Simple Constructions of Almost k‐wise Independent Random Variables

Alon, Noga; Goldreich, Oded; Håstad, Johan; Peralta, René

doi:10.1002/rsa.3240030308

Cited by 446 publications

(373 citation statements)

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“…The large size of the hash function that has to be communicated for privacy amplification can be reduced by using "almost universal" hash functions based on almost k-wise independent random variables that can be constructed efficiently [1]. Such functions g : X --4 y can be described with about 51ogl~ I instead of log IXI bits.…”

mentioning

confidence: 99%

Unconditional security against memory-bounded adversaries

Cachin

Maurer

1997

Lecture Notes in Computer Science

144

135

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Abstract. We propose a private-key cryptosystem and a protocol for key agreement by public discussion that are unconditionally secure based on the sole assumption that an adversary's memory capacity is limited. No assumption about her computing power is made. The scenario assumes that a random bit string of length slightly larger than the adversary's memory capacity can be received by all parties. The random bit string can for instance be broadcast by a satellite or over an optical network, or transmitted over an insecure channel between the communicating parties. The proposed schemes require very high bandwidth but can nevertheless be practical. IntroductionOne of the most important properties of a cryptographic system is a proof of its security under reasonable and general assumptions. However, every design involves a trade-off between the strength of the security and further important qualities of a cryptosystem, such as efficiency and practicality. The security of all currently used cryptosystems is based on the difficulty of an underlying computational problem, such as factoring large numbers or computing discrete logarithms in the case of many public-key systems. Security proofs for these systems show that the ability of the adversary to defeat the cryptosystem with significant probability contradicts the assumed difficulty of the problem [24]. Although the hardness of these problems is unquestioned at the moment, it can be dangerous to base the security of the global information economy on a very small number of mathematical problems. Recent advances in quantum computing show that precisely these two problems, factoring and discrete logarithm, could be solved efficiently if quantum computers could be built [27].An alternative to proofs in the computational security model is offered by the stronger notion of information-theoretic or unconditional security where no limits on an adversary's computational power are assumed. The first informationtheoretic definition of perfect secrecy by Shannon [26] led immediately to his famous impracticality theorem, which states, roughly, that the shared secret key in any perfectly secure cryptosystem must be at least as long as the plaintext * Current address:

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mentioning

confidence: 99%

Unconditional security against memory-bounded adversaries

Cachin

Maurer

1997

Lecture Notes in Computer Science

144

135

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“…k-wise independent random variables were first studied in probability theory [23] and then in complexity theory [13,2,28,29] mainly for derandomization purposes. Constructions of almost k-wise independent distributions were studied in [31,3,6,17,10]. Construction results of non-uniform k-wise independent distributions were given in [24,26].…”

Section: Other Related Researchmentioning

confidence: 99%

Testing Non-uniform k-Wise Independent Distributions over Product Spaces

Rubinfeld

Xie

2010

Automata, Languages and Programming

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Abstract. A distribution D over Σ1 × · · · × Σn is called (non-uniform) k-wise independent if for any set of k indices {i1, . . . , i k } and for any z1We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al.[1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.A full version of this paper is available at

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“…We should mention that there are well known constructions (e.g. [2,14,22,23,26]) for the much stronger property requiring that all possible 2 k patterns appear in the submatrix formed by arbitrary k rows. However, in such examples, N ≤ M or 2 k ≤ M must trivially hold.…”

Section: Lemmamentioning

confidence: 99%