Primality of the number of points on an elliptic curve over a finite field

1988/01/01 by Neal Koblitz · 2 citations
Mathematics · Computer Science · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Coding theory and cryptography #Mathematics #Primality test #Finite field #Field (mathematics) #Elliptic curve #Schoof's algorithm #Pure mathematics #Discrete mathematics #Combinatorics #Prime (order theory) #Quarter period

paper · pdf · doi:10.2140/pjm.1988.131.157

openalex publication_date 1988/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/05

Abstract

Given a fixed elliptic curve E defined over Q having no rational torsion points, we discuss the probability that the number of points on E mod/? is prime as the prime p varies. We give conjectural asymptotic formulas for the number of p < n for which this number is prime, both in the case of a complex multiplication and a non-CM curve E. Numerical evidence is given supporting these formulas.

Citations

• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Use of Elliptic Curves in Cryptography
• Elliptic curve cryptosystems
• Primitive points on elliptic curves
• Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes

Cited by

• Almost Prime Orders of Elliptic Curves Over Prime Power Fields
• Sato–Tate conjecture [wikipedia]