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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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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.
Proving Primality In Essentially Quartic Expected Time
, 2003
"... This paper presents a randomized algorithm that, given a prime n, nds and veri es a proof of the primality of n in expected time (lg n) . ..."
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This paper presents a randomized algorithm that, given a prime n, nds and veri es a proof of the primality of n in expected time (lg n) .
Doublyfocused enumeration of pseudosquares and pseudocubes
 In Proceedings of the 7th International Algorithmic Number Theory Symposium (ANTS VII
, 2006
"... Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doublyfocused enumer ..."
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Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doublyfocused enumeration — is examined. This technique, first described by D. J. Bernstein, allowed us to obtain recordsetting sieve computations in software on general purpose computers. 1
RP3
"... Abstract — We implement the AgrawalKayalSaxena primality testing algorithm. We discuss optimizations to the implementation that resulted in improved performance over the initial implementation. We further discuss methods of obtaining faster runtimes for candidate primes of increasing size. ..."
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Abstract — We implement the AgrawalKayalSaxena primality testing algorithm. We discuss optimizations to the implementation that resulted in improved performance over the initial implementation. We further discuss methods of obtaining faster runtimes for candidate primes of increasing size.
ELLIPTIC PERIODS AND PRIMALITY PROVING (EXTENTED VERSION)
, 2009
"... 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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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.
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.