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

Part of: Computational number theory Diophantine equations Multiplicative number theory Communication, information

Published online by Cambridge University Press:  22 November 2012

JOSHUA HOLDEN  and
MARGARET M. ROBINSON
JOSHUA HOLDEN
Affiliation:
Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA (email: holden@rose-hulman.edu)
MARGARET M. ROBINSON*
Affiliation:
Department of Mathematics and Statistics, Mount Holyoke College, 50 College Street, South Hadley, MA 01075, USA (email: robinson@mtholyoke.edu)
*
✉For correspondence; e-mail: robinson@mtholyoke.edu
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Abstract

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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.

Keywords

Brizolisdiscrete logarithmdiscrete exponentialHensel’s lemmap-adic interpolationfixed pointstwo-cyclescollisions

MSC classification

Secondary: 11D88: $p$-adic and power series fields 11N37: Asymptotic results on arithmetic functions 11Y16: Algorithms; complexity 94A60: Cryptography
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 163 - 178
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first-named author thanks the Hutchcroft Fund at Mount Holyoke College for support.

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