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Date: Fri, 05 Sep 2014 22:07:13 +0100 From: Samuel Neves <sneves@....uc.pt> To: discussions@...sword-hashing.net Subject: Re: [PHC] A review per day - Makwa On 05-09-2014 21:37, Thomas Pornin wrote: > I am too lazy and rushed right now to > compute the probability that, by generating random 1000-bit integers, > you hit a value whose largest prime factor is less than 2^80, but it > is negligible (less than 1 in 2^200). Approximating Dickman's rho as u^-u, the probability that a randomly-selected 1000-bit integer is 2^80-smooth is rho(log(2^1000)/log(2^80)), or roughly 2^-45. That said, I agree that worrying about safe primes in unwarranted. By all measures, cycling and p+-1 attacks are a terrible factorization method for general numbers; any serious attacker will be better served by trying ECM or the number field sieve. I recommend Rivest and Silverman's 1999 paper on this precise subject, specifically Section 9: http://people.csail.mit.edu/rivest/pubs/RS01.version-1999-11-22.pdf
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