Pseudoprime reductions of Elliptic curves

Abstract

Let $E$ be an elliptic curve over $\F_p$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\F_p) |$. Let $Q_{E,b}(x)$ be the number of primes $p\le x$ such that $b^{n_E(p)}\equiv b(\mathrm{mod}{n}_E(p))$, and ${\pi }_{E,b}^{\mathrm{pseu}}(x)$ be the number of {\it compositive} $n_E(p)$ such that $b^{n_E(p)}\equiv b(\mathrm{mod}{n}_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address in this paper the problem of finding upper bounds for $Q_{E,b}(x)$ and ${\pi }_{E,b}^{\mathrm{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes \cite{Erdos56, Pomerance81} to this new setting.

References in corpus (2)

• Analytic problems for elliptic curves
• The Large Sieve and Galois Representations