Deterministic elliptic curve primality proving for a special sequence of numbers

TL;DR

This paper presents a fast, deterministic elliptic curve-based primality proving algorithm for a specific sequence of numbers, enabling the verification of extremely large primes without relying on classical tests.

Contribution

The authors introduce a novel elliptic curve primality test tailored for a special sequence, capable of proving primality for numbers with over a million bits efficiently.

Findings

Proved primality of several large integers with over 100,000 digits.

Largest known prime proven without partial factorization of N-1 or N+1.

Algorithm achieves quasi-quadratic bit complexity in log N.

Abstract

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical "N-1" or "N+1" primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. At the time it was found, it was the largest proven prime N for which no significant partial factorization of N-1 or N+1 is known.