By Rosario Gennaro, Daniele Micciancio (auth.), Lars R. Knudsen (eds.)
This e-book constitutes the refereed complaints of the overseas convention at the concept and alertness of Cryptographic suggestions, EUROCRYPT 2002, held in Amsterdam, The Netherlands, in April/May 2002.
The 33 revised complete papers awarded 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 conception and new types, implementational research, circulation ciphers, electronic signatures, key trade, modes of operation, traitor tracing and id-based encryption, multiparty and multicast, and symmetric cryptology.
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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
Chaining these isogenies together we obtain the desired map from E1 to E2 . In practice we do not actually write down the isogeny but simply evaluate the isogeny on the points of interest. √ The ideal b in Stage 2 will have norm at most O( ∆). The smooth representation of the ideal equivalent to a will have at most O(log ∆) not necessarily distinct factors in it, each factor corresponding to an isogeny of degree at most L. Hence, the mapping of points from E1 to E2 , given the smooth representation of the ideal equivalent to a, can be performed in time polynomial in log q n .
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. Therefore the complexity of Stage 2 is q n/4+ .
Cohen, A course in computational number theory. Springer GTM 138 1993. 5. -M. Couveignes. Computing l-isogenies using the p-torsion. Algorithmic Number Theory Symposium- ANTS II, Springer-Verlag LNCS 1122, 59–65, 1996. 6. G. Frey. How to disguise an elliptic curve. Talk at ECC’ 98, Waterloo. 7. G. Frey and H. R¨ uck. A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. , 62, 865–874, 1994. 8. D. Galbraith. Constructing isogenies between elliptic curves over ﬁnite ﬁelds.