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Proving primality in essentially quartic random time
 Math. Comp
, 2003
"... Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1. ..."
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Cited by 18 (0 self)
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Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1.
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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Cited by 12 (1 self)
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
It Is Easy to Determine Whether a Given Integer Is
, 2005
"... Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wis ..."
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Cited by 6 (0 self)
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Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
Cyclotomy primality proofs and their certificates. Mathematica Goettingensis
, 2006
"... Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein ..."
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Cited by 2 (1 self)
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Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein
ELLIPTIC PERIODS AND PRIMALITY PROVING (EXTENTED VERSION)
, 810
"... Abstract. We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion. ..."
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Abstract. We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion.
Fast Primality Proving on Cullen Numbers
, 2009
"... We present a unconditional deterministic primality proving algorithm for Cullen numbers. The expected running time and the worst case running time of the algorithm are Õ(log2 N) bit operations and Õ(log 3 N) bit operations, respectively. 1 ..."
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We present a unconditional deterministic primality proving algorithm for Cullen numbers. The expected running time and the worst case running time of the algorithm are Õ(log2 N) bit operations and Õ(log 3 N) bit operations, respectively. 1
Elliptic periods and primality proving
, 2009
"... We define the ring of elliptic periods modulo an integer n and give an elliptic version of the AKS primality criterion. ..."
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We define the ring of elliptic periods modulo an integer n and give an elliptic version of the AKS primality criterion.