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Period of the power generator and small values of Carmichael’s function. (English) Zbl 1029.11043

The paper considers the pseudorandom number generator

u n =u n-1 e (modm),0u n m-1,n=1,2,,

where the modulus m, the initial state u 0 =v and the exponent e are given. One particularly interesting case is when the modulus is of the form m=pl where p and l are different primes of the same magnitude. The authors show that for almost all choices of p, l it holds for almost all choices of v, e that the period of the generator exceeds (pl) 1-ε . From earlier work by some of the authors it follows that this implies that the power generator is uniformly distributed.

One application of the results are a rigorous proof that the cycling attack on the RSA cryptosystem has a negligible chance to be efficient.

In the Corrigendum a corrected proof of Theorem 8 is given.


MSC:
11K45Pseudo-random numbers; Monte Carlo methods
11B50Sequences (mod m)
11N56Rate of growth of arithmetic functions
11T71Algebraic coding theory; cryptography
94A60Cryptography