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J. Cryptology (1988) 1:53-64 Journal of Cryptology 9 1988 International Asso ciation for Cryptologic Research The Generation of Random Numbers That Are Probably Prime Pierre Beauchemin and Gil les Brassa rd ~ D~partement d'informatique et de recherche oprrationnelle, Universit6 de Montrral, C.P. 6128, Succ. "A", Montreal, Qurbec, Canada H3C3J7 Claude Crrpeau 2 Massachusetts Institute of Technology, Department of Computer Science, 545 Technology Square, Cambridge, MA 02139, U.S.A. Claude Goutier Centre de Calcul, Universit6 de M ontrral, C.P. 6128, Succ. "A", Montr6al, Qurbec, Canada H3C3J7 Carl Pomerance 3 Departm ent of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. Abstract. In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probab ility of failing only o n composite numbers tha t can actually be split in expected polynomial time. There- fore, factoring would be easy if Rabin's test systematically failed with a 25% probability on each composite integer (which, of course, it does not). The second observation is more fundamental because it is n o t restricted to primality testing: it has consequences for the entire field of probabilistic algorithms. The failure prob- ability when using a probabilistic algorithm for the purpose of testing some property is compared with that when using it for the purpose of obtaining a random element hopefully having this property. More specifically, we investigate the ques- tion of how reliable Rabin's test is when used to generate a random integer that is probably prime, rather than to test a specific integer for primality. Key words. Factorization, False witnesses, Primality testing, Probabilistic algo- rithms, Rabin's test. 1. A Brief Survey of Primality Testing How difficult is it to distinguish prime numbers from composite numbers? This is perhaps the singl e most important problem in computational number theory. We 1 Supported in part by NSERC grant A4107. Part of the research was performed while this author was at the CWI, Amsterdam. 2 Supported in part by an NSER C Posgraduate Scholarship. Part of the research was performed while

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