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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

${u}_{n}={u}_{n-1}^{e}\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right),\phantom{\rule{1.em}{0ex}}0\le {u}_{n}\le m-1,\phantom{\rule{4pt}{0ex}}n=1,2,\cdots ,$

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 ${\left(pl\right)}^{1-\epsilon }$. 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:
 11K45 Pseudo-random numbers; Monte Carlo methods 11B50 Sequences (mod $m$) 11N56 Rate of growth of arithmetic functions 11T71 Algebraic coding theory; cryptography 94A60 Cryptography