J. Hastad. Pseudo-random generation under uniform assumptions. In Proceedings of the 22-nd ACM Symposium on Theory of Computing, pages 395--404, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....family of pseudorandom generators G = fG k : f0;1g k f0;1g 2k g computable in P poly, for every e 0, for sufficiently large values of k, H(G k ) 2 k e . This follows from the above three lemmas and Theorem 2. From the known equivalence of strong PSRGs and strong one way functions (see [10, 8, 9, 25]) we also have: Corollary 8 If for some g 0 there exists a one way function of security 2 n g , then P poly does not have measure zero in EXP. 3.1 Non measurability of P poly We strengthen the conclusion of our main result from not measure zero to not measurable at all, after observing ....

J. Hastad. Pseudorandom generation under uniform assumptions. In Proc. 22ndSTOC, pages 395--404, 1990.

....problem. The work of Yao and Blum and Micali focused on these hard problems. A central open question then was to know if it was possible to construct a pseudorandom generator based on any one way function. Building on nearly a decade of research, Impagliazzo, Levin, Luby [ILL89] and Hastad [Has90] showed that this task was indeed possible. Once again, the single most important idea used in their work was that of a 2 universal hash function. Suppose we have a random element x from a set S of n bit strings, where jSj 2 n . The entropy, or the randomness content, of the string x is log ....

....Turing machine running in time 2 O(n c ) i.e. L 2 EXP. Thus we have constructed a language L 2 P poly EXP on which k does not succeed, whence it follows that (P polyjEXP) 6= 0. From the known equivalence of strong pseudorandom generators and strong one way functions (see [ILL89, Has90, HILL91] we also have: Corollary 3.5 If for some g 0 there exists a one way function of security 2 n g , then P poly does not have measure zero in EXP. Based on assumptions about the hardness of the subset sum problem, Impagliazzo and Naor [IN89] show how to construct a pseudorandom ....

J. Hastad. Pseudorandom generation under uniform assumptions. In Proc. 22nd Annual ACM Symposium on the Theory of Computing, pages 395--404, 1990.

....zero or NP is not measurable at all within EXP. Hence Lutz s conjecture is really that NP is not measurable. Up to now there have not been any general techniques for showing classes C to be non measurable. Our paper gives such a technique, based on the theory of pseudorandom generators (PSRGs) [5, 10, 8, 9]) PSRGs that have exponential hardness , meaning that for some 0 they are unbreakable by 2 n sized circuits, are widely believed to exist. Indeed, the smallest circuits known to break PSRGs based on the discrete logarithm problem have size just short of 2 n 1=2 . Our main theorem, ....

....n ) h(n) A well known robustness theorem (see [5, 9] states that so long as (n) n O(1) H(G n ) is invariant up to constant factors. As Razborov and Rudich do, we work with PSRGs that stretch n bits to 2n bits. We mention in passing the results of Hastad, Impagliazzo, Levin, and Luby [9, 10, 8] proving that for any resource bound class R between n O(1) and 2 n o(1) that is invariant under polynomial scaling, there exists a PSRG of hardness greater than R iff there exists a one way function that is secure against R bounded adversaries. The equivalence holds also in the uniform case ....

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J. Hastad. Pseudorandom generation under uniform assumptions. In Proc. 22nd STOC, pages 395--404, 1990.

....of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 2000 USA. Both authors were supported in part by NSF Grant CCR 9409104 every polynomial p(n) the bias multiplied by p(n) still converges to zero, is an important concept in the study of pseudorandom generators and one way functions (see [13, 11, 12, 24, 25]) We show, in fact, that for a measure one class of languages L 2 E, not only is L weakly stochastic over the same nonuniform classes as in [17, 19] but also the bias vanishes as 1=2 an , where a can be chosen as any constant less than 1 2. Section 2 defines the needed concepts from ....

....and fl, WS[2 cn ; cn; 2 fln ] j E) 1. Our improvement has to do with the speed of convergence toward 1 2, which their results and proofs do not address. This opens up a connection to the important notion of hardness used in research on pseudorandom generators (PSRGs) and one way functions [13, 11, 12, 24, 25]. Here we define an appropriate notion of a language being hard (as compared with a PSRG or one way function being hard) for time advice bounded machines to gain bias (n) Definition 4. a) A language A is hard for time t(n) machines with advice q(n) to achieve bias (n) if for every t(n) time ....

J. Hastad. Pseudorandom generation under uniform assumptions. In Proc. 22nd STOC, pages 395--404, 1990.

....on input n 2 N outputs uniformly at random a description of h 2 H n . It is assumed that all hash functions considered here are both polynomial time computable and accessible. Universal hash functions , first introduced in [CW79] play essential rules in many recent key results in cryptography [Has90, ILL89, Rom90] and theoretical computer science. Definition 6 Let k be a fixed positive integer, and H a hash function compressing (n) bit input into m(n) bit output strings. H is a (strong) universal k hash function if for all n, for all k (distinct) strings x 1 ; x 2 ; x k 2 Sigma (n) and all ....

J. Hastad: "Pseudo-random generation under uniform assumptions", Proceedings of the 22-nd ACM Symposium on Theory of Computing, 1990, pp.395-404.

....secure against all possible attacks consistent with quantum physics. Nevertheless, an interesting protocol results if we are satisfied with computational security. Indeed, it is well known that computationally secure bit commitments are possible under the assumption that one way functions exist [20, 19, 24]. Therefore, quantum physics provides for an OT protocol that is computationally secure against unrestricted technology (including the ability to perform coherent measurements) under the sole assumption that one way functions exist. This is interesting because Impagliazzo and Rudich have proved ....

Hastad, J., "Pseudo-random generation under uniform assumptions", Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 1990, pp. 395 -- 440.

....tosses of A. The concept of pseudo random functions were first introduced by Goldreich, Goldwasser and Micali in [5] In the same paper they have also shown that pseudo random function families can be constructed from pseudo random string generators [5] By a result of Impagliazzo, Levin and Luby [7, 6], the existence of one way functions is sufficient for the construction of pseudo random function families. We are interested in a particular type of pseudo random function families F = S n2N F n , where F n can be represented by F n = ff idx jidx 2 Sigma n ; f idx : Sigma (n) Sigma ....

....DES ( DES 56 ) and transforms an (i 8) bit plaintext into an (i 8) bit ciphertext using an i bit key. 2. 2 Universal Hash Function Families Universal hash function families (UHFFs) 4, 13] play an essential role in many recent major results in cryptography and theoretical computer science [6, 7, 9]. Let H = S n2N H n be a family of functions mapping (n) bit input into m(n) bit output strings. For two strings x; y 2 Sigma (n) with x 6= y, we say that x and y collide with each other under h 2 H n or x and y are siblings under h 2 H n , if h(x) h(y) Definition 2 Let H = S n2N H ....

J. Hastad. Pseudo-random generation under uniform assumptions. In Proceedings of the 22-nd ACM Symposium on Theory of Computing, pages 395--404, 1990.

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J. Hastad. Pseudo-random generation under uniform assumptions. In Proceedings of the 22-nd ACM Symposium on Theory of Computing, pages 395--404, 1990.

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J. Hastad: "Pseudo-random generation under uniform assumptions", Proceedings of the 22-nd ACM Symposium on Theory of Computing, 1990, pp.395-404.

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