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Simple Extractors for All MinEntropies and a New PseudoRandom Generator
 Journal of the ACM
, 2001
"... A “randomness extractor ” is an algorithm that given a sample from a distribution with sufficiently high minentropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Minentropy is a measure of the amount of randomness in a distribution). We present a ..."
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Cited by 113 (30 self)
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A “randomness extractor ” is an algorithm that given a sample from a distribution with sufficiently high minentropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Minentropy is a measure of the amount of randomness in a distribution). We present a simple, selfcontained extractor construction that produces good extractors for all minentropies. Our construction is algebraic and builds on a new polynomialbased approach introduced by TaShma, Zuckerman, and Safra [TSZS01]. Using our improvements, we obtain, for example, an extractor with output length m = k/(log n) O(1/α) and seed length (1 + α) log n for an arbitrary 0 < α ≤ 1, where n is the input length, and k is the minentropy of the input distribution. A “pseudorandom generator ” is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform. Our technique also gives a new way to construct pseudorandom generators from functions that require large circuits. Our pseudorandom generator construction is not based on the NisanWigderson generator [NW94], and turns worstcase hardness directly into pseudorandomness. The parameters of our generator match those in [IW97, STV01] and in particular are strong enough to obtain a new proof that P = BP P if E requires exponential size circuits.
On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields
, 2000
"... In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, defined on the points of an elliptic curve over a prime finite field. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over Fq among certain ..."
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Cited by 17 (12 self)
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In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, defined on the points of an elliptic curve over a prime finite field. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over Fq among certain sets of bounded size. We apply this estimate to obtain a deterministic O(q 1=2+" ) algorithm for finding generators of the group in echelon form, and in particular to determine its group structure.
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 8 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
EXPLICIT CONSTRUCTIONS OF RIP MATRICES AND RELATED PROBLEMS
"... Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε> 0, large N and any n satisfying N 1−ε ≤ n ≤ N, we construct RIP matrices of order k 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proo ..."
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Cited by 4 (1 self)
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Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε> 0, large N and any n satisfying N 1−ε ≤ n ≤ N, we construct RIP matrices of order k 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1 ≤ k ≤ N (Turán’s power sum problem), which improves upon known explicit constructions when (log N) 1+o(1) ≤ n ≤ (log N) 4+o(1). This latter construction produces elementary explicit examples of n × N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (log N) 1+o(1) ≤ n ≤ (log N) 5/2+o(1). 1.
Approximate Constructions In Finite Fields
"... this paper are new, we do not give complete detailed proofs but indicate the underlying ideas. Here we present a list of possible applications (which is certainly incomplete). We start from pointing out some general purpose applications: ffl Coding Theory : AP1, AP3, AP6 ..."
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Cited by 1 (1 self)
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this paper are new, we do not give complete detailed proofs but indicate the underlying ideas. Here we present a list of possible applications (which is certainly incomplete). We start from pointing out some general purpose applications: ffl Coding Theory : AP1, AP3, AP6
Open Problems on Exponential and Character Sums
, 2009
"... This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of selfreferences in the bibliography. By no means should this be considered as an indication of anything e ..."
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This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of selfreferences in the bibliography. By no means should this be considered as an indication of anything else than
GENERATORS OF ELLIPTIC CURVES OVER FINITE
"... Abstract. We prove estimates on character sums on the subset of points of an elliptic curve over IFq n with xcoordinate of the form α + t where t ∈ IFq varies and fixed α is such that IFq n = IFq(α). We deduce that, for a suitable choice of α, this subset has a point of maximal order in E(IFq n). T ..."
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Abstract. We prove estimates on character sums on the subset of points of an elliptic curve over IFq n with xcoordinate of the form α + t where t ∈ IFq varies and fixed α is such that IFq n = IFq(α). We deduce that, for a suitable choice of α, this subset has a point of maximal order in E(IFq n). This provides a deterministic algorithm for finding a point of maximal order which for a very wide class of finite fields is faster than other available algorithms. 1.
Fast reductions from RAMs to . . .
, 2012
"... Succinct arguments for NP are proof systems that allow a weak verifier to retroactively check computation done by a more powerful prover. These protocols prove membership in languages (consisting of succinctlyrepresented very large constraint satisfaction problems) that, alas, are unnatural in the s ..."
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Succinct arguments for NP are proof systems that allow a weak verifier to retroactively check computation done by a more powerful prover. These protocols prove membership in languages (consisting of succinctlyrepresented very large constraint satisfaction problems) that, alas, are unnatural in the sense that the problems that arise in practice are not in such form. For general computation tasks, the most natural and efficient representation is typically as randomaccess machine (RAM) algorithms, because such a representation can be obtained very efficiently by applying a compiler to code written in a highlevel programming language. We thus study efficient reductions from RAM to other problem representations for which succinct arguments are known. Specifically, we construct reductions from the correctness of computation of a Tstep nondeterministic randomaccess machine to: 1. (succinct) circuit satisfiability with O(log T) overhead, and 2. (succinct) algebraic constraint satisfaction with O(log 2 T) overhead. On the latter problem representation, the best known Probabilistically Checkable Proofs can be directly invoked. Our constructions are explicit and do not hide large constants. To attain these, we develop a set of tools (both unconditional and leveraging computational assumptions) for generically and efficiently structuring and arithmetizing the computation of randomaccess machines.
Fast reductions from RAMs to delegatable . . .
, 2012
"... Succinct arguments for NP are proof systems that allow a weak verifier to retroactively check computation done by a powerful prover. Constructions of such protocols prove membership in languages consisting of very large yet succinctlyrepresented constraint satisfaction problems that, alas, are unna ..."
Abstract
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Succinct arguments for NP are proof systems that allow a weak verifier to retroactively check computation done by a powerful prover. Constructions of such protocols prove membership in languages consisting of very large yet succinctlyrepresented constraint satisfaction problems that, alas, are unnatural in the sense that the problems that arise in practice are not in such form. For general computation tasks, the most natural representation is typically as randomaccess machine (RAM) algorithms, because such a representation can be obtained very efficiently by applying a compiler to code written in a highlevel programming language. Thus, understanding the efficiency of reductions from RAM computations to other NPcomplete problem representations for which succinct arguments (or proofs) are known is a prerequisite to a more complete understanding of the applicability of these arguments. Existing succinct argument constructions rely either on circuit satisfiability or (in PCPbased constructions) on algebraic constraint satisfaction problems. In this paper, we present new and more efficient reductions from RAM (and parallel RAM) computations to both problems that (a) preserve succinctness (i.e., do not “unroll” the computation of a machine), (b) preserve zeroknowledge