Pseudoprime reductions of elliptic curves

Abstract

Let b ≥ 2 be an integer and let E/ be a fixed elliptic curve. In this paper, we estimate the number of primes p ≤ x such that the number of points nE(p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of nE(p).

The square sieve and a Lang–Trotter question for generic abelian varieties

“…How often is the order of the group of rational points, #A p (F p ) prime, nearly-prime, or pseudoprime (to a fixed base)? On the analogous questions for elliptic curves, see for instance [Kob88;MM01;CLS09], and see [BCD11] for the study of the primality of #E p (F p ) on average. Question 7.4.…”

Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects

“…In fact one can even obtain rather strong bounds on the number of reductions of E with pseudoprime cardinalities, see [41,52,111].…”

Infinitude of Elliptic Carmichael Numbers

“…is an elliptic Carmichael number for all elliptic curves E/Q with complex multiplication by Q( 2,3,7,11,19,43,67, 163}, then we call N an elliptic Carmichael number.…”

“…Since L has fewer than log L log x prime factors, we may assume [11] Infinitude of elliptic Carmichael numbers 55 that y is so large that L has fewer than x 1/4 prime factors. Further, the reciprocal sum of these primes is log log y/ log y, so we may assume that y is so large that this reciprocal sum is smaller than 1/60.…”

“…We mention that there have been some other notions of elliptic pseudoprimes in the literature; see [11,13,25,41] .…”