Pseudoprime Reductions of Elliptic Curves

Chantal David (1) , J Wu (2)

1 Concordia University [Montreal]
2 IECN - Institut Élie Cartan de Nancy

Let E be an elliptic curve over Q without complex multiplication, and for each prime p of good reduction, let n E (p) = |E(F p)|. For any integer b, we are studying in this paper elliptic pseudoprimes to the base b. More precisely, let Q E,b (x) be the number of primes p x such that b n E (p) ≡ b (mod n E (p)), and π pseu E,b (x) be the number of compositive n E (p) such that b n E (p) ≡ b (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 π pseu E,b (x), generalising some of the literature for the classical pseudoprimes [6, 17] to this new setting.

Domaines: Théorie des nombres [math.NT]

Chantal David, J Wu. Pseudoprime Reductions of Elliptic Curves. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, 2012, ⟨10.4153/CJM-2011-044-x⟩. ⟨hal-01278385⟩