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On the uniformity of distribution of the RSA pairs

Authors:	Igor E Shparlinski

Institution:	Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia

Abstract:	Let be a product of two distinct primes and . We show that for almost all exponents with $\operatorname{gcd} (e, \varphi(m))= 1$ the RSA pairs are uniformly distributed modulo when runs through $\bullet$ the group of units $\mathbb{Z}_m^$ modulo (that is, as in the classical RSA scheme); $\bullet$ the set of -products $x = a_{i_1}\cdots a_{i_k}$ , $1 \le i_1 < \cdots < i_k \le n$ , where $a_1, \cdots, a_n \in \mathbb{Z}_m^$ are selected at random (that is, as in the recently introduced RSA scheme with precomputation). These results are based on some new bounds of exponential sums.

Keywords:	RSA cryptosystem uniform distribution precomputation exponential sums

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