Infinite Sets of Primes with Fast Primality Tests and Quick Generation of Large Primes

Infinite sets P and Q of primes are described, P C Q. For any natural number n it can be decided if n e P in (deterministic) time 0((logn)9). This answers affirmatively the question of whether there exists an infinite set of primes whose membership can be tested in polynomial time, and is a main result of the paper. Also, for every n € Q, we show how to randomly produce a proof of the primality of n. The expected time is that needed for l| exponentiations mod n. We also show how to randomly generate fc-digit integers which will be in Q with probability proportional to A;-1. Combined with the fast verification of n € Q just mentioned, this gives an 0(k4) expected time algorithm to generate and certify primes in a given range and is probably the fastest method to generate large certified primes known to belong to an infinite subset. Finally, it is important that P and Q are relatively dense (at least en2/3/logn elements less than n). Elements of Q in a given range may be generated quickly, but it would be costly for an adversary to search Q in this range, a property that could be useful in cryptography.