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Implementing Coppersmith Freivald’s Algorithm in Java
  • Difficulty Level : Expert
  • Last Updated : 27 Dec, 2020

Concept: Coppersmith Freivald’s Algorithm is to check whether the matrix A multiplied by matrix B equals the given matrix C. It is used to verify matrix multiplication. It is verified with the help of an equation which stands A*(B*r)-(C*r)=0, where r is a random column vector consisting of 0/1 only.



Enter the dimensions of the matrices:  


Enter the 1st matrix:  

-2 1

0  4

Enter the 2st matrix:  

6  5

-7  1

Enter the result matrix:  

-19  9

-28  4

Output: Yes, The matrix multiplication is correct.


Take the size of the matrix as input from the user.

Goal: According to the equation we need to verify matrix A * matrix B = matrix C.

Take inputs of matrix A(n*n) matrix B(n*n) and the resultant matrix C(n*n) as input.

1) Take array r[n][1] randomly which consists of elements of 0/1 only.

2) Compute matrix B*r, matrix C*r and then matrix A*(matrix B*r) for evaluating the expression matrix A*(matrix B * r) – (matrix C*r)

3) Check if the equation matrix A*(matrix B * r) – (matrix C*r)=0 or not.

4)If it is zero then print “Yes” else print “No”.

Implementation: Input should be taken in the order shown above, else it will lead to wrong results. Below is the example for consideration


// Importing class to create objects
// generating pseudo random numbers
import java.util.Random;
// Inporting class to take input from user
import java.util.Scanner;
public class GFG {
    public static double[][] multiplyVector(double[][] a,
                                            double[][] b,
                                            int n)
    // Method to check the result of the equation.
        double result[][] = new double[n][1];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < 1; j++) {
                for (int k = 0; k < n; k++) {
                        = result[i][j] + a[i][k] * b[i][j];
        return result;
    public static void main(String args[])
        // Driver main method
        Scanner input = new Scanner(;
            " Enter the dimensions of the matrix");
        int n = input.nextInt();
        // n- size or dimensions of matrix
        System.out.println("Enter the 1st matrix:");
        // Taking input for 1st matrix
        double a[][] = new double[n][n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                a[i][j] = input.nextDouble();
        System.out.println("Enter the 2nd matrix");
        double b[][] = new double[n][n];
        // Taking input for second matrix
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                b[i][j] = input.nextDouble();
        // Covering up Resultant matrix
        System.out.println("Enter the resultant matrix");
        double c[][] = new double[n][n];
        // the resultant matrix
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i][j] = input.nextDouble();
        // generating random matrix r consisting of 0/1 only
        double[][] r = new double[n][1];
        Random random = new Random();
        for (int i = 0; i < n; i++) {
            r[i][0] = random.nextInt(2);
        // testing of the standard equation A*(B*r)-(C*r)=0
        double br[][] = new double[n][1];
        double cr[][] = new double[n][1];
        double abr[][] = new double[n][1];
        br = multiplyVector(b, r, n);
        cr = multiplyVector(c, r, n);
        abr = multiplyVector(a, br, n);
        // check for all zeroes in abr
        boolean flag = true;
        // Setting flag wth true
        for (int i = 0; i < n; i++) {
            if (abr[i][0] == 0)
                // Set flag to false(change flag)
                flag = false;
        // Boolean comparison resulting in message printing
        if (flag == true)
                "Yes,The matrix multiplication is correct");
                "No,The matrix multiplication is wrong");


Output: Custom input for 2 random matrices of order 2


Time Complexity : O(kN^2) where N is the size of the matrix.


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