By Rosario Gennaro, Daniele Micciancio (auth.), Lars R. Knudsen (eds.)

ISBN-10: 3540435530

ISBN-13: 9783540435532

ISBN-10: 3540460357

ISBN-13: 9783540460350

This ebook constitutes the refereed lawsuits of the foreign convention at the conception and alertness of Cryptographic ideas, EUROCRYPT 2002, held in Amsterdam, The Netherlands, in April/May 2002.

The 33 revised complete papers provided have been conscientiously reviewed and chosen from a complete of 122 submissions. The papers are equipped in topical sections on cryptanalysis, public-key encryption, details concept and new versions, implementational research, move ciphers, electronic signatures, key alternate, modes of operation, traitor tracing and id-based encryption, multiparty and multicast, and symmetric cryptology.

**Read or Download Advances in Cryptology — EUROCRYPT 2002: International Conference on the Theory and Applications of Cryptographic Techniques Amsterdam, The Netherlands, April 28 – May 2, 2002 Proceedings PDF**

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**Extra info for Advances in Cryptology — EUROCRYPT 2002: International Conference on the Theory and Applications of Cryptographic Techniques Amsterdam, The Netherlands, April 28 – May 2, 2002 Proceedings**

**Example text**

J. V´elu. Isog´enies entre courbes elliptiques. Comptes Rendus l’Acad. Sci. Paris, Ser. A, 273, 238-241 1971. Appendix: A More Reﬁned Search Strategy Recall that the goal is to produce a representative of each isogeny class of elliptic curves with small values for m. Since r and n are odd we may assume that all elliptic curves have the form E : y 2 + xy = x3 + ax2 + b with a ∈ {0, 1}. If E is an elliptic curve which has a small value for m then E π : y 2 + xy = x3 + ax2 + π(b) is an isogenous elliptic curve.

Lemma 1. Let the notation be as above. Write h(x) = (xn − 1)/f (x) and deﬁne α = h(σ)α. Then (2) B = {g(σ)α : g(x) ∈ Fq [x]}. , over all elements of Fq [x] of degree less than deg(f (x))). Lemma 1 gives an eﬃcient algorithm to compute representatives for each set B which has running time O(#B). Clearly #B = q deg(f (x)) . By taking the union of these sets we obtain all the values for b which we require. In the notation of Theorem 2 we have s values for i and deg(fi (x)) = t, therefore #Bi = q t+1 for each index i and, since the intersection of any two Bi has size q it follows that #(∪i Bi ) = qs(q t − 1) + q as claimed in Theorem 2.

Hence, we require polynomial storage to hold the isogeny as a smooth ideal. To estimate the running time we need to examine the probability of obtaining √ a smooth number. e. the norm of a reduced ideal, factors over a factor base of integers less than L . There is an optimal choice for L , but to obtain our result it is enough to take L = (log(q n ))2 . Standard estimates give an asymptotic smoothness probability of approximately u−u where u = log(∆)/ log(L). In our case the probability is u−u ≈ q n(−1+c/(log log q))/4 for some constant c.

### Advances in Cryptology — EUROCRYPT 2002: International Conference on the Theory and Applications of Cryptographic Techniques Amsterdam, The Netherlands, April 28 – May 2, 2002 Proceedings by Rosario Gennaro, Daniele Micciancio (auth.), Lars R. Knudsen (eds.)

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