Fast Monte-Carlo Primality Evidence Shown in the Dark

We construct an e cient proof of knowledge protocol for the demonstration of Monte-Carlo evidence that a number n is the product of two odd primes of roughly equal size without the prime factors being disclosed. The cost for a proof amounts to 12k log 2 n multiplications of integers of size of n where k is a security parameter which controls the error probability of the proof under 2 . With the same security parameter this error probability improves from the previous result of e k=74 for n in the general case of the two-prime-product structure. We also prove the security of our protocol with respect to a decision-Di e-Hellman problem.