Integer Factorization

1. Integer Factorization • Various approaches • trial divisions up to B find factors up to B2 • probabilistic approaches find factors up to B4 • Pollard’s rho heutistic • neither the running time nor success is guaranteed • any divisor it finds will be correct, but it may never report any results • in practice, it is the most effective means of factorization currently known • it will print the factor p after approximately p iterations; thus it finds small factors quickly

2. Pollard’s rho heuristic • The while loop searches indefinitely for factors generating a new xi each time • Lines 1-4 are for initialization • The xi values saved in y are when i = 1,2,4,8,16, … • d is the gcd of y- xi and n; if it is nontrivial then it is printed as a factor of n • If n is composite, we expect to find enough divisors to factor n after approximately n1/4 updates • Zn is finite, so we need to estimate how long until the xis repeat themselves

3. How long until the xis repeat? • An analysis • let p be a nontrivial factor of n so gcd(p,n/p)=1 • the xi induce a sequence xi’ mod p where xi’ = xi mod p • by the Chinese remainder theoremxi+1’2 = (xi’2 - 1 )mod p since (x mod n)mod p=x mod p • by the birthday-paradox analysis (section 6.6.1) we expect repetition in aprx. O(p) • once k becomes large enough we will make a complete cycle around the loop without changing y, then a factor is discovered since xi  y (mod p) • the factor will be p or a multiple of p

4. The Big Picture

5. The rho diagrams • (a) is generated by the xi starting at 2 for n = 1387 • The factor 19 (since 1387 = 19 * 73) is discovered when the xi is 177, this is before the value 1186 is repeated • (b) show the recurrence for mod 19, every xi in part (a) is equivalent to the xi‘ mod 19 • (c) shows the recurrence for mod 73, again every xi in part (a) is equivalent to the xi” mod 73 • By the Chinese remainder theorem, each node in (a) corresponds to a pair of nodes in (b) and (c)