This preview shows page 1. Sign up to view the full content.
Unformatted text preview: primes is a one-way function; it’s easy to multiply the numbers to get a product but hard to factor the product and recover the two large primes (see Section 11.3). Public-key cryptography uses this idea to make a trap-door one-way function. Actually, that’s a lie; factoring is conjectured to be a hard problem (see Section 11.4). As far as anyone knows, it seems to be. Even if it is, no one can prove that hard problems are actually hard. Most everyone assumes that factoring is hard, but it has never been mathematically proven one way or the other. This is worth dwelling on. It is easy to imagine that 50 years in the future we will all sit around, reminiscing about the good old days when people used to think factoring was hard, cryptography was based on factoring, and companies actually made money from this stuff. It is easy to imagine that future developments in number theory will make factoring easier or that developments in complexity theory will make factoring trivial. There’s no reason to believe this will happen—and most people who know enough to have an opinion will tell you that it is unlikely—but there’s also no reason to believe it won’t. In any case, today’s dominant public-key encryption algorithms are based on the difficulty of factoring large numbers that are the product of two large primes. (Other algorithms are based on something called the Discrete Logarithm Problem, but for the moment assume the same discussion applies.) These algorithms are also susceptible to a brute-force attack, but of a different type. Breaking these algorithms does not involve trying every possible key; breaking these algorithms involves trying to factor the large number (or taking discrete logarithms in a very large finite field—a similar problem). If the number is too small, you have no security. If the number is large enough, you have security against all the computing power in the world working from now until the sun goes nova—given today’s understanding of the mathe...
View Full Document
This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
- Fall '10