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# Pseudoprime Reductions of Elliptic Curves

Published:2011-06-27
Printed: Feb 2012
• C. David,
Department of Mathematics and Statistics, Concordia University, Montréal, QC, H3G 1M8
• J. Wu,
Institut Elie Cartan Nancy, CNRS, Université Henri Poincaré (Nancy 1), INRIA, 54506 Vandœuvre-lès-Nancy, France
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## Abstract

Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.
 Keywords: Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes
 MSC Classifications: 11N36 - Applications of sieve methods 14H52 - Elliptic curves [See also 11G05, 11G07, 14Kxx]

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