By Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)
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"The e-book comprises 39 articles approximately computational algebraic quantity conception, mathematics geometry and cryptography. … The articles during this ebook mirror the wide curiosity of the organizing committee and the individuals. The emphasis lies at the mathematical conception in addition to on computational effects. we suggest the e-book to scholars and researchers who are looking to examine present learn in quantity thought and mathematics geometry and its applications." (R. Carls, Nieuw Archief voor Wiskunde, Vol. 6 (3), 2005)
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Extra info for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings
One of its main drawback is the fact that in some cases it does not reach the optimal value for r deﬁned above. On the other hand, the Tate pairing always reaches this optimal value. For this reason, Frey, M¨ uller and R¨ uck proposed in  to use it as a replacement for the Weil pairing. Moreover, it is somewhat less costly from a computational point of view (see [3,17]). In cryptographic applications, the terms of Weil and Tate pairings are also used, somewhat abusively, to denote various modiﬁed pairings based upon the original ones.
P ≡ 3 p+1 2 p ≡ 2 p+1 2 p ≡ 2 p2 − p + 1 3 n ≡ ±1 3n + 3 n+1 2 +1 6 n ≡ ±5 3n − 3 n+1 2 +1 6 n ≡ ±1 3n − 3 n+1 2 +1 6 n ≡ ±5 3n + 3 n+1 2 +1 6 Fig. 1. Some supersingular curves and their distorsions Another approach is to use non-supersingular curves. In that case, the pairings are still deﬁned, however, in general, the security parameter r is so large that computations in Fqr cannot be performed. Yet, using complex multiplication techniques, it is possible to construct curves with reasonably small values of r.
Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings by Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)