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VSH and multiplicative modular relations between small primes with polynomial exponents

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Abstract

We study the frequency of the positive integers \(n\) that are products of two primes of the same order of magnitude and such that the congruence

$$\begin{aligned} \prod _{i=1}^k p_i^{f_i (n)} \equiv 1 \pmod {n} \end{aligned}$$

holds with some fixed nonzero polynomials \(f_1(X), \ldots , f_k(X) \in {\mathbb {Z}}[X]\), where \(p_i\) denotes the \(i\)th prime. The question is motivated by collision finding in the so-called Very Smooth Hash function, introduced by Contini et al. (Lecture notes in computer science, vol. 4004. Springer, Berlin, pp 165–182, 2006).

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References

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Acknowledgments

During the preparation of this paper, I. Blake was supported by NSERC Grant 7382, F. Luca was supported in part by Grants PAPIIT 104512 (UNAM) and a Marcos Moshinsky fellowship, and I.E. Shparlinski was supported in part by Australian Research Council Grants DP110100628 and DP130100237.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

    Ian F. Blake

  2. Mathematical Institute, UNAM Juriquilla, 76230 , Santiago de Querétaro, Mexico

    Florian Luca

  3. School of Mathematics, University of the Witwatersrand, P.O. Box Wits 2050, Johannesburg, South Africa

    Florian Luca

  4. Department of Computing, Macquarie University, North Ryde, Sydney, NSW, 2109, Australia

    Igor E. Shparlinski

Authors
  1. Ian F. Blake
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  2. Florian Luca
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  3. Igor E. Shparlinski
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Correspondence to Igor E. Shparlinski.

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Blake, I.F., Luca, F. & Shparlinski, I.E. VSH and multiplicative modular relations between small primes with polynomial exponents. AAECC 25, 181–188 (2014). https://doi.org/10.1007/s00200-014-0219-2

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  • DOI: https://doi.org/10.1007/s00200-014-0219-2

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