Close to Uniform Prime Number Generation With Fewer Random Bits

In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime p less than x, the basic idea is to fix a constant q ∝ x
^1-ε
, pick a uniformly random a <; q coprime to q, and choose p of the form a + t · q, where only t is updated if the primality test fails. We prove that variants of this approach provide prime generation algorithms requiring a few random bits and whose output distribution is close to uniform, under less and less expensive assumptions: first a relatively strong conjecture by H. Montgomery, made precise by Friedlander and Granville; then the Extended Riemann Hypothesis; and finally fully unconditionally using the Barban-Davenport-Halberstam theorem. We argue that this approach has a number of desirable properties compared with the previous algorithms, at least in an asymptotic sense. In particular: 1) it uses much fewer random bits than both the “trivial algorithm” (testing random numbers less than x for primality) and Maurer's almost uniform prime generation algorithm; 2) the distance of its output distribution to uniform can be made arbitrarily small, unlike algorithms like PRIMEINC (studied by Brandt and Damgård), which we show exhibit significant biases; and 3) all quality measures (number of primality tests, output entropy, randomness, and so on) can be obtained under standard conjectures or even unconditionally, whereas most previous nontrivial algorithms can only be proved based on stronger, less standard assumptions like the Hardy- Littlewood prime tuple conjecture. Note, however, that our analysis involves non-explicit constants, and therefore does not establish the superiority of our approach for concrete parameter sizes.