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Pseudorandom Generators Without the XOR Lemma£

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Pseudorandom Generators Without the XOR Lemma£ Pseudorandom generators without the XOR Lemma£ [Extended Abstract] Þ Ü Madhu SudanÝ Luca Trevisan Salil Vadhan ½ ÁÒØÖÓ Impagliazzo and Wigderson [IW97] have recently shown that if This paper continues the exploration of hardness versus random- Ç ´Òµ there exists a decision problem solvable in time ¾ and hav- ness trade-offs, that is, results showing that randomized algorithms Òµ ª´ can be efficiently simulated deterministically if certain complexity- Ò È ing circuit complexity ¾ (for all but finitely many ) then theoretic assumptions are true. We present two new approaches BÈÈ . This result is a culmination of a series of works showing con- nections between the existence of hard predicates and the existence to proving the recent result of Impagliazzo and Wigderson [IW97] Ç ´Òµ of good pseudorandom generators. that, if there is a decision problem computable in time ¾ and ª´Òµ Ò The construction of Impagliazzo and Wigderson goes through having circuit complexity ¾ for all but finitely many , then BÈÈ three phases of “hardness amplification” (a multivariate polyno- È . Impagliazzo and Wigderson prove their result by pre- mial encoding, a first derandomized XOR Lemma, and a second senting a “randomness-efficient amplification of hardness” based derandomized XOR Lemma) that are composed with the Nisan– on a derandomized version of Yao’s XOR Lemma. The hardness- Wigderson [NW94] generator. In this paper we present two dif- amplification procedure is then composed with the Nisan–Wigderson ferent approaches to proving the main result of Impagliazzo and (NW) generator [NW94] and this gives the result. The hardness Wigderson. In developing each approach, we introduce new tech- amplification goes through three steps: an encoding using multi- niques and prove new results that could be useful in future improve- variate polynomials (from [BFNW93]), a first derandomized XOR ments and/or applications of hardness-randomness trade-offs. Lemma (from [Imp95]) and a second derandomized XOR Lemma Our first result is that when (a modified version of) the Nisan- (which is the technical contribution of [IW97]). Wigderson generator construction is applied with a “mildly” hard In our first result, we show how to construct a “pseudoentropy predicate, the result is a generator that produces a distribution indis- generator” starting from a predicate with “mild” hardness. Roughly tinguishable from having large min-entropy. An extractor can then speaking, a pseudoentropy generator takes a short random seed be used to produce a distribution computationally indistinguishable as input and outputs a distribution that is indistinguishable from from uniform. This is the first construction of a pseudorandom gen- having high min-entropy. Combining our pseudoentropy generator erator that works with a mildly hard predicate without doing hard- with an extractor, we obtain a pseudorandom generator. Interest- ness amplification. ingly, our pseudoentropy generator is (a modification of) the NW We then show that in the Impagliazzo–Wigderson construction generator itself. Along the way we prove that, when built out of only the first hardness-amplification phase (encoding with multi- a mildly hard predicate, the NW generator outputs a distribution variate polynomial) is necessary, since it already gives the required that is indistinguishable from having high Shannon entropy, a result average-case hardness. We prove this result by (i) establishing a that has not been observed before. The notion of a pseudoentropy connection between the hardness-amplification problem and a list- generator, and the idea that a pseudoentropy generator can be con- decoding problem for error-correcting codes based on multivariate verted into a pseudorandom generator using an extractor, are due polynomials; and (ii) presenting a list-decoding algorithm that im- to H˚astad et al. [HILL98].½ Our construction is the first construc- proves and simplifies a previous one by Arora and Sudan [AS97]. tion of a pseudorandom generator that works using a mildly hard predicate and without hardness amplification. £ The full version of this paper appears as [STV98]. Ý We then revisit the hardness amplification problem, as consid- Laboratory for Computer Science, 545 Technology Square, MIT, Cambridge, MA 02141. E-mail: [email protected]. ered in [BFNW93, Imp95, IW97], and we show that the first step Þ Department of Computer Science, Columbia University, 500W 120th St., New alone (encoding with multivariate polynomials) is sufficient to am- York, NY 10027. Email: [email protected]. Work done at MIT. plify hardness to the desired level, so that the derandomized XOR Ü Laboratory for Computer Science, 545 Technology Square, MIT, Cam- Lemmas are not necessary in this context. Our proof is based on bridge, MA 02141. E-mail: [email protected]. URL: a list-decoding algorithm for multivariate polynomial codes and http://theory.lcs.mit.edu/˜salil. Supported by a DOD/NDSEG grad- uate fellowship and partially by DARPA grant DABT63-96-C-0018. exploits a connection between the list-decoding and the hardness- amplification problems. The list-decoding algorithm described in this paper is quantitatively better than a previous one by Arora and Sudan [AS97], and has a simpler analysis. ½ To be accurate, the term extractor comes fron [NZ96] and postdates the paper of H˚astad et al. [HILL98]. ÓÚÖÚÛ Ó ÔÖÚÓÙ× Ö×ÙÐØ׺ Ò The works of Blum and in increasingly sophisticated forms, in [Imp95, IW97]. Likewise, Micali [BM84] and Yao [Yao82] introduce the notion of a cryp- the NW generator and its original analysis have always been used tographically strong pseudorandom generator (csPRG) and show in conditional derandomization results since.4 Future progress in how to construct pseudorandom generators based on the existence the area will probably require a departure from this observance of of one-way permutations. A csPRG is a polynomial-time algorithm the NW methodology, or at least a certain amount of revisitation of ¯ that on input a randomly selected string of length Ò produces an its main parts. output of length Ò that is computationally indistinguishable from In this paper, we give two new ways to build pseudorandom Òµ ¯ uniform by any adversary of Ô ÓÐÝ ´ size, where is an arbitrar- generators with seeds of logarithmic length. Both approaches by- ily small constant. Yao also observes that a given polynomial-time pass the need for the XOR Lemma, and instead use tools (such as randomized algorithm can be simulated deterministically using a list decoding, extractors, and pseudoentropy generators) that did ¯ Ò ¾ Ô ÓÐÝ ´Òµ csPRG in time ¡ by trying all the seeds and taking the not appear in the sequence of works from [NW94] to [IW97]. For majority answer. a diagram illustrating the steps leading up to the results of [IW97] In a seminal work, Nisan and Wigderson [NW94] explore the and how our techniques depart from that framework, see Figure 1. use of a weaker type of pseudorandom generator (PRG) in order to Both of our approaches are described in more detail below. derandomize randomized algorithm. They observe that, for the pur- pose of derandomization, one can consider generators computable Ø µ Ô ÓÐÝ ´Øµ Ø in time Ô ÓÐÝ ´¾ (instead of ) where is the length of the Worst−case hard seed, since the derandomization process cycles through all the seeds, polynomial Ø [BFNW93] and this induces an overhead factor ¾ anyway. They also observe encoding that one can restrict to generators that are good against adversaries whose running time is bounded by a fixed polynomial, instead of polynomial Mildly hard encoding Thm 10 XOR [Imp95] every polynomial. They then show how to construct a pseudo- + noisy Lemma random generator meeting this relaxed definition under weaker as- reconstruction (Thm 19) Constant hardness sumptions than those used to build cryptographically strong pseu- XOR Pseudoentropy Lemma [IW97] dorandom generators. Furthermore, they show that, under a suf- Generator ficiently strong assumption, one can build a PRG that uses seeds Extremely hard of logarithmic length (which would be impossible for a csPRG). [NW94] extractor Such a generator can be used to simulate randomized algorithms (Lem 13) BÈÈ in polynomial time, and its existence implies È . The con- Pseudorandom Generator dition under which Nisan and Wigderson prove the existence of a PRG with seeds of logarithmic length is the existence of a decision Ò f¼ ½g ! f¼ ½g problem (i.e., a predicate È ) solvable in time ´Òµ ¯Ò Ç Figure 1: A comparison of our approach with previous ones. Dou- ¯ ¾ ¾ such that for some positive constant no circuit of size ¯Ò ble arrows indicate our results. ¾ · ¾ can solve the problem on more than a fraction ½ of the inputs.¾ This is a very strong hardness requirement, and it is of interest to obtain similar conclusions under weaker assumptions. È×ÙÓ ÒØÖÓÔÝ ÒÖØÓÖº An example of a weaker assumption is the existence of a mildly Nisan and Wigderson show hard predicate. We say that a predicate is mildly hard if for some that when their generator is constructed using a very hard-on-average ¯Ò ¼ ¾ fixed ¯ no circuit of size can decide the predicate on more predicate, then the output of the generator is indistinguishable from ½Ô ÓÐÝ ´Òµ than a fraction ½ of the inputs. Nisan and Wigder- the uniform distribution. It is a natural question to ask what hap- son prove that mild hardness suffices to derive a pseudorandom pens if there are stronger or weaker conditions on the predicate. In ¾ ´ÐÓ Òµ generator with seed of Ç length, which in turn implies a this paper we consider the question of what happens if the predicate quasi-polynomial deterministic simulation of BÈÈ. This result is is only mildly hard. Specifically we are interested in whether expo- proved by using Yao’s XOR Lemma [Yao82] (see, e.g., [GNW95] nential average-case hardness is really necessary for direct pseu- for a proof) to convert a mildly hard predicate over Ò inputs into dorandom generation.
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