1. 7. Key Length Public key length Kim Hyoung-Shick

2. Contents 1. Introduction 2. Results

3. Contents 1. Introduction 2. Results

4. - Is factoring hard ? - Is factoring in NP ? - Is factoring NP-complete ? Questions

5. - RSA are based on the factoring problem. - Elgamal are based on the discrete logarithm problem. Today’s dominant public key systems

6. - Finding key - Solving the base problem Brute-force Attack

7. One-way function: f() is one way if –For any x, y = f(x) is easy to compute –For any (or almost all) y, it is hard to find an x such that f(x) = y Trapdoor one-way function f s () –given s and y, it is easy to compute an x such that f s (x) = y Building Public Key Systems

8. Quadratic sieve – below 110 bits General number field sieve – above 110 bits Special number field sieve – special form Factoring Algorithm

9. RSA Laboratories continues its sponsorship of the RSA Factoring Challenge To help users of the RSA public-key cryptosystem in choosing suitable key lengths for an appropriate level of security. New RSA Factoring Challenge http://www.rsasecurity.com/rsalabs/challenges/factoring/index.html

10. How much does it cost to factor a large number? Number Length (bits) MachinesMemory 4301trivial 760215,0004 Gb 1020342,000,000170 Gb 16201.6 x 10 15 120 Tb For one year, the machines column is the number of 500 MHz Pentium

11. New Records - Factorization of RSA-155 digit (512 bit) by distributed computing, 1999

12. Recommended Key Length (Recommendation by RSA Laboratories, RSA Data Security, Inc.'s research arm) 768-bit and 1024-bit keys as the minimum for achieving reliable security

13. Distributed computing DNA computing Quantum computing Future Attacks