Question

Project 2 (RSA Signature Scheme): Recall that in the RSA cryptosystem, Bob creates a public key (e, n), so that other parties can encrypt a message M as C = M e mod n. In the RSA signature scheme, Bob instead encrypts a message M using his secret key d as follows: S = M d mod n Any third party can verify this signature by testing the following condition: Is it true that M = S e mod n The verification method follows from the fact that de mod (n) = 1. Indeed, we have: S e mod n = M de mod n = M de mod (n) mod n = M1 mod n = M In addition, the verification of the RSA signature scheme involves the same algorithm as RSA encryption and uses the same public key (e, n) for Bob. Provide an implementation of the RSA signature scheme. More precisely, write a program that implements the RSA digital signatures. The implementation of the RSA digital signatures should be as structured as possible.

Answer 1

Check the explanation

Step-by-step explanation:

Below is the Java code with output screenshots

Code :

import java.io.DataInputStream;

import java.io.IOException;

import java.math.BigInteger;

import java.util.Random;

public class RSAImplement

{

private BigInteger p; // p and q are two prime numbers

private BigInteger q;

private BigInteger N; // n = p*q

private BigInteger phi; // phi = (p-1)*(q-1)

private BigInteger e; // e is a relative prime number where 1<e<phi

private BigInteger d; //private key - (d*e mod phi = 1)

private int bitlength = 1024;

private Random r;

// public key = (n,e)

public RSAImplement()

{

r = new Random();

p = BigInteger. probablePrime(bitlength, r);

q = BigInteger. probablePrime(bitlength, r);

N = p.multiply(q);

phi = p.subtract(BigInteger. ONE).multiply(q.subtract(BigInteger. ONE));

e = BigInteger. probablePrime(bitlength / 2, r);

while (phi.gcd(e). compareTo(BigInteger. ONE) > 0 && e.compareTo(phi) < 0)

{

e. add(BigInteger. ONE);

}

d = e. modInverse(phi);

}

public RSAImplement(BigInteger e, BigInteger d, BigInteger N)

{

this. e = e;

this. d = d;

this. N = N;

}

public static void main(String[] args) throws IOException

{

RSAImplement rsaImplement = new RSAImplement();

DataInputStream in = new DataInputStream(System.in);

String teststring;

System.out.println("Enter the message you want to send :");

teststring = in. readLine();

System.out.println("Encrypting Message: " + teststring);

System.out.println("Message in Bytes: "

+ bytesToString(teststring.getBytes()));

// encryptMessage

byte[] encrypted = rsaImplement. encryptMessage(teststring. getBytes());

// decryptMessage

byte[] decrypted = rsaImplement. decryptMessage(encrypted);

System.out.println("Decrypting Bytes: " + bytesToString(decrypted));

System.out.println("Decrypted Message: " + new String(decrypted));

}

private static String bytesToString(byte[] encrypted)

{

String test = "";

for (byte b : encrypted)

{

test += Byte. toString(b);

}

return test;

}

// method which Encrypt message

public byte[] encryptMessage(byte[] message)

{

return (new BigInteger(message)).modPow(e, N). toByteArray();

}

// method which Decrypt message

public byte[] decryptMessage(byte[] message)

{

return (new BigInteger(message)).modPow(d, N). toByteArray();

}

}

Kindly check the attached image below for the code output.