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Ebook A course in number theory and cryptography (2E): Part 2

(BQ) Part 2 book "A course in number theory and cryptography" has contents: Primality and Factoring, pseudoprimes, the rho method, fermat factorization and factor bases, elliptic curve cryptosystems, elliptic curve primality test, elliptic curve factorization,.and other contents. | v Primality and Factoring There are many situations where one wants to know if a large number n is prime. For example, in the RSA public key cryptosystem and in various cryptosystems based on the discrete log problem in finite fields, we need to find a large "random" prime. One interpretation of what this means is to choose a large odd integer n0 using a generator of random digits and then test no' no + 2, . . . for primality until we obtain the first prime which is 2:: n0 • A second type of use of primality testing is to determine wheth.er an integer of a certain very special type is a prime. For example, for some large prime f we might want to know whether 2 1 - 1 is a Mersenne prime. If we're working in the field of 2 1 elements, we saw that every element "I= 0, 1 is a generator of F ; , if ( and only if ) 2 1 - 1 is prime ( see Ex. 13 ( a) of § ILl) . A primality test is a criterion for a number n not t o be prime. If n ''passes" a primality test, then it may be prime. If it passes a whole lot of primality tests, then it is very likely to be prime. On the other hand, if n fails any single primality test, then it is definitely composite. But that leaves us with a very difficult problem: finding the prime factors of n. In general, it is much more time-consuming to factor a large number once it is known to be composite ( because it fails a primality test) than it is to find a prime number of the same order of magnitude. ( This is an empirical statement, not a theorem; no assertion of this sort has been proved. ) The security of the RSA cryptosystem is based on the assumption that it is much easier for someone to find two extremely large primes p and q than it is for someone else, knowing n = pq but not p or q, to find the two factors in n. After discussing primality tests in § 1, we shall describe three different factorization methods in §§2-5. 126 V. Primality and Factoring 1 Pseudoprimes Have you ever noticed that there's no attempt being made to find really