A NOTE ON MONTE CARLO PRIMALITY TESTS AND ALGORITHMIC INFORMATION THEORY [chapter]

GREGORY J. CHAITIN, JACOB T. SCHWARTZ
1987 Information, Randomness & Incompleteness  
Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coin-flipping and whose con- G. J. Chaitin clusions are only probably correct. On the other hand, algorithmic information theory provides a precise mathematical definition of the notion of random or patternless sequence. In this paper we shall describe conditions under which if the sequence of coin tosses in the Solovay-Strassen and Miller-Rabin algorithms is replaced by a sequence of
more » ... and tails that is of maximal algorithmic information content, i.e., has maximal algorithmic randomness, then one obtains an error-free test for primality. These results are only of theoretical interest, since it is a manifestation of the Gödel incompleteness phenomenon that it is impossible to "certify" a sequence to be random by means of a proof, even though most sequences have this property. Thus by using certified random sequences one can in principle, but not in practice, convert probabilistic tests for primality into deterministic ones. 1 The second author has been supported by US DOE, Contract EY-76-C-02-3077*000. We wish to thank John Gill III and Charles Bennett for helpful discussions. Reproduction in whole or in part is permitted for any purpose of the United States Government. 2 We could equally well have used in this paper the newer formalism of [7], in which programs are "self-delimiting."
doi:10.1142/9789814434058_0016 fatcat:apf7bbkpcbbsngzm4dytoskfla