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Pseudoprime Reductions of Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

C. David  and
J. Wu
C. David
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, QC, H3G 1M8 email: cdavid@mathstat.concordia.ca
J. Wu
Affiliation:
Institut Elie Cartan Nancy, CNRS, Université Henri Poincaré (Nancy 1), INRIA, 54506 Vandoeuvre-lès- Nancy, France email: wujie@iecn.u-nancy.fr
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Abstract

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Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication, and for each prime $p$ of good reduction, let ${{n}_{E}}(p)\,=\,\left| E\left( {{\mathbb{F}}_{p}} \right) \right|$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let ${{Q}_{E,B}}(x)$ be the number of primes $p\le x$ such that ${{b}^{{{n}_{E}}(p)}}\,\equiv \,b\left( \bmod \,{{n}_{E}}\left( p \right) \right)$, and let $\pi _{E,b}^{\text{pseu}}(x)$ be the number of compositive${{n}_{E}}(p)$ such that ${{b}^{{{n}_{E}}(p)}}\,\equiv \,b\left( \bmod \,{{n}_{E}}\left( p \right) \right)$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for ${{Q}_{E,B}}(x)$ and $\pi _{E,b}^{\text{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

Keywords

11N3614H52Rosser-Iwaniec sievegroup order of elliptic curves over finite fieldspseudoprimes
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 1 , 01 February 2012 , pp. 81 - 101
Copyright
Copyright © Canadian Mathematical Society 2012

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