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53
Computing Discrete Logarithms In Quadratic Orders
 J. Cryptology
, 2000
"... . We present efficient algorithms for computing discrete logarithms in the class group of a quadratic order and for principality testing in a real quadratic order, based on the work of Dullmann and Abel. We show how the idea of generating relations with sieving can be applied to improve the performa ..."
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. We present efficient algorithms for computing discrete logarithms in the class group of a quadratic order and for principality testing in a real quadratic order, based on the work of Dullmann and Abel. We show how the idea of generating relations with sieving can be applied to improve the performance of these algorithms. Computational results are presented which demonstrate that our new techniques yield a significant increase in the sizes of discriminants for which these discrete logarithm problems can be solved. 1. Introduction It is wellknown that finite Abelian groups offer an excellent setting for cryptographic protocols [15], in particular, groups G in which the discrete logarithm problem (DLP) is intractable. That is, given g; a 2 G; it should be beyond the reach of an adversary to recover an integer x such that g x = a; or determine that no such x exists. Several types of finite Abelian groups have been proposed for this purpose, including the original idea of the multipl...
Factoring Large Numbers with Programmable Hardware
 ACM/SIGDA International Symposium on FPGAs
, 2000
"... This paper develops and evaluates an architecture for highspeed number factoring on a configurable computing system based on field programmable gate arrays (FPGA) 1. Currently, the primary interest in factoring large integers is to test the integrity of a number ..."
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This paper develops and evaluates an architecture for highspeed number factoring on a configurable computing system based on field programmable gate arrays (FPGA) 1. Currently, the primary interest in factoring large integers is to test the integrity of a number
Cryptanalysis of RSA using the ratio of the primes
 In: B. Preneel (Ed.) Africacrypt 2009, LNCS 5580
, 2009
"... Abstract. Let N = pq be an RSA modulus, i.e. the product of two large unknown primes of equal bitsize. In the X9.311997 standard for public key cryptography, Section 4.1.2, there are a number of recommendations for the generation of the primes of an RSA modulus. Among them, the ratio of the primes ..."
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Abstract. Let N = pq be an RSA modulus, i.e. the product of two large unknown primes of equal bitsize. In the X9.311997 standard for public key cryptography, Section 4.1.2, there are a number of recommendations for the generation of the primes of an RSA modulus. Among them, the ratio of the primes shall not be close to the ratio of small integers. In this paper, we show that if the public exponent e satisfies an equation eX − (N − (ap + bq))Y = Z with suitably small integers X, Y, Z, where a q is an unknown convergent of the continued fraction expansion of b p, then N can be factored efficiently. In addition, we show that the number of such exponents is at least N 3 4 −ε where ε is arbitrarily small for large N.
On Quadratic Polynomials for the Number Field Sieve
 Australian Computer Science Communications
, 1997
"... . The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this pa ..."
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. The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this paper we examine the smoothness properties of integer values taken by these polynomials. Given a quadratic NFS polynomial f , let \Delta be its discriminant. We show that a prime p can divide values taken by f only if (\Delta=p) = 1. We measure the effect of this residuosity property on the smoothness of fvalues by adapting a parameter ff, developed for analysis of MPQS, to quadratic NFS polynomials. We estimate the yield of smooth values for these polynomials as a function of ff, and conclude that practical changes in ff might bring significant changes in the yield of smooth and almost smooth polynomial values. Keywords: integer factorisation, number field sieve 1
Integer Factoring
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
The Number of Relations in the Quadratic Sieve Algorithm
, 1996
"... The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relatio ..."
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The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relations is then computed by a known formula. Hence practical hints for the optimal choice of the parameter values can be derived. We further compare theoretical estimates for the total number of smooth integers in an interval with countings in practice. AMS Subject Classification (1991): 11A51, 11Y05 CR Subject Classification (1991): F.2.1 Keywords & Phrases: Factorization, Multiple Polynomial Quadratic Sieve, Vector supercomputer, Cluster of work stations 1. Introduction We assume that the reader is familiar with the multiple polynomial quadratic sieve algorithm [Bre89, Pom85, PST88, Sil87, RLW89]. We consider the single large prime variation of the algorithm and write MPQS for short. If we ...
Implementation of the Hypercube Variation of the Multiple Polynomial Quadratic Sieve
, 1995
"... We discuss the implementation of the Hypercube variation of the Multiple Polynomial Quadratic Sieve (HMPQS) integer factorization algorithm. HMPQS is a variation on Pomerance's Quadratic Sieve algorithm which inspects many quadratic polynomials looking for quadratic residues with small prime factors ..."
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We discuss the implementation of the Hypercube variation of the Multiple Polynomial Quadratic Sieve (HMPQS) integer factorization algorithm. HMPQS is a variation on Pomerance's Quadratic Sieve algorithm which inspects many quadratic polynomials looking for quadratic residues with small prime factors. The polynomials are organized as the nodes of an ndimensional cube. Since changing polynomials on the hypercube is cheap, the optimal value for the size of the sieving interval is much smaller than in other implementations of the Multiple Polynomial Quadratic Sieve (MPQS). This makes HMPQS substantially faster than MPQS. We also describe a relatively fast way to find good parameters for the single large prime variation of the algorithm. Finally, we report on the performance of our implementation on factoring several large numbers for the Cunningham Project. Supported by National Science Foundation grant No. CCR9207204 1 Introduction Integer factorization algorithms are usually cate...
Factorization beyond the googol with MPQS on a single computer
 CWI Quarterly
, 1991
"... For the first time a number of more than 100 decimal digits has been factorized on a single computer by means of the Multiple Polynomial Quadratic Sieve method of Kraïtchik and Pomerance (with improvements by Montgomery and Silverman). This method (MPQS) is the best one known to handle numbers which ..."
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For the first time a number of more than 100 decimal digits has been factorized on a single computer by means of the Multiple Polynomial Quadratic Sieve method of Kraïtchik and Pomerance (with improvements by Montgomery and Silverman). This method (MPQS) is the best one known to handle numbers which are the product of two large, approximately equal prime factors. These numbers are being used in cryptography as keys in publickey cryptosystems. The safety of such cryptosystems depends on our ability to factorize these keys. The computer used is the fourprocessor Cray YMP4/464 which was installed
Sieving Methods for Class Group Computation
 PROCEEDINGS OF ALGORITHMIC ALGEBRA AND NUMBER THEORY
, 1997
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