Abstract
We give a new construction of pseudorandom generators from any one-way function. The construction achieves better parameters and is simpler than that given in the seminal work of H\aastad, Impagliazzo, Levin, and Luby [SIAM J. Comput., 28 (1999), pp. 1364--1396]. The key to our construction is a new notion of next-block pseudoentropy, which is inspired by the notion of “inaccessible entropy” recently introduced in [I. Haitner, O. Reingold, S. Vadhan, and H. Wee, Proceedings of the $41$st Annual ACM Symposium on Theory of Computing (STOC), 2009, pp. 611--620]. An additional advantage over previous constructions is that our pseudorandom generators are parallelizable and invoke the one-way function in a nonadaptive manner. Using [B. Applebaum, Y. Ishai, and E. Kushilevitz, SIAM J. Comput., 36 (2006), pp. 845--888], this implies the existence of pseudorandom generators in NC$^0$ based on the existence of one-way functions in NC$^1$.
Highlights
The result of Hastad, Impagliazzo, Levin, and Luby [14] that one-way functions imply pseudorandom generators is one of the centerpieces of the foundations of cryptography and the theory of pseudorandomness.From the perspective of cryptography, it shows that a very powerful and useful cryptographic primitive can be constructed from the minimal assumption for complexity-based cryptography
Numerous other cryptographic primitives can be constructed from one-way functions, such as privatekey cryptography [5, 21], bit-commitment schemes [22], zero-knowledge proofs for NP [6], and identification schemes [3]
Harnik, and Reingold [10] show how to save a factor of n in the enumeration step to obtain a seed length of O(n7), but still all of the steps remain
Summary
The result of Hastad, Impagliazzo, Levin, and Luby [14] that one-way functions imply pseudorandom generators is one of the centerpieces of the foundations of cryptography and the theory of pseudorandomness. From the perspective of cryptography, it shows that a very powerful and useful cryptographic primitive (namely, pseudorandom generators) can be constructed from the minimal assumption for complexity-based cryptography (namely, one-way functions). From the perspective of pseudorandomness, it provides strong evidence that pseudorandom bits can be generated very efficiently, with smaller computational resources than the “distinguishers” to whom the bits should look random. Such kinds of pseudorandom generators are needed, for example, for hardness results in learning [27] and the natural proofs barrier for circuit lower bounds [23]. The key to our construction is a new notion of next-block pseudoentropy, which is inspired by the recently introduced notion of “inaccessible entropy” [13]
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