1. RSA basics 2. Key generation 3. What it would take to break RSA

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1. RSA basics 2. Key generation 3. What it would take to break RSA RSA encryption 1. RSA basics 2. Key generation 3. What it would take to break RSA

1. RSA basics Two large prime numbers (p, q): n = pq Small number e relatively prime to (p-1)(q-1) (e,n) is the public key Number d (multiplicative inverse of d, modulo (p-1)(q-1)) d e = 1 ( modulo (p-1)(q-1) ) (d,n) is the secret key It works because (Me)d = Med = M , [1] [1] Proof of RSA: http://cactus.eas.asu.edu/Partha/Teaching/Crypto.2000S/RSA-Proof.htm

1. RSA basics C  decrypt: M = SecretKey(C) = Cd (mod n) M  encrypt: C = PublicKey(M) = Me (mod n) M Public Key (e,n) C Secret key (d,n) M C  decrypt: M = SecretKey(C) = Cd (mod n)

2. Key generation Each pair public/private key requires two large primes (around 512 bits) RSA widely used  needs lots of large primes Primality tests: - try all possible factors (good for small numbers) - probable tests (may be enough) - recently shown (2002): primality can be proven in just polynomial time in the number of digits [2] [2] Agrawal, M., N. Kayal, and N. Saxena. Preprint. Primes is in P. Available at http://www.cse.iitk.ac.in/primality.pdf.

3. What it would take to break RSA What does “breaking RSA” mean? A: Factor n  find d C = Me (mod n)  computing e-th roots (mod n) B: Guessing the message C: Attacking a particular implementation

Experiment: Breaking DES using brute force Project statistics: Start of contest: January 29, 1997 Announcement of DESCHALL project: February 18, 1997 End of contest: June 17, 1997 Size of keyspace: 72,057,594,037,927,936 Keys searched: 17,731,502,968,143,872 Peak keys/day: 601,296,394,518,528 Peak keys/second: 7,000,000,000 (approx) Peak clients/day: 14,000 (approx, based on IP address) Total clients, since start: 78,000 (approx, based on IP address) The computer that found the key: CPU: Pentium 90 RAM: 16 megabytes Operating System: FreeBSD 2.2.1 Speed (keys/second): 250,000 (approx) Client: FreeBSD v0.214, built March 12, 1997 Owner: iNetZ Corporation, Salt Lake City, Utah Operator: Michael K. Sanders