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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 O((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 E Q, we show how to randomly produce a proof of the primality of n. The expected time is that needed for 1 exponentiations mod n. We also show how to randomly generate k-digit integers which will be in Q with probability proportional to k-1. Combined with the fast verification of n E 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 cn2/3/ log n 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.

Infinite sets of primes with fast primality tests and quick generation of large primes