By Oded Goldreich (auth.), Oded Goldreich (eds.)

ISBN-10: 3642226698

ISBN-13: 9783642226694

This e-book provides a suite of 36 items of clinical paintings within the components of complexity thought and foundations of cryptography: 20 learn contributions, thirteen survey articles, and three programmatic and reflective perspective statements. those to this point officially unpublished items have been written via Oded Goldreich, a few in collaboration with different scientists.

The articles integrated during this e-book basically mirror the topical scope of the clinical profession of Oded Goldreich now spanning 3 many years. particularly the themes handled comprise average-case complexity, complexity of approximation, derandomization, expander graphs, hashing services, in the community testable codes, machines that take recommendation, NP-completeness, one-way features, probabilistically checkable proofs, proofs of information, estate trying out, pseudorandomness, randomness extractors, sampling, trapdoor diversifications, zero-knowledge, and non-iterative zero-knowledge.

All in all, this potpourri of experiences in complexity and cryptography constitutes a most beneficial contribution to the sector of theoretical desktop technological know-how established round the own achievements and perspectives of 1 of its awesome representatives.

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Then, for every S ⊂ {0, 1}n of cardinality 2k ≤ 2 , there exists h ∈ H such that 1. No value has more than n preimages under h; that is, |h−1 (β) ∩ S| ≤ n, for every β ∈ {0, 1} . 2. At most 22k− values have more than one preimage under h; that is, |{β ∈ {0, 1} : |h−1 (β) ∩ S| > 1}| ≤ 22k− . Proof: Fixing an arbitrary 2k -subset, S, and uniformly selecting h ∈ H, we consider the probability that the two items (above) hold. Firstly, we consider the probability that h maps n elements of S to the same image.

We then choose m random lattice points u1 , · · · um ∈ C. To do that, we use the basis B = {b1 , · · · , bn } of the lattice. To choose each point, we take a linear combination of the basis vectors c bi with large enough integer coeﬃcients (say, in the range [0, 2n · max(S, |B|)] for some constant c). This gives us some lattice point p. We then “reduce p mod C”. By this we mean that we look at a tiling of the space Rn with the pseudo-cube C, and we compute the location vector of p in its surrounding pseudo-cube.

Since we chose the parameters such that m > n log q, there are collisions in hM . As we will argue below, however, it is infeasible to ﬁnd any of these collisions unless some well known lattice problems have good approximation in the worst case. , a vector s ∈ {0, 1}m in the solution space). O. : Studies in Complexity and Cryptography, LNCS 6650, pp. 30–39, 2011. c Springer-Verlag Berlin Heidelberg 2011 Collision-Free Hashing from Lattice Problems 31 Remark. Using our notation, the candidate one-way function introduced by Ajtai def is f (M, s) = (M, hM (s)).

### Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation: In Collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, M by Oded Goldreich (auth.), Oded Goldreich (eds.)

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