Spring 2008, CSC395 Special Topics: Information Security Project 3

Transcription

1 Spring 2008, CSC395 Special Topics: Information Security Project 3 Due: March 31, 2008, 23:59. How: 1 copy in your digital drop box, and 1 copy in your Unix directory (\CSC395\Project-3). No excuse if you do not meet this requirement. Objectives: Familiar with theory of public key cryptography (RSA) and its implementations. The RSA Cipher: Invented by Rivest, Shamir, and Adleman. A public key cryptography. RSA works like this: 1. The receiver of a message generates two large strong prime, p and q, forms their product, say n = pq, and makes public the value of n. 2. The receiver then chooses an integer e < n, such that gcd (e, (p-1)(q-1)) = 1. The value of e is made public. 3. The receiver also computes a decryption key, d, which is an inverse of e modulo (p-1)(q-1). This inverse exists since e was chosen relatively prime to (p-1)(q-1). That is, d must satisfy the congruence ed 1 (mod(p-1)(q-1)): 1 is the remainder of (ed) divided by (p-1)(q-1). 4. The sender of the message can send a message P < n by computing with the enciphering transformation C = P e (mod n) 0 C < n: C is the remainder of P e divided by n The receiver gets the cipher-text message C, and can retrieve the plaintext by computing P = C d (mod n): P is the remainder of C d divided by n Example: To establish a public and private key, and individual first selects two primes, say p = 563 and q = So, n = 563 x 2357 = Finally, he selects an integer e = 3 relatively prime to (p-1)(q-1) = , and computes the inverse of e modulo (p-1)(q-1) by solving 3d 1 mod ( ) for d. This yields d (mod ). The values for n and e are made public; d, p, and q remain privates. Suppose someone wants to send the message (regarded as an integer) P = to the aforementioned individual. The must simply calculate and send the cipher-text C = = (mod ).

2 To decrypt, the recipient uses the decryption key to derive the plaintext thus: P = = (mod ) Implementation: BigIntegerMath.java: included. PrimeGenerator.java: included. RSA modulus: n = p x q, where p and q are two strong prime. This need to be implemented in your java console/applet main function. sr=new SecureRandom(); PrimeGenerator pg=new PrimeGenerator(modByteLength*4+1,10,sr); p=pg.getstrongprime(); q=pg.getstrongprime(); modulus=p.multiply(q); Keys: o Public enciphering key e (exponent): This need to be implemented in your java console/applet main function. o BigInteger pminusone=p.subtract(bigintegermath.one); BigInteger qminusone=q.subtract(bigintegermath.one); do { exponent=new BigInteger(modulus.bitLength()-1,10,sr); while (!exponent.gcd(pminusone.multiply(qminusone)).equals(bigintegermath.one)); o Private deciphering key d (dexponent). This need to be implemented in your java console/applet main function. dexponent=exponent.modinverse(pminusone.multiply(qminusone)); RSA encipher: message, RSA modulus n, and public enciphering key exponent need to be passed (located in Ciphers.java). public static byte[] RSAEncipher(byte[] msg,biginteger e,biginteger n) { //Compute the plaintext block size int blocksize=(n.bitlength()-1)/8; byte[][] ba=block(pad(msg,blocksize),blocksize); //Begin the enciphering for (int i=0;i<ba.length;i++) ba[i]=getbytes(new BigInteger(1,ba[i]).modPow(e,n)); //Return to a 1D array. The ciphertext block size is one byte greater than plaintext block size. return unblock(ba,blocksize+1); RSA decipher: message, RSA modulus n, and public deciphering key dexponent need to be passed (located in Ciphers.java). public static byte[] RSADecipher(byte[] msg,biginteger d,biginteger n) { //Compute the ciphertext block size int blocksize=(n.bitlength()-1)/8+1; byte[][] ba=block(msg,blocksize); //Begin the deciphering for (int i=0;i<ba.length;i++) ba[i]=getbytes(new BigInteger(1,ba[i]).modPow(d,n)); //Go from blocks to a 1D array, and remove padding; return this return unpad(unblock(ba,blocksize-1),blocksize-1);

3 RSA encipher/decipher salts: o Salt simply refers to adding random data to the end of each block. public static byte[] RSAEncipherWSalt(byte[] msg,biginteger e,biginteger n,securerandom sr) { //Compute the plaintext block size int blocksize=(n.bitlength()-1)/8; if (blocksize<5) throw new IllegalArgumentException("Block size must be >= 5 bytes"); byte[][] ba=block(pad(msg,blocksize-4),blocksize-4); //Begin the enciphering for (int i=0;i<ba.length;i++) { ba[i]=addsalt(ba[i],sr); ba[i]=getbytes(new BigInteger(1,ba[i]).modPow(e,n)); //Return to a 1D array. The ciphertext block size is one byte greater than plaintext block size. return unblock(ba,blocksize+1); public static byte[] RSADecipherWSalt(byte[] msg,biginteger d,biginteger n) { //Compute the ciphertext block size int blocksize=(n.bitlength()-1)/8+1; byte[][] ba=block(msg,blocksize); //Begin the deciphering for (int i=0;i<ba.length;i++) { ba[i]=getbytes(new BigInteger(1,ba[i]).modPow(d,n)); ba[i]=removesalt(ba[i]); //Go from blocks to a 1D array, and remove padding; return this return unpad(unblock(ba,blocksize-5),blocksize-5); Requirements: Implement a Java console or Java applet by yourself.. Output should look like following:

4 Modulus n Encipher key e Decipher key d Must following the structure (platform) provided. o Adds the necessary codes. Must be compilable and executable. Must be implemented in Java. Must be implemented in Unix/Linux environment. Grading criteria: Following instances will be grade 0: o Cheating & Plagiarism. o Late or not hand in. o Can not be compiled. o Can not be executed. Certain penalty will be given if following instances occur: o Not follow the platform provided. o Compilable and executable but Certain functionalities are not met the requirements. Certain outputs are not correct.

5 Plateform: BigintegerMath.java (included). PrimeGenerator.java (included). Ciphers.java import java.math.*; import java.security.*; import java.util.*; public class Ciphers { public static byte[] RSAEncipher(byte[] msg,biginteger e,biginteger n) { public static byte[] RSADecipher(byte[] msg,biginteger d,biginteger n) { public static byte[] RSAEncipherWSalt(byte[] msg,biginteger e,biginteger n,securerandom sr) { public static byte[] RSADecipherWSalt(byte[] msg,biginteger d,biginteger n) { //Go from blocks to a 1D array, and remove padding; return this return unpad(unblock(ba,blocksize-5),blocksize-5); //Method to add salt to blocks private static byte[] addsalt(byte[] b,securerandom random) { byte[] answer=new byte[b.length+4]; byte[] salt=new byte[4]; random.nextbytes(salt); //Put salt in front System.arraycopy(salt,0,answer,0,4); //Copy the message over System.arraycopy(b,0,answer,4,b.length); return answer; private static byte[] removesalt(byte[] b) { byte[] answer=new byte[b.length-4]; //Copy the message over System.arraycopy(b,4,answer,0,answer.length); return answer; //Method to rectify the extra bit problem of the tobytearray() method private static byte[] getbytes(biginteger big) { byte[] bigbytes=big.tobytearray(); if (big.bitlength()%8!=0) return bigbytes; else { byte[] smallerbytes=new byte[big.bitlength()/8]; System.arraycopy(bigBytes,1,smallerBytes,0,smallerBytes.length); return smallerbytes; //Pads message using PKCS#5 private static byte[] pad(byte[] msg,int blocksize) { //Check that block size is proper for PKCS#5 padding

6 if (blocksize<1 blocksize>255) throw new IllegalArgumentException("Block size must be between 1 and 255."); //Pad the message int numbertopad=blocksize-msg.length%blocksize; byte[] paddedmsg=new byte[msg.length+numbertopad]; System.arraycopy(msg,0,paddedMsg,0,msg.length); for (int i=msg.length;i<paddedmsg.length;i++) paddedmsg[i]=(byte)numbertopad; return paddedmsg; //Turns the message into a 2D array of blocks private static byte[][] block(byte[] msg,int blocksize) { //Create a 2D array of bytes corresponding to the message-all blocks should be full int numberofblocks=msg.length/blocksize; byte[][] ba=new byte[numberofblocks][blocksize]; for (int i=0;i<numberofblocks;i++) for (int j=0;j<blocksize;j++) ba[i][j]=msg[i*blocksize+j]; return ba; //Takes the blocks and returns the message as a linear array private static byte[] unblock(byte[][] ba,int blocksize) { //Create the 1D array in which to place the enciphered blocks byte[] m2=new byte[ba.length*blocksize]; //Place the blocks in the 1D array for (int i=0;i<ba.length;i++) for (int j=blocksize-1,k=ba[i].length-1;k>=0;k--,j--) m2[i*blocksize+j]=ba[i][k]; return m2; private static byte[] unpad(byte[] msg,int blocksize) { //Determine the amount of padding-just look at last block int numberofpads=(msg[msg.length-1]+256)%256; //Chop off the padding and return the array byte[] answer=new byte[msg.length-numberofpads]; System.arraycopy(msg,0,answer,0,answer.length); return answer;