By Shai Halevi

ISBN-10: 3642033555

ISBN-13: 9783642033551

This publication constitutes the refereed complaints of the twenty ninth Annual foreign Cryptology convention, CRYPTO 2009, held in Santa Barbara, CA, united states in August 2009.
The 38 revised complete papers offered have been rigorously reviewed and chosen from 213 submissions. Addressing all present foundational, theoretical and learn points of cryptology, cryptography, and cryptanalysis in addition to complex functions, the papers are equipped in topical sections on key leakage, hash-function cryptanalysis, privateness and anonymity, interactive proofs and zero-knowledge, block-cipher cryptanalysis, modes of operation, elliptic curves, cryptographic hardness, merkle puzzles, cryptography within the actual international, assaults on signature schemes, mystery sharing and safe computation, cryptography and game-theory, cryptography and lattices, identity-based encryption and cryptographers’ toolbox.

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Additional info for Advances in Cryptology - CRYPTO 2009: 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2009, Proceedings (Lecture ... Computer Science / Security and Cryptology)

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14 to compute gcd(23S, 124) including its representation. 15. 4. 18 Let a > b > O. 14 requires O(log b) iterations to compute gcd(a, b). 19 Let a > b > o. Prove that the number ofiterations that the euclidean algorithm needs to compute gcd(a, b) depends only on the ratio a/b. 20 Find a sequence (ai)i~l of positive integers such that the euclidean algorithm needs exactly i iterations to compute gcd(ai+l' ai). 21 Prove that gcd(a, m) = 1 and gcd(b, m) = 1 implies gcd(ab, m) = 1. 22 Compute the prime factorization of 37800.

For all b = (b I , ... , bk) E {O, I}k, we determine k Gb = ngt'· i=l Then O::::j < n. We analyze this algorithm. The computation of the Gb, bE {O, I}k requires Zk - Z multiplications in G. 14. Computation of Element Orders 49 requires n - 1 squarings and multiplications in G. Therefore, the following result is proved. 1 Let kEN, gi E G, ei E Z>o, 1 ::::: i ::::: k, and let n be the maximal binary length of the ei. Then the power product n~=l g:l can be computed using 2k + n - 3 multiplications and n - 1 squarings in G.

Therefore, n k ge = gL~=oe,2' = n(g2')e, = i=O 2' g. O:::;i:::;k,e, =l From this formula, we obtain the following idea: 1. Compute the successive squares g2', a :s i :s k. 2. Determine ge as the product of those g2' for which ei = 1. Observe that 2' 2 g 2'+1 =(g). Therefore, g2'+! can be computed from g2' by one squaring. Before we explain the algorithm in more detail, we give an example to show that this method is much faster than the naive one. 1 We determine 673 mod 100. We write the binary expansion of the exponent: 73 = 1 + 2 3 + 2 6 .

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Advances in Cryptology - CRYPTO 2009: 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2009, Proceedings (Lecture ... Computer Science / Security and Cryptology) by Shai Halevi


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