Journal Article10.1080/10586458.1999.10504623
Sieving in function fields
Ralf Flassenberg,Sachar Paulus +1 more
TL;DR: The first implementation of sieving techniques in the context of function fields is presented, combining the algorithm of Hafner and McCurley with sieving ideas known from factoring to compute in class groups of quadratic congruence function fields.
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Abstract: We present the first implementation of sieving techniques in the context of function fields. More precisely, we compute in class groups of quadratic congruence function fields by combining the algorithm of Hafner and McCurley with sieving ideas known from factoring. We apply our methods to the computation of generators and relations of the Jacobian variety of hyperelliptic curves over fin ite fields. The algorithms introduced here were implemented in C++ with the help of LEDA and LiDIA. We provide examples of running times and comparisons with earlier algorithms.
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Citations
Advances in Cryptology — EUROCRYPT ’99
Jacques Stern
- 01 Jan 1999
TL;DR: This work shows that if the private exponent d used in the RSA public-key cryptosystem is less than N then the system is insecure.
972
Cryptography from Pairings
Kenneth G. Paterson
- 01 Apr 2005
TL;DR: This chapter presents a survey of positive applications of pairings in cryptography, presenting the small number of schemes that the author considers to be the high points in the area and which are likely to have a significant impact on future research.
Weak Fields for ECC
Alfred Menezes,Edlyn Teske,Annegret Weng +2 more
- 23 Feb 2004
TL;DR: It is demonstrated that some finite fields, including \(\mathbb{F}_{{2}^{210}}\), are weak for elliptic curve cryptography in the sense that any instance of the elliptic curves discrete logarithm problem for any elliptic Curve over these fields can be solved in significantly less time than it takes Pollard’s rho method to solve the hardest instances.
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•Posted Content
Weak Fields for ECC.
TL;DR: In this paper, it was shown that some finite fields, such as √ F √ n, are weak for elliptic curve cryptography in the sense that any instance of the ECC discrete logarithm problem over these fields can be solved in significantly less time than it takes Pollard's rho method to solve the hardest instances.
57
On the performance of hyperelliptic cryptosystems
Nigel P. Smart
- 02 May 1999
TL;DR: It is concluded that hyperelliptic curves offer no performance advantage over elliptic curves, and the implementation of the group law on such curves and how to generate suitable curves for use in cryptography is covered.
References
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J. Van Leeuwen
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TL;DR: The Handbook of Theoretical Computer Science provides professionals and students with a comprehensive overview of the main results and developments in this rapidly evolving field.
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Computing in the Jacobian of a hyperelliptic curve
TL;DR: A reduction algorithm is presented which is asymptotically faster than that of Gauss when the genus g is very large and the Jacobian of a hyperelliptic curve is studied.
Explicit bounds for primality testing and related problems
TL;DR: In this article, it was shown that if the Extended Riemann Hypothesis holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log 2m.
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Algorithms in number theory
Arjen K. Lenstra,Hendrik W. Lenstra +1 more
- 02 Jan 1991
TL;DR: This chapter discusses algorithms that solve two basic problems in computational number theory—factoring integers into prime factors and finding discrete logarithms.
Algorithms in number theory
Arjen K. Lenstra,Hendrik W. Lenstra +1 more
- 01 Jan 1990
TL;DR: In this article, the authors discuss algorithms that solve two basic problems in computational number theory (factoring integers into prime factors and finding discrete logarithm) and their analyses depend on many different parts of number theory.
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