Weil Descent of Elliptic Curves over Finite Fields of Characteristic Three

Arita, Seiko

doi:10.1007/3-540-44448-3_19

Cited by 15 publications

(15 citation statements)

References 14 publications

(12 reference statements)

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“…But this proved not to be feasible. On the other hand, an approach based on the Weil restriction process [17,14,2,20] produced important results: taking as input a discrete logarithm problem in an elliptic curve defined over an extension field, it is possible to transport it into the Jacobian of a curve of larger genus, but defined over a smaller base field than the initial field. Since there exist sub-exponential algorithms for discrete logarithms in Jacobians of high genus curves [1,9,3,21,6,7,10], in some cases this yields a faster attack than Pollard's Rho [27,25].…”

Section: Introductionmentioning

confidence: 99%

Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem

Gaudry

2009

Journal of Symbolic Computation

138

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International audienceWe propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties of small dimension. The main difference with the previous approaches is that we do not make use of any embedding into the Jacobian of a well-suited curve. We apply this algorithm to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields. In particular, our attack can solve an elliptic curve discrete logarithm problem defined over GF(q^3) in heuristic asymptotic running time O~(q^(4/3)); and an elliptic problem over GF(q^4) or a genus 2 problem over GF(q^2) in heuristic asymptotic running time O~(q^(3/2))

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Section: Introductionmentioning

confidence: 99%

Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem

Gaudry

2009

Journal of Symbolic Computation

138

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“…Recently elliptic curve based cryptosystems have emerged as a competing alternative to RSA. [8,11,17] …”

Section: Introductionmentioning

confidence: 99%

Comparative Study of Arithmetic and Huffman Data Compression Techniques for Koblitz Curve Cryptography

Rao¹,

Setty²

2011

IJCA

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Over the past 20 years, numerous papers have been written on various aspects of ECC implementation. In this paper we investigate the superiority of the Arithmetic data compression technique over the Huffman data compression technique in reducing the channel bandwidth and the transmission time. The main purpose of data compression is to reduce the memory space or transmission time, while that of cryptography is to ensure the security of the data. Applying Data compression techniques not only reduces the bandwidth but also enhances the strength of the cryptosystem. It is also observed that even if the given string is doubled i.e. AAAA (4A's) to AAAAAAAA (8A's), the compression ratio remains constant. Further in Arithmetic Data Compression the compression ratio is 50% more when compared to the Huffman Data Compression and the ratio increases with increasing string length.

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“…For efficiency reasons, one usually considers supersingular curves. An alternative Weil descent construction for ordinary elliptic curves in characteristic three is described in [1].…”

Section: Characteristic Threementioning

confidence: 99%

“…Since conjugation by elements of U 1ν maps H ν and H to themselves, H ν U 1ν and H U 1ν are subgroups of G. Furthermore, H U 1ν is in fact the normaliser of H ν in G, because G = H U 1 and H ν is normal in H . The factor group H U 1ν /H ν is then a semidirect product of H/H ν and H ν U 1ν /H ν .…”

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confidence: 99%

Generalising the GHS Attack on the Elliptic Curve Discrete Logarithm Problem

Heß

2004

LMS J. Comput. Math.

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The Weil descent construction of the GHS attack on the elliptic curve discrete logarithm problem (ECDLP) is generalised in this paper, to arbitrary Artin-Schreier extensions. A formula is given for the characteristic polynomial of Frobenius for the curves thus obtained, as well as a proof that the large cyclic factor of the input elliptic curve is not contained in the kernel of the composition of the conorm and norm maps. As an application, the number of elliptic curves that succumb to the basic GHS attack is considerably increased, thereby further weakening curves over F 2 155 . Other possible extensions or variations of the GHS attack are discussed, leading to the conclusion that they are unlikely to yield further improvements.

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Weil Descent of Elliptic Curves over Finite Fields of Characteristic Three

Cited by 15 publications

References 14 publications

Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem

Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem

Comparative Study of Arithmetic and Huffman Data Compression Techniques for Koblitz Curve Cryptography

Generalising the GHS Attack on the Elliptic Curve Discrete Logarithm Problem

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