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Smooth ideals in hyperelliptic function fields
 Math.Comp., posted on October 4, 2001, PII
"... Abstract. Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of highgenus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently de ..."
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Cited by 9 (7 self)
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Abstract. Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of highgenus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not exploit analytic properties of generating functions. 1.
Irreducible Polynomials of Given Forms
, 1999
"... We survey under a unified approach on the number of irreducible polynomials of given forms: x + g(x) where the coefficient vector of g comes from an affine algebraic variety over Fq . For instance, all but 2 log n coefficients of g(x) are prefixed. The known results are mostly for large q and little ..."
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Cited by 7 (4 self)
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We survey under a unified approach on the number of irreducible polynomials of given forms: x + g(x) where the coefficient vector of g comes from an affine algebraic variety over Fq . For instance, all but 2 log n coefficients of g(x) are prefixed. The known results are mostly for large q and little is know when q is small or fixed. We present computer experiments on several classes of polynomials over F 2 and compare our data with the results that hold for large q. We also mention some related applications and problems of (irreducible) polynomials with special forms.
Faster index calculus for the medium prime case. application to 1175bit and 1425bit finite fields. Cryptology ePrint Archive, Report 2012/720, 2012. http: //eprint.iacr.org
"... Abstract. Many index calculus algorithms generate multiplicative relations between smoothness basis elements by using a process called Sieving. This process allows to filter potential candidate relations very quickly, without spending too much time to consider bad candidates. However, from an asympt ..."
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Cited by 6 (3 self)
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Abstract. Many index calculus algorithms generate multiplicative relations between smoothness basis elements by using a process called Sieving. This process allows to filter potential candidate relations very quickly, without spending too much time to consider bad candidates. However, from an asymptotic point of view, there is not much difference between sieving and straightforward testing of candidates. The reason is that even when sieving, some small amount time is spend for each bad candidates. Thus, asymptotically, the total number of candidates contributes to the complexity. In this paper, we introduce a new technique: Pinpointing, which allows us to construct multiplicate relations much faster, thus reducing the asymptotic complexity of relations ’ construction. Unfortunately, we only know how to implement this technique for finite fields which contain a mediumsized subfield. When applicable, this method improves the asymptotic complexity of the index calculus algorithm in the cases where the sieving phase dominates. In practice, it gives a very interesting boost to the performance of stateoftheart algorithms. We illustrate the feasability of the method with a discrete logarithm record in medium prime finite fields of sizes 1175 bits and 1425 bits. 1
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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Cited by 3 (1 self)
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
Discrete logarithms: The past and the future
 Designs, Codes, and Cryptography
, 2000
"... The first practical public key cryptosystem to be published, the DiffieHellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. ..."
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The first practical public key cryptosystem to be published, the DiffieHellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. While there have been substantial advances in discrete log algorithms in the last two decades, in general the discrete log still appears to be hard, especially for some groups, such as those from elliptic curves. Unfortunately no proofs of hardness are available in this area, so it is necessary to rely on experience and intuition in judging what parameters to use for cryptosystems. This paper presents a brief survey of the current state of the art in discrete logs. 1. Introduction Many of the popular public key cryptosystems are based on discrete exponentiation. If G is a group, such as the multiplicative group of a finite field or the group of points on an elliptic curve, and g is an elem...
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"... A new index calculus algorithm with complexity L(1/4 + o(1)) in very small characteristic ..."
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A new index calculus algorithm with complexity L(1/4 + o(1)) in very small characteristic
Evaluation Report on the Discrete Logarithm Problem over finite fields
"... This document is an evaluation of the discrete logarithm problem over finite fields (DLP), as a basis for designing cryptographic schemes. It relies on the analysis of numerous research papers on the subject. The present report is organized as follows: firstly, we review the DLP and several ..."
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This document is an evaluation of the discrete logarithm problem over finite fields (DLP), as a basis for designing cryptographic schemes. It relies on the analysis of numerous research papers on the subject. The present report is organized as follows: firstly, we review the DLP and several
on Factoring Polynomials Over Finite Fields: A Survey
"... This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem. c ○ 2001 Academic Press 1. ..."
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This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem. c ○ 2001 Academic Press 1.
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"... A new index calculus algorithm with complexity L(1/4 + o(1)) in small characteristic ..."
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A new index calculus algorithm with complexity L(1/4 + o(1)) in small characteristic