Number-Theoretic Algorithms

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Presentation transcript:

Number-Theoretic Algorithms Chapter 31, CLRS book

Modular Arithmetic

The Chinese Remainder Problem A problem described in an ancient Chinese arithmetic book, Sun Tze Suan Ching, by Sun Tze (around 300AD, author of The Art of War). Problem: We have a number of objects, but we do not know exactly how many. If we count them by threes we have two left over. If we count them by fives we have three left over. If we count them by sevens we have two left over. How many objects are there?

Example: Chinese remainder theorem

Algorithms

The RSA Cryptosystem

Attacks on RSA

RSA-200 = 27,997,833,911,221,327,870,829,467,638, 722,601,621,070,446,786,955,428,537,560, 009,929,326,128,400,107,609,345,671,052, 955,360,856,061,822,351,910,951,365,788, 637,105,954,482,006,576,775,098,580,557, 613,579,098,734,950,144,178,863,178,946, 295,187,237,869,221,823,983.

Generating large primes To set up an RSA cryptosystem, we need two large primes p and q.

Integer Factorization Reference on quadratic sieve: http://blogs.msdn.com/b/devdev/archive/2006/06/19/637332.aspx