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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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On the distribution of the power generator
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by John B. Friedlander and Igor E. Shparlinski PDF
Math. Comp. 70 (2001), 1575-1589 Request permission

Abstract:

We present a new method to study the power generator of pseudorandom numbers modulo a Blum integer $m$. This includes as special cases the RSA generator and the Blum–Blum–Shub generator. We prove the uniform distribution of these, provided that the period $t\ge m^{3/4 + \delta }$ with fixed $\delta > 0$ and, under the same condition, the uniform distribution of a positive proportion of the leftmost and rightmost bits. This sharpens and generalizes previous results which dealt with the RSA generator, provided the period $t\ge m^{23/24 + \delta }$. We apply our results to deduce that the period of the binary sequence of the rightmost bit has exponential length.
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Additional Information
  • John B. Friedlander
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
  • Email: frdlndr@math.toronto.edu
  • Igor E. Shparlinski
  • Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia
  • MR Author ID: 192194
  • Email: igor@ics.mq.edu.au
  • Received by editor(s): April 30, 1999
  • Received by editor(s) in revised form: November 10, 1999
  • Published electronically: October 17, 2000
  • Additional Notes: The first author was supported in part by NSERC grant A5123 and by an NEC grant to the Institute for Advanced Study.
    The second author was supported in part by ARC grant A69700294.
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 1575-1589
  • MSC (2000): Primary 11L07, 11T71, 94A60; Secondary 11T23, 11K45
  • DOI: https://doi.org/10.1090/S0025-5718-00-01283-7
  • MathSciNet review: 1836920