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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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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Constructing Isogenies Between Elliptic Curves Over Finite Fields
 LMS J. Comput. Math
, 1999
"... Let E 1 and E 2 be ordinary elliptic curves over a finite field Fp such that #E1 (Fp ) = #E2 (Fp ). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp . The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny. ..."
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Cited by 30 (3 self)
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Let E 1 and E 2 be ordinary elliptic curves over a finite field Fp such that #E1 (Fp ) = #E2 (Fp ). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp . The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny.
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.
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
ECM on Graphics Cards
"... Abstract. This paper reports recordsetting performance for the ellipticcurve method of integer factorization: for example, 604.99 curves/second for ECM stage 1 with B1 = 8192 for 280bit integers on a single PC. The stateoftheart GMPECM software handles 171.42 curves/second for ECM stage 1 with ..."
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Cited by 13 (4 self)
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Abstract. This paper reports recordsetting performance for the ellipticcurve method of integer factorization: for example, 604.99 curves/second for ECM stage 1 with B1 = 8192 for 280bit integers on a single PC. The stateoftheart GMPECM software handles 171.42 curves/second for ECM stage 1 with B1 = 8192 for 280bit integers using all four cores of a 2.4GHz Core 2 Quad Q6600. The extra speed takes advantage of extra hardware, specifically two NVIDIA GTX 280 graphics cards, using a new ECM implementation introduced in this paper. Our implementation uses Edwards curves, relies on new parallel addition formulas, and is carefully tuned for the highly parallel GPU architecture. On a single GTX 280 the implementation performs 22.66 million modular multiplications per second for a general 280bit modulus. GMPECM, using all four cores of a Q6600, performs 17.91 million multiplications per second. This paper also reports speeds on other graphics processors: for example,
ON MORDELLWEIL GROUPS OF ELLIPTIC CURVES INDUCED BY DIOPHANTINE TRIPLES
"... Dedicated to Professor Sibe Mardeˇsić on the occasion of his 80th birthday Abstract. We study the possible structure of the groups of rational points on elliptic curves of the form y 2 = (ax + 1)(bx + 1)(cx + 1), where a, b, c are nonzero rationals such that the product of any two of them is one le ..."
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Cited by 12 (11 self)
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Dedicated to Professor Sibe Mardeˇsić on the occasion of his 80th birthday Abstract. We study the possible structure of the groups of rational points on elliptic curves of the form y 2 = (ax + 1)(bx + 1)(cx + 1), where a, b, c are nonzero rationals such that the product of any two of them is one less than a square. 1.
Integer Factorization
, 2006
"... Factorization problems are the “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,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the sc ..."
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Cited by 10 (1 self)
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Factorization problems are the “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,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “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.” But what exactly is the problem? It turns out that there are many different factorization problems, as we will discuss in this paper.
The Probability That The Number Of Points On An Elliptic Curve Over A Finite Field Is Prime
 Journal of the London Mathematical Society
"... . The paper gives a formula for the probability that a randomly chosen elliptic curve over a nite eld has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen ..."
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Cited by 10 (1 self)
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. The paper gives a formula for the probability that a randomly chosen elliptic curve over a nite eld has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a nite eld has kq points where k is a small number and where q is a prime. 1. Introduction Cryptographic and computational applications have recently motivated the study of several questions in the theory of elliptic curves over nite elds. For instance, the analysis of the elliptic curve factoring method leads to estimates ([7], [8]) for the probability that the number of points on an elliptic curve is smooth. In this paper, motivated by the use of elliptic curves in public key cryptosystems, we consider the \opposite" problem. More specically, we ask the question: What is the probability that a randomly chosen elliptic curve over F p has kq points, where k is sm...
AreaTime Efficient Hardware Architecture for Factoring Integers with the Elliptic Curve Method
 IEE Proceedings Information Security
, 2005
"... Abstract: Since the introduction of public key cryptography, the problem of factoring large composites has been of increased interest. The security of the most popular asymmetric cryptographic scheme RSA depends on the hardness of factoring large numbers. The best known method for factoring large in ..."
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Cited by 9 (3 self)
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Abstract: Since the introduction of public key cryptography, the problem of factoring large composites has been of increased interest. The security of the most popular asymmetric cryptographic scheme RSA depends on the hardness of factoring large numbers. The best known method for factoring large integers is the general number field sieve (GNFS). One important step within the GNFS is the factorization of midsize numbers for smoothness testing, an efficient algorithm for which is the elliptic curve method (ECM). Since smoothness testing is also suitable for parallelization, the implementation of ECM in hardware is promising. We show that massive parallel and costefficient ECM hardware engines can improve the area–time product of the RSA moduli factorization via the GNFS considerably. The computation of ECM is a classic example of an algorithm that can be significantly accelerated through specialpurpose hardware. We thoroughly analyse the prerequisites for an area–time efficient hardware architecture for ECM. We present an implementation of ECM to factor numbers up to 200 bits, which is also scalable to other bit lengths. ECM is realized as a software–hardware codesign on a fieldprogrammable gate array (FPGA) and an embedded microcontroller (systemonchip). Furthermore, we provide estimates for stateoftheart CMOS implementation of the design and for the application of massive parallel ECM engines to the GNFS. This appears to be the first publication of a realized hardware implementation of ECM, and the first description of GNFS acceleration through hardwarebased ECM. 1
Diophantine triples and construction of highrank elliptic curves over Q with three nontrivial 2torsion points
 Rocky Mountain J. Math.30
"... Let E be an elliptic curve over Q. The famous theorem of MordellWeil states that E(Q) ≃ E(Q)tors × Z r, ..."
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Cited by 7 (6 self)
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Let E be an elliptic curve over Q. The famous theorem of MordellWeil states that E(Q) ≃ E(Q)tors × Z r,