Results 1 
4 of
4
Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
Abstract

Cited by 162 (22 self)
 Add to MetaCart
The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
DISTRIBUTED PRIMALITY PROVING AND THE PRIMALITY OF (2^3539+ 1)/3
, 1991
"... We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the fir ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the first ordinary Titanic prime.
Easy numbers for the Elliptic Curve Primality Proving Algorithm
, 1992
"... We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given. ..."
Abstract
 Add to MetaCart
We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given.
DETERMINISTIC ELLIPTIC CURVE PRIMALITY PROVING FOR A SPECIAL SEQUENCE OF NUMBERS
"... 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 ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/ ..."
Abstract
 Add to MetaCart
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 ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasiquadratic 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. We believe that this is the largest proven prime N for which no significant partial factorization of N − 1 or N + 1 is known. 1.