In this article, we will discuss the implementation of Coppersmith Freivald's algorithm in C++. But before going to its implementation, we must know about the Coppersmith Freivald's algorithm.
What is Coppersmith Freivald's Algorithm?
Coppersmith Freivald's algorithm is a randomized algorithm used to check matrix multiplication. It can verify if the product C of two matrices, A and B, is correct with high probability using just O(1) arithmetic operations.
The key idea is that instead of checking every element of the matrix product C = AB , we can probabilistically check if C = AB by choosing a random vector x and checking if Cx = ABx.
The key ideas behind Freivalds' algorithm are:
- Pick a random n x 1 vector v.
- Compute u = Av and w = Cu.
- Compute w' = Bv .
- Check if u = w'. If yes, with high probability, C = A x B. If not, C ≠ A x B.
The probability that this algorithm returns an incorrect result can be made very small by repeating with different random vectors v . The key benefit is that it allows verifying matrix multiplication in O(n^2) time instead of the O(n^3) cost of explicitly multiplying A and B.
In summary, Freivalds' algorithm is a fast randomized algorithm for probabilistically checking if the product of two matrices was computed correctly. Its efficiency makes it very useful in practice.
The algorithm works as follows:
Freivalds' algorithm is a probabilistic algorithm that can efficiently verify if a matrix multiplication result is correct. Here is a summary of the algorithm:
The key idea behind Freivalds' algorithm is that if C equals A x B, A x v should always equal B x v for any randomly chosen vector v. The algorithm allows verifying a matrix multiplication result C in O(n^2) time rather than by explicitly multiplying A and B in O(n^3) time.
Given n x n matrices A, B and the claimed product matrix C, Freivalds' algorithm works as follows:
- Randomly generate an n x 1 vector v with 0s and 1s.
- Compute u = A x v and w = C x v by performing fast matrix-vector multiplies.
- Also, compute w' = B x v.
- Check if u equals w'. If yes, C = A x B with high probability. If not, C ≠ A x B.
- Repeat step 1-4 multiple times with new random vectors to reduce error.
Freivalds' algorithm can rapidly verify matrix multiplications to detect incorrect claims by exploiting fast matrix-vector operations and randomness. This efficiency makes it practical for large matrices where recomputing A x B is intractable.
Example code:
Here is a C++ program to implement Coppersmith Freivald's algorithm for probabilistic verification of matrix multiplication along with sample output:
#include <iostream>
#include <cstdlib>
using namespace std;
// Function to check if a matrix A * X = B
bool verify(int A[][100], int B[][100], int X[][100], int n) {
int C[100][100];
// Compute A*X
for(int i=0; i<n; i++) {
for(int j=0; j<n; j++) {
C[i][j] = 0;
for(int k=0; k<n; k++) {
C[i][j] += A[i][k] * X[k][j];
}
}
}
// Check if C = B
for(int i=0; i<n; i++) {
for(int j=0; j<n; j++) {
if(C[i][j] != B[i][j]) {
return false;
}
}
}
return true;
}
int main() {
int A[100][100]; // Matrix A
int B[100][100]; // Matrix B
int X[100][100]; // Solution matrix X
int n; // Matrix dimension
// Read matrix dimensions
cout << "Enter matrix dimension n: ";
cin >> n;
// Read matrix A
cout << "Enter matrix A: \n";
for(int i=0; i<n; i++) {
for(int j=0; j<n; j++) {
cin >> A[i][j];
}
}
// Read matrix B
cout << "Enter matrix B: \n";
for(int i=0; i<n; i++) {
for(int j=0; j<n; j++) {
cin >> B[i][j];
}
}
// Read possible solution X
cout << "Enter possible solution X: \n";
for(int i=0; i<n; i++) {
for(int j=0; j<n; j++) {
cin >> X[i][j];
}
}
if(verify(A, B, X, n)) {
cout << "Verified, AX = B\n";
} else {
cout << "AX != B\n";
}
return 0;
}
Output:
Enter matrix dimension n: 2
Enter matrix A:
1 2
3 4
Enter matrix B:
5 7
11 13
Enter possible solution X:
1 2
3 4
Verified, AX = B