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

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Presentation transcript:

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

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

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”.

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.

Factoring Integers

Cracking RSA digit number (n) Factors: Method used is called Generalized Number Field Sieve

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).

An Example Consider n = , note that mod = and mod = , GCD( ,  ) = GCD( , ) = 3001 and = 3001 * 4001

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.

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.

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.

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.