Abstract

F13.54> k = 100, the algorithm answers correctly with an overwhelming probability: 1 \Gamma 2 \Gamma100 . Also observe that the running time is O(kn 3 ), since for each of the k a's, we compute each of the at most n u i 's by a simple squaring. The idea behind the proof of claim 1 is to distinguish two special cases: 1. N is a prime power, N = p b , b 2 and 2-1 2-2 Lecture 2 : March 8, 1995 2. N has at least two distinct prime factors. Proof: (In case 1.) Here, we are almost home free, since this case won't pass the criterion in Fermat's little theorem, i.e. for most a we have a N \Gamma1 6j 1 (mod N ). Assume N = p b

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