Advances in Cryptology — EUROCRYPT 2000: International by Stefania Cavallar, Bruce Dodson, Arjen K. Lenstra, Walter

By Stefania Cavallar, Bruce Dodson, Arjen K. Lenstra, Walter Lioen, Peter L. Montgomery (auth.), Bart Preneel (eds.)

This booklet constitutes the refereed lawsuits of the overseas convention at the concept and alertness of Cryptographic suggestions, EUROCRYPT 2000, held in Bruges, Belgium, in may well 2000. The 39 revised complete papers offered have been rigorously chosen from a complete of a hundred and fifty submissions in the course of a hugely aggressive reviewing strategy. The ebook is split in topical sections of factoring and discrete logarithm, electronic signatures, inner most details retrieval, key administration protocols, threshold cryptography, public-key encryption, quantum cryptography, multi-party computation and knowledge idea, zero-knowledge, symmetric cryptography, Boolean features and undefined, vote casting schemes, and circulation ciphers and block ciphers.

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Example text

Indeed the expected running time is O ( 2 ) for curves of small genus and therefore it is faster than Pollard Rho as soon as the genus is greater than 4, as explained in the following table: 1 Rho Index 2 1/2 2 2 3 4 5 6 7 3/2 2 5/2 3 7/2 2 2 2 2 2 Practical experiments have shown that this algorithm is efficient in practice, and a genus 6 example was broken by this technique. Hence it seems that there is no point in using hyperelliptic cryptosystem with genus other than 2, 3 or 4, because for a higher genus, the size of the key has to be chosen larger in order to guarantee a given level of security.

This construction provides then the so-called Frobenius automorphism defined by → q , which can be applied to each coordinate of a point of the curve and gives therefore an automorphism of order n. Another construction, which is a bit harder than the previous (see [42] [7] [3], comes from the theory of complex multiplication. This theory allows to build a curve starting from its ring of endomorphisms. In some cases, this ring contains units of finite order, and then there is an automorphism on the curve corresponding to this unit.

The Hasse-Weil √ √ bound gives a precise interval for this order: ( −1)2g ≤ #J (C) ≤ ( +1)2g . In [4], Cantor gave an efficient algorithm for the computation of the group law. We do not recall his method, but we recall the representation of the elements. = of Proposition 1 In every class of divisors in J (C), there exists an unique divisor = 1 + · · ·+ g − ∞, such that for all i= , i and j are not symmetric points. Such a divisor is called reduced, and there is a unique representation of by two polynomials [u v], such that deg v deg u ≤ , and u divides v2 + v− f.

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