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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. ..."
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 RAPPORT DE RECHERCHE 911, INRIA, OCTOBRE
, 1988
"... We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implem ..."
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Cited by 9 (7 self)
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We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number.
The Lucas–Pratt primality tree
 Math. Comp
"... Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtai ..."
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Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1.
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/ ..."
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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.
Atkin's test: news from the front
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... We make an attempt to compare the speed of some primality testing algorithms for certifying 100digit prime numbers. ..."
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We make an attempt to compare the speed of some primality testing algorithms for certifying 100digit prime numbers.
PRIME CHAINS AND PRATT TREES
, 2009
"... We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x ..."
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We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x.
A Distributed Approach To Proving Large Numbers Prime
, 1992
"... of a Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Computer Science The original of the complete thesis is on file in the Rensselaer Polytechnic Institute Library Approved ..."
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of a Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Computer Science The original of the complete thesis is on file in the Rensselaer Polytechnic Institute Library Approved by the Examining Committee: Erich Kaltofen, Thesis Adviser Sam Kim, Member Edith H. Luchins, Member Robert McNaughton, Member Noriko Yui, Member Rensselaer Polytechnic Institute Troy, New York August 1992 (For Graduation December 1992) c fl Copyright 1992 by Thomas Valente All Rights Reserved ii CONTENTS LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : v LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi ACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x 1. INTRODUCTION AND HISTORICAL BACKGROUND : : : : : : : : : : 1...