diff --git a/.gitignore b/.gitignore
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+NTL/
+.DS_Store
diff --git a/Makefile b/Makefile
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+CXX = clang++
+# Added Homebrew include path
+CXXFLAGS = -O3 -std=c++17 -I./NTL/include -I/opt/homebrew/include
+# Added Homebrew library path
+LDFLAGS = -L./NTL/src -L/opt/homebrew/lib -lntl -lgmp
+
+TARGET = phase1
+SRC = phase1.cpp
+
+all: $(TARGET)
+
+$(TARGET): $(SRC)
+	$(CXX) $(CXXFLAGS) $(SRC) $(LDFLAGS) -o $(TARGET)
+
+clean:
+	rm -f $(TARGET)
diff --git a/conv_impl.cpp b/conv_impl.cpp
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diff --git a/phase1 b/phase1
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diff --git a/phase1.cpp b/phase1.cpp
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+++ b/phase1.cpp
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+#include<NTL/ZZ_p.h>
+#include<NTL/tools.h>
+#include<NTL/vec_ZZ.h>
+#include<NTL/vec_ZZ_p.h>
+#include<NTL/vector.h>
+#include<iostream>
+#include<sstream>
+#include<iomanip>
+#include<string>
+
+constexpr const long PRIME_BIT  = 60;
+constexpr const long ERR_BOUND =  80;
+double total_time{};
+
+using namespace std;
+using namespace NTL;
+
+std::vector<std::pair<long,long double>>global;
+/* O(n^2) compute for toeplitz matrix with input vector Benchmarking against Cooley method */
+/* from 1d -> 2d we use V[(n-1) + ( j- i) * xj)] j: 0 -> n */
+
+vec_ZZ_p mul_norm(const vec_ZZ_p &input, const  vec_ZZ_p &tope_mat, long n){
+    vec_ZZ_p res;
+    res.SetLength(n);
+
+    for(long i{};i<n;++i){
+        ZZ_p sum{};
+        for(long j{};j < n;++j){
+            long v  = (n - 1) + (j - i);
+            ZZ_p prod;
+            mul(prod, tope_mat[v], input[j]);
+            add(sum, sum, prod);
+        }
+        res[i] = sum;
+    }
+    return res;
+}
+
+#if 0
+/* optimized mutliplation of tope_mat * input vector. O(n log n). Would be benchmark against mul_norm */
+vec_ZZ_p cooley_mul(const vec_ZZ_p &input, const vec_ZZ_p &tope_mat, long n ){
+    vec_zz_p res;
+
+    for(long i{};i<n;++i){
+
+    }
+    return res;
+}
+
+#endif
+
+/* Function the benchmark the time and the performance */
+/* generic so we can pass any type of Multiplication Function */
+
+template<typename Func>
+void routine_check(const vec_ZZ_p &toep_mat, const vec_ZZ_p &input, long n, Func mul_function){
+
+    long double curr_time = GetTime();
+    vec_ZZ_p mat  = mul_function(input,toep_mat, n);
+    long double final_time = GetTime() - curr_time;
+    global.push_back({n,final_time});
+    std::cout << "[ time:  " << final_time << "]" << std::endl;
+}
+
+
+void print_matrix(const vec_ZZ_p &toep_mat, long n) {
+    const int width = 30;
+
+    auto print_separator = [&]() {
+        std::cout << "+";
+        for(long i = 0; i < n; ++i)
+            std::cout << std::string(width, '-') << "+";
+        std::cout << "\n";
+    };
+
+    std::cout << "\n--- Toeplitz Matrix Verification (n=" << n << ") ---\n";
+
+    print_separator();
+
+    for(long i = 0; i < n; ++i) {
+        std::cout << "|";
+
+        for(long j = 0; j < n; ++j) {
+            long index = (n-1) + (j-i);
+
+            std::ostringstream oss;
+            oss << toep_mat[index];
+            std::string s = oss.str();
+
+            std::cout << std::setw(width) << s << "|";
+        }
+
+        std::cout << "\n";
+        print_separator();
+    }
+}
+
+int main(){
+
+  ZZ p;
+  GenPrime(p ,PRIME_BIT ,ERR_BOUND);
+  ZZ_p::init(p);
+  long t = 8;
+  SetSeed(to_ZZ(time(nullptr)));
+
+  long start,step,end;
+
+  std::cout << "Please Enter the Start value: " << std::endl;
+  std::cin >> start;
+  std::cout << "Please Enter the Step value: " <<  std::endl;
+  std::cin >> step;
+  std::cout << "Please Enter the End value: "  <<  std::endl;
+  std::cin >> end;
+
+
+  while(start != end){
+
+       vec_ZZ_p input;
+       input.SetLength(start);
+
+       for(long i{};i< start; ++i){
+         random(input[i]);
+       }
+       vec_ZZ_p toep_mat;
+
+       toep_mat.SetLength(2*start-1);
+
+       for(long i{};i<2*start-1;++i){
+         random(toep_mat[i]);
+       }
+       print_matrix(toep_mat, start);
+       routine_check(toep_mat,input, start, mul_norm);
+       start += step;
+  }
+
+  std::cout << "\n";
+
+  std::cout << "FINAL SUMMARY ---------------------------------- " << std::endl;
+  for(auto&[n,final_time]:global){
+      total_time += final_time;
+      std::cout << "For n: " << n <<  "      " << "   Time taken: " << final_time << std::endl;
+  }
+  std::cout << "\n";
+  std::cout << "Total time taken: " << " "  <<  total_time << std::endl;
+
+  return 0;
+}