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16
Speeding Up The Computations On An Elliptic Curve Using AdditionSubtraction Chains
 Theoretical Informatics and Applications
, 1990
"... We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up acco ..."
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Cited by 100 (4 self)
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We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up accordingly the factorization and primality testing algorithms using elliptic curves. 1. Introduction. Recent algorithms used in primality testing and integer factorization make use of elliptic curves defined over finite fields or Artinian rings (cf. Section 2). One can define over these sets an abelian law. As a consequence, one can transpose over the corresponding groups all the classical algorithms that were designed over Z/NZ. In particular, one has the analogue of the p \Gamma 1 factorization algorithm of Pollard [29, 5, 20, 22], the Fermatlike primality testing algorithms [1, 14, 21, 26] and the public key cryptosystems based on RSA [30, 17, 19]. The basic operation performed on an elli...
Parallel Algorithms for Integer Factorisation
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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Cited by 41 (17 self)
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiplepolynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.
SPRNG: A Scalable Library for Pseudorandom Number Generation
"... In this article we present background, rationale, and a description of the Scalable Parallel Random
Number Generators (SPRNG) library. We begin by presenting some methods for parallel pseudorandom number generation. We will focus on methods based on parameterization, meaning that we will not conside ..."
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Cited by 29 (6 self)
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In this article we present background, rationale, and a description of the Scalable Parallel Random
Number Generators (SPRNG) library. We begin by presenting some methods for parallel pseudorandom number generation. We will focus on methods based on parameterization, meaning that we will not consider splitting methods such as the leapfrog or blocking methods. We describe in detail
parameterized versions of the following pseudorandom number generators: (i) linear congruential
generators, (ii) shiftregister generators, and (iii) laggedFibonacci generators. We briey describe
the methods, detail some advantages and disadvantages of each method, and recount results from
number theory that impact our understanding of their quality in parallel applications.
SPRNG was designed around the uniform implementation of dierent families of parameterized random number
generators. We then present a short description of
SPRNG. The description contained within this
document is meant only to outline the rationale behind and the capabilities of SPRNG. Much more
information, including examples and detailed documentation aimed at helping users with putting
and using SPRNG on scalable systems is available at the URL:
http://sprng.cs.fsu.edu/RNG. In this description of SPRNG we discuss the random number generator library as well as the suite of
tests of randomness that is an integral part of SPRNG. Random number tools for parallel Monte
Carlo applications must be subjected to classical as well as new types of empirical tests of ran
domness to eliminate generators that show defects when used in scalable environments.
Factorization Of The Tenth Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
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Cited by 22 (10 self)
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We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
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...
On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) ..."
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Cited by 14 (5 self)
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For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) = C 2 n nxD 2 n ; Here A n (x); : : : ; D n (x) are polynomials with integer coecients. We show how these coef cients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n (x); : : : ; D n (x), and illustrate the application to integer factorization with some numerical examples.
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 impl ..."
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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. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...
Answers To Frequently Asked Questions About Today's Cryptography
, 1993
"... this document, authentication will generally refer to the use of digital signatures, which play a function for digital documents similar to that played by handwritten signatures for printed documents: the signature is an unforgeable piece of data asserting that a named person wrote or otherwise agre ..."
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Cited by 8 (0 self)
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this document, authentication will generally refer to the use of digital signatures, which play a function for digital documents similar to that played by handwritten signatures for printed documents: the signature is an unforgeable piece of data asserting that a named person wrote or otherwise agreed to the document to which the signature is attached. The recipient, as well as a third party, can verify both that the document did indeed originate from the person whose signature is attached and that the document has not been altered since it was signed. A secure digital signature system thus consists of two parts: a method of signing a document such that forgery is infeasible, and a method of verifying that a signature was actually generated by whomever it represents. Furthermore, secure digital signatures cannot be repudiated; i.e., the signer of a document cannot later disown it by claiming it was forged.
Some Methods Of Parallel Pseudorandom Number Generation
 in Proceedings of the IMA Workshop on Algorithms for Parallel Processing
, 1997
"... . We detail several methods used in the production of pseudorandom numbers for scalable systems. We will focus on methods based on parameterization, meaning that we will not consider splitting methods. We describe parameterized versions of the following pseudorandom number generation: 1. linear cong ..."
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Cited by 5 (1 self)
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. We detail several methods used in the production of pseudorandom numbers for scalable systems. We will focus on methods based on parameterization, meaning that we will not consider splitting methods. We describe parameterized versions of the following pseudorandom number generation: 1. linear congruential generators 2. linear matrix generators 3. shiftregister generators 4. laggedFibonacci generators 5. inversive congruential generators We briefly describe the methods, detail some advantages and disadvantages of each method and recount results from number theory that impact our understanding of their quality in parallel applications. Several of these methods are currently part of scalable library for pseudorandom number generation, called the SPRNG package available at the URL: www.ncsa.uiuc.edu/Apps/CMP/RNG. Key words. pseudorandom number generation, parallel computing, linear congruential, laggedFibonacci, inversive congruential, shiftregister AMS(MOS) subject classifications....