1. Strength of Cryptographic Systems Dr. C F Chong, Dr. K P Chow Department of Computer Science and Information Systems The University of Hong Kong

2. Outline of Presentation Introduction Cracking RSA –Overview of RSA –Factoring integers –Number theory Cracking DES –Overview of DES –Brute force attack on DES

3. Introduction A cryptographic algorithm is usually well- known and security is provided by keeping secret some private information: some private key. A cryptographic system is said to be strong if it is very difficult to “break”.

4. Overview of RSA Encrypt a message using public key and decrypt using secret private key. Public key is a pair of integers (e, n) where n is the product of 2 large prime numbers. Private key is the pair of integers (d, n) where n = p * q, and e * d  1 mod ((p  1) * (q  1)) The length of n is an indication of the strength of RSA and is usually expressed in number of bits.

5. Factoring Integers

6. Cracking RSA-130 130 digit number (n) 18070 82088 68740 48059 51656 16440 59055 66278 10251 67694 01349 17012 70214 50056 66254 02440 48387 34112 75908 12303 37178 18879 66563 18201 32148 80557 Factors: 39685 99945 95974 54290 16112 61628 83786 06757 64491 12810 06483 25551 57243 45534 49864 67359 72188 40368 68972 74408 86435 63012 63205 06960 09990 44599 Method used is called Generalized Number Field Sieve

7. The Theory Theorem: Given a non-prime integer n, if x 2  y 2 mod n, and x   y mod n, then GCD(x+y, n) is a proper factor of n, and so is GCD(x  y, n).

8. An Example Consider n = 12007001, note that 9815310 2 mod 12007001 = 626422 and 1247455 2 mod 12007001 = 626422, GCD(12007001, 9815310  1247455) = GCD(12007001, 8567855) = 3001 and 12007001 = 3001 * 4001

9. Our Efforts on Factorization Primary purpose is to learn how to implement factorization algorithms. Still in very early stage, currently working on an implementation of Multiple Polynomial Quadratic Sieve.

10. Overview of DES Based on a 64-bit secret key which is used both for encryption and decryption. The actual key is 56-bit since one of the bits in each 8-bit units is actually a parity bit and is not used for encryption/decryption. Messages are encrypted in blocks of 64-bit units.

11. Brute Force Attacks on DES Try all possible keys until a “match” is found. Early 1998, 40 days using 50,000 CPUs on the Internet, about 85% of the key space searched. July 1998, 56 hours using specially designed hardware (EFF DES Cracker) which costs about US$250,000.

12. Our Efforts on Cracking 40-bit DES Demonstration only, not optimized. Check first 4 bytes of “decrypted” message against a set of known “headers” for conformance, “yes” means highly probable that message is cracked. Use idle times of about 50 workstations in the Department (most more than 3 years old), about 80% of key space searched in about 15 days.