By Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)
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"The booklet comprises 39 articles approximately computational algebraic quantity concept, mathematics geometry and cryptography. … The articles during this booklet replicate the huge curiosity of the organizing committee and the individuals. The emphasis lies at the mathematical conception in addition to on computational effects. we suggest the ebook to scholars and researchers who are looking to examine present examine in quantity concept and mathematics geometry and its applications." (R. Carls, Nieuw Archief voor Wiskunde, Vol. 6 (3), 2005)
Read Online or Download Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings PDF
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Extra info for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings
Org, 2002. Number 2002/008. 4. D. Boneh and M. Franklin. Identity-based encryption from the Weil pairing. In J. Kilian, editor, Proceedings of CRYPTO’2001, volume 2139 of Lecture Notes in Comput. , pages 213–229. Springer, 2001. 5. D. Boneh, B. Lynn, and H. Shacham. Short signatures from the Weil pairing. In C. Boyd, editor, Proceedings of ASIACRYPT’2001, volume 2248 of Lecture Notes in Comput. , pages 514–532. Springer, 2001. Updated version available from the authors. 6. S. Brands. An eﬃcient oﬀ–line electronic cash system based on the representation problem.
Indeed, if we could ﬁnd a group morphism from a third group G3 to (one of the many possible) G1 , deciding DDH in G3 would become easy. This would become extremely interesting if G3 could be chosen as the multiplicative subgroup of order of Fqr . Indeed, this would give a partial solution to solve DDH in ﬁnite ﬁeld and would have a wide impact on many cryptographic schemes. Such an “attack” was recently proposed in . It requires the construction of a special auxiliary curve, whose existence is conjectured by Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems 29 the authors of .
Let P = P0 and P = mP . Let t0 = x(P0 ), t = x(P ), and t = x(P ). Then conditions (1), (2), and (3) in the deﬁnition of S are satisﬁed, and (4) and (5) follow from Lemmas 9 and 11, respectively. Hence m2 ∈ S. Now suppose that μ ∈ S. We wish to show that μ ∈ OF . Fix P0 , P , P , t0 , t, t satisfying (1) through (5). By (4) and Lemma 9, P = mP for some nonzero m ∈ Z. By Lemma 11, den(t) | num((t/t − m2 )2 ). On the other hand, (5) says that den(t) | num((t/t − μ)2 ). Therefore den(t)1/2 | num(μ − m2 ) = (μ − m2 ).
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.)