)].clone());
+        let mut actual = Vec::new();
+        for col in 0..3 {
+            for row in 0..3 {
+                actual.push(lattice[(col, row)].clone());
             }
         }
+        assert_eq!(actual, expected);
+    }
 
-        assert_eq!(actual_flat, expected_flat);
+    #[test]
+    fn test_rational_lattice() {
+        let ciphertexts = vec![rational!(5), rational!(8), rational!(12)];
+        let noise_bits = 2;
+        let lattice = Matrix::new_lattice(noise_bits, ciphertexts).unwrap();
+
+        let two_pow = Rational::from(Integer::from(2u64).pow(3));
+        assert_eq!(lattice[(0, 0)], two_pow);
+        assert_eq!(lattice[(0, 1)], rational!(8));
+        assert_eq!(lattice[(0, 2)], rational!(12));
     }
 }
diff --git a/src/vector.rs b/src/vector.rs
index 503a8b2..c1b6ce7 100644
--- a/src/vector.rs
+++ b/src/vector.rs
@@ -1,42 +1,53 @@
-use rug::Integer;
 use std::{
     fmt,
+    iter::Sum,
     ops::{Add, Index, IndexMut, Mul, Sub},
 };
 
-macro_rules! int {
-    ($x:expr) => {
-        rug::Integer::from($x)
-    };
-}
-
 #[derive(Clone, PartialEq)]
-pub struct IntVector {
-    elements: Vec<Integer>,
+pub struct Vector<T> {
+    pub elements: Vec<T>,
 }
 
-impl IntVector {
+impl<T: Default> Vector<T> {
     pub fn init(size: usize) -> Self {
-        Self {
-            elements: vec![Default::default(); size],
+        let mut elements = Vec::with_capacity(size);
+        for _ in 0..size {
+            elements.push(T::default());
         }
+        Self { elements }
     }
+}
 
-    pub fn from_vec(elements: Vec<Integer>) -> Self {
+impl<T> Vector<T> {
+    pub fn from_vec(elements: Vec<T>) -> Self {
         Self { elements }
     }
 
     pub fn size(&self) -> usize {
         self.elements.len()
     }
+}
 
-    pub fn mul_scalar(&self, other: &Integer) -> Self {
-        let n = self.size();
-        Self::from_vec((0..n).map(|i| int!(&self.elements[i] * other)).collect())
+impl<T> Vector<T>
+where
+    T: Mul<Output = T> + Clone,
+{
+    pub fn mul_scalar(&self, scalar: &T) -> Self {
+        let elements = self
+            .elements
+            .iter()
+            .cloned()
+            .map(|e| e * scalar.clone())
+            .collect();
+        Self { elements }
     }
 }
 
-impl Add for IntVector {
+impl<T> Add for Vector<T>
+where
+    T: Add<Output = T>,
+{
     type Output = Self;
     fn add(self, other: Self) -> Self::Output {
         assert_eq!(self.size(), other.size());
@@ -46,11 +57,14 @@ impl Add for IntVector {
             .zip(other.elements)
             .map(|(a, b)| a + b)
             .collect();
-        IntVector::from_vec(elements)
+        Vector::from_vec(elements)
     }
 }
 
-impl Sub for IntVector {
+impl<T> Sub for Vector<T>
+where
+    T: Sub<Output = T>,
+{
     type Output = Self;
     fn sub(self, other: Self) -> Self::Output {
         assert_eq!(self.size(), other.size());
@@ -60,36 +74,40 @@ impl Sub for IntVector {
             .zip(other.elements)
             .map(|(a, b)| a - b)
             .collect();
-        IntVector::from_vec(elements)
+        Vector::from_vec(elements)
     }
 }
 
-impl Mul for IntVector {
-    type Output = Integer;
-    fn mul(self, other: Self) -> Self::Output {
-        let n = self.size();
-        assert_eq!(n, other.size());
-        (0..n)
-            .map(|i| Integer::from(&self.elements[i] * &other.elements[i]))
+impl<T> Mul for Vector<T>
+where
+    T: Mul<Output = T> + Sum,
+{
+    type Output = T;
+    fn mul(self, other: Self) -> T {
+        assert_eq!(self.size(), other.size());
+        self.elements
+            .into_iter()
+            .zip(other.elements)
+            .map(|(a, b)| a * b)
             .sum()
     }
 }
 
-impl Index<usize> for IntVector {
-    type Output = Integer;
+impl<T> Index<usize> for Vector<T> {
+    type Output = T;
 
-    fn index(&self, index: usize) -> &Integer {
+    fn index(&self, index: usize) -> &T {
         &self.elements[index]
     }
 }
 
-impl IndexMut<usize> for IntVector {
-    fn index_mut(&mut self, index: usize) -> &mut Integer {
+impl<T> IndexMut<usize> for Vector<T> {
+    fn index_mut(&mut self, index: usize) -> &mut T {
         &mut self.elements[index]
     }
 }
 
-impl fmt::Debug for IntVector {
+impl<T: fmt::Debug> fmt::Debug for Vector<T> {
     fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
         write!(f, "{:?}", self.elements)
     }
@@ -98,10 +116,23 @@ impl fmt::Debug for IntVector {
 #[cfg(test)]
 mod tests {
     use super::*;
+    use rug::{Integer, Rational};
+
+    macro_rules! int {
+        ($x:expr) => {
+            Integer::from($x)
+        };
+    }
+
+    macro_rules! rational {
+        ($x:expr) => {
+            Rational::from(Integer::from($x))
+        };
+    }
 
     #[test]
-    fn test_from_vec() {
-        let v = IntVector::from_vec(vec![int!(1), int!(2), int!(3)]);
+    fn test_integer_vector_from_vec() {
+        let v = Vector::from_vec(vec![int!(1), int!(2), int!(3)]);
         assert_eq!(v.size(), 3);
         assert_eq!(v[0], int!(1));
         assert_eq!(v[1], int!(2));
@@ -109,9 +140,18 @@ mod tests {
     }
 
     #[test]
-    fn test_add_vectors() {
-        let v1 = IntVector::from_vec(vec![int!(1), int!(2), int!(3)]);
-        let v2 = IntVector::from_vec(vec![int!(4), int!(5), int!(6)]);
+    fn test_rational_vector_from_vec() {
+        let v = Vector::from_vec(vec![rational!(1), rational!(2), rational!(3)]);
+        assert_eq!(v.size(), 3);
+        assert_eq!(v[0], rational!(1));
+        assert_eq!(v[1], rational!(2));
+        assert_eq!(v[2], rational!(3));
+    }
+
+    #[test]
+    fn test_add_integer_vectors() {
+        let v1 = Vector::from_vec(vec![int!(1), int!(2), int!(3)]);
+        let v2 = Vector::from_vec(vec![int!(4), int!(5), int!(6)]);
         let result = v1 + v2;
         assert_eq!(result[0], int!(5));
         assert_eq!(result[1], int!(7));
@@ -119,9 +159,9 @@ mod tests {
     }
 
     #[test]
-    fn test_sub_vectors() {
-        let v1 = IntVector::from_vec(vec![int!(5), int!(7), int!(9)]);
-        let v2 = IntVector::from_vec(vec![int!(4), int!(5), int!(6)]);
+    fn test_sub_integer_vectors() {
+        let v1 = Vector::from_vec(vec![int!(5), int!(7), int!(9)]);
+        let v2 = Vector::from_vec(vec![int!(4), int!(5), int!(6)]);
         let result = v1 - v2;
         assert_eq!(result[0], int!(1));
         assert_eq!(result[1], int!(2));
@@ -129,8 +169,8 @@ mod tests {
     }
 
     #[test]
-    fn test_scalar_multiplication() {
-        let v = IntVector::from_vec(vec![int!(2), int!(3), int!(4)]);
+    fn test_scalar_multiplication_integer() {
+        let v = Vector::from_vec(vec![int!(2), int!(3), int!(4)]);
         let scalar = int!(5);
         let result = v.mul_scalar(&scalar);
         assert_eq!(result[0], int!(10));
@@ -139,17 +179,37 @@ mod tests {
     }
 
     #[test]
-    fn test_dot_product() {
-        let v1 = IntVector::from_vec(vec![int!(1), int!(2), int!(3)]);
-        let v2 = IntVector::from_vec(vec![int!(4), int!(5), int!(6)]);
+    fn test_dot_product_integer() {
+        let v1 = Vector::from_vec(vec![int!(1), int!(2), int!(3)]);
+        let v2 = Vector::from_vec(vec![int!(4), int!(5), int!(6)]);
         let dot = v1 * v2;
-        assert_eq!(dot, int!(32)); // 1*4 + 2*5 + 3*6 = 32
+        assert_eq!(dot, int!(32));
     }
 
     #[test]
-    fn test_indexing_mut() {
-        let mut v = IntVector::from_vec(vec![int!(1), int!(2), int!(3)]);
+    fn test_indexing_mut_integer() {
+        let mut v = Vector::from_vec(vec![int!(1), int!(2), int!(3)]);
         v[1] += int!(10);
         assert_eq!(v[1], int!(12));
     }
+
+    #[test]
+    fn test_add_rational_vectors() {
+        let v1 = Vector::from_vec(vec![rational!(1), rational!(2), rational!(3)]);
+        let v2 = Vector::from_vec(vec![rational!(4), rational!(5), rational!(6)]);
+        let result = v1 + v2;
+        assert_eq!(result[0], rational!(5));
+        assert_eq!(result[1], rational!(7));
+        assert_eq!(result[2], rational!(9));
+    }
+
+    #[test]
+    fn test_scalar_multiplication_rational() {
+        let v = Vector::from_vec(vec![rational!(2), rational!(3), rational!(4)]);
+        let scalar = rational!(5);
+        let result = v.mul_scalar(&scalar);
+        assert_eq!(result[0], rational!(10));
+        assert_eq!(result[1], rational!(15));
+        assert_eq!(result[2], rational!(20));
+    }
 }