Probabilistic analyses on finding optimal combinations of primality tests in real applications

Generating a prime is an iterative application of generating a random number r and testing the primality of r until r is a prime. Among them, the primality test on r is much more time-consuming than the random number generation and thus it occupies most of the running time of the prime generation. To reduce the running time of the primality test, real applications combine several primality test methods. The most widely used combination is the combination of the trial division and the probabilistic primality test. Although this combination is widely used in practice, few analyses were given on finding the optimal combination, i.e., on finding the optimal number of small primes used in trial division that minimizes the expected running time of this combination. In this paper, we present probabilistic analyses on finding the optimal combinations of the trial division and the probabilistic primality test. Using these analyses, we present three optimal combinations. One is for the primality test and the others are for the safe primality test. The optimal combinations are universal in that they are presented as functions of div and ppt where div is the time required for dividing the random number r by a small prime and ppt is the time required for the probabilistic primality test of r. Thus, in any situation that div and ppt can be measured, the optimal combinations can be calculated from these functions. The experimental results show that our probabilistic analyses predict the optimal combinations well. The predicted optimal combinations can be used as useful guidelines in designing a primality or a safe primality test. The usefulness of the optimal combinations is more evident when the primality test is implemented on embedded systems or crypto-processors because finding optimal combinations using experiments is very time-consuming and inefficient.