162,796 research outputs found
Near-Optimal Cayley Expanders for Abelian Groups
We give an efficient deterministic algorithm that outputs an expanding generating set for any finite abelian group. The size of the generating set is close to the randomized construction of Alon and Roichman [Alon and Roichman, 1994], improving upon various deterministic constructions in both the dependence on the dimension and the spectral gap. By obtaining optimal dependence on the dimension we resolve a conjecture of Azar, Motwani, and Naor [Azar et al., 1998] in the affirmative. Our technique is an extension of the bias amplification technique of Ta-Shma [Ta-Shma, 2017], who used random walks on expanders to obtain expanding generating sets over the additive group of ???. As a consequence, we obtain (i) randomness-efficient constructions of almost k-wise independent variables, (ii) a faster deterministic algorithm for the Remote Point Problem, (iii) randomness-efficient low-degree tests, and (iv) randomness-efficient verification of matrix multiplication
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
Efficient non-malleable codes and key derivation for poly-size tampering circuits
Non-malleable codes, defined by Dziembowski, Pietrzak, and Wichs (ICS '10), provide roughly the following guarantee: if a codeword c encoding some message x is tampered to c' = f(c) such that c' ≠c , then the tampered message x' contained in c' reveals no information about x. The non-malleable codes have applications to immunizing cryptosystems against tampering attacks and related-key attacks. One cannot have an efficient non-malleable code that protects against all efficient tampering functions f. However, in this paper we show 'the next best thing': for any polynomial bound s given a-priori, there is an efficient non-malleable code that protects against all tampering functions f computable by a circuit of size s. More generally, for any family of tampering functions F of size F ≤ 2s , there is an efficient non-malleable code that protects against all f in F . The rate of our codes, defined as the ratio of message to codeword size, approaches 1. Our results are information-theoretic and our main proof technique relies on a careful probabilistic method argument using limited independence. As a result, we get an efficiently samplable family of efficient codes, such that a random member of the family is non-malleable with overwhelming probability. Alternatively, we can view the result as providing an efficient non-malleable code in the 'common reference string' model. We also introduce a new notion of non-malleable key derivation, which uses randomness x to derive a secret key y = h(x) in such a way that, even if x is tampered to a different value x' = f(x) , the derived key y' = h(x') does not reveal any information about y. Our results for non-malleable key derivation are analogous to those for non-malleable codes. As a useful tool in our analysis, we rely on the notion of 'leakage-resilient storage' of Davì, Dziembowski, and Venturi (SCN '10), and, as a result of independent interest, we also significantly improve on the parameters of such schemes
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