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02:33 < gmaxwell> http://www.openwall.com/lists/oss-security/2016/02/01/4  fun.. sipa: any idea what the probablity is that a 1023 bit number will be non-prime but have no factors smaller than about 200 bits?
02:35 < nickler> gmaxwell: https://twitter.com/esizkur/status/694310046994407426
02:37 < gmaxwell> nickler: yea, I found the small factor and have had ECM running for a long time.
02:37 < gmaxwell> which is where my 'about 200 bits' question came from.
02:39 < gmaxwell> A cute backdoor approach would be to pick two 500-ish bit primes for the modulus (and maybe the small factors to make it pass some broken primality test?)... with knoweldge of the factorization solving a discrete log problem would be tractable.
02:39 < gmaxwell> so a trapdoor.
02:40 < gmaxwell> perhaps an obvious test, assuming stupiditiy but non-malice, would be to take all the well known 512bit dh moduluses and see if the divide it.
02:47 < sipa> only 1/log(x) numbers in the neighbourhood of x are prime
02:47 < sipa> so for a 1023-bit number, that's a small minority
02:48 < sipa> so the answer to your question can likely be simplified to "what ratio of numbers around 2^1023 have only prime factors above 2^200"
02:52 < sipa> gmaxwell: my guess is that if it is actually an arbitrary 1023 bit number, very very low
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