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COUNTING FIXED POINTS, TWO-CYCLES, AND COLLISIONS OF THE DISCRETE EXPONENTIAL FUNCTION USING p-ADIC METHODS

  • JOSHUA HOLDEN (a1) and MARGARET M. ROBINSON (a2)
Abstract
Abstract

Brizolis asked for which primes p greater than 3 there exists a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Various authors have contributed to the understanding of this problem. In this paper, we use p-adic methods, primarily Hensel’s lemma and p-adic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.

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For correspondence; e-mail: robinson@mtholyoke.edu
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The first-named author thanks the Hutchcroft Fund at Mount Holyoke College for support.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] A. Balog , K. A. Broughan and I. E. Shparlinski , ‘On the number of solutions of exponential congruences’, Acta Arith. 148 (2011), 93103.

[10] L. Glebsky and I. Shparlinski , ‘Short cycles in repeated exponentiation modulo a prime’, Des. Codes Cryptogr. 56 (2010), 3542.

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[14] J. Holden , ‘Fixed points and two-cycles of the discrete logarithm’, in: Algorithmic Number Theory (ANTS 2002), Lecture Notes in Computer Science, 2369 (eds. C. Fieker and D. R. Kohel ) (Springer, Berlin, 2002), pp. 405415.

[15] J. Holden and P. Moree , ‘Some heuristics and results for small cycles of the discrete logarithm’, Math. Comp. 75 (2006), 419449.

[17] N. Koblitz , p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edn (Springer, New York, 1984).

[18] M. Levin , C. Pomerance and K. Soundarajan , ‘Fixed points for discrete logarithms’, in: Algorithmic Number Theory (ANTS-IX), Lecture Notes in Computer Science, 6197 (eds. G. Hanrot , F. Morain and E. Thomé ) (Springer, Berlin, 2010), pp. 615.

[19] A. J. Menezes , P. C. van Oorschot and S. A. Vanstone , Handbook of Applied Cryptography (CRC Press, Boca Raton, FL, 1996).

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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