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Smooth Orders and Cryptographic Applications

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Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

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Abstract

We obtain rigorous upper bounds on the number of primes p ≤ x for which p- 1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.

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Author information

Authors and Affiliations

  1. Department of Fundamental Mathematics, Bell Laboratories, Murray Hill, NJ, 07974-0636, USA

    Carl Pomerance

  2. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Igor E. Shparlinski

Authors
  1. Carl Pomerance
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  2. Igor E. Shparlinski
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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© 2002 Springer-Verlag Berlin Heidelberg

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Pomerance, C., Shparlinski, I.E. (2002). Smooth Orders and Cryptographic Applications. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_27

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  • DOI: https://doi.org/10.1007/3-540-45455-1_27

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43863-2

  • Online ISBN: 978-3-540-45455-7

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