Biblio

A long-standing open question in computational learning theory is to prove NP-hardness of learning efficient programs, the setting of which is in between proper learning and improper learning. Ko (COLT’90, SICOMP’91) explicitly raised this open question and demonstrated its difficulty by proving that there exists no relativizing proof of NP-hardness of learning programs. In this paper, we overcome Ko’s relativization barrier and prove NP-hardness of learning programs under randomized polynomial-time many-one reductions. Our result is provably non-relativizing, and comes somewhat close to the parameter range of improper learning: We observe that mildly improving our inapproximability factor is sufficient to exclude Heuristica, i.e., show the equivalence between average-case and worst-case complexities of N P. We also make progress on another long-standing open question of showing NP-hardness of the Minimum Circuit Size Problem (MCSP). We prove NP-hardness of the partial function variant of MCSP as well as other meta-computational problems, such as the problems MKTP* and MINKT* of computing the time-bounded Kolmogorov complexity of a given partial string, under randomized polynomial-time reductions. Our proofs are algorithmic information (a.k. a. Kolmogorov complexity) theoretic. We utilize black-box pseudorandom generator constructions, such as the Nisan-Wigderson generator, as a one-time encryption scheme secure against a program which “does not know” a random function. Our key technical contribution is to quantify the “knowledge” of a program by using conditional Kolmogorov complexity and show that no small program can know many random functions.

Randomness extractors and error correcting codes are fundamental objects in computer science. Recently, there have been several natural generalizations of these objects, in the context and study of tamper resilient cryptography. These are seeded non-malleable extractors, introduced by Dodis and Wichs; seedless non-malleable extractors, introduced by Cheraghchi and Guruswami; and non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs. Besides being interesting on their own, they also have important applications in cryptography, e.g, privacy amplification with an active adversary, explicit non-malleable codes etc, and often have unexpected connections to their non-tampered analogues. However, the known constructions are far behind their non-tampered counterparts. Indeed, the best known seeded non-malleable extractor requires min-entropy rate at least 0.49; while explicit constructions of non-malleable two-source extractors were not known even if both sources have full min-entropy, and was left as an open problem by Cheraghchi and Guruswami. In this paper we make progress towards solving the above problems and other related generalizations. Our contributions are as follows. (1) We construct an explicit seeded non-malleable extractor for polylogarithmic min-entropy. This dramatically improves all previous results and gives a simpler 2-round privacy amplification protocol with optimal entropy loss, matching the best known result. In fact, we construct more general seeded non-malleable extractors (that can handle multiple adversaries) which were used in the recent construction of explicit two-source extractors for polylogarithmic min-entropy. (2) We construct the first explicit non-malleable two-source extractor for almost full min-entropy thus resolving the open question posed by Cheraghchi and Guruswami. (3) We motivate and initiate the study of two natural generalizations of seedless non-malleable extractors and non-malleable codes, where the sources or the codeword may be tampered many times. By using the connection found by Cheraghchi and Guruswami and providing efficient sampling algorithms, we obtain the first explicit non-malleable codes with tampering degree t, with near optimal rate and error. We call these stronger notions one-many and many-manynon-malleable codes. This provides a stronger information theoretic analogue of a primitive known as continuous non-malleable codes. Our basic technique used in all of our constructions can be seen as inspired, in part, by the techniques previously used to construct cryptographic non-malleable commitments.

We explicitly construct an extractor for two independent sources on n bits, each with polylogarithmic min-entropy. Our extractor outputs one bit and has polynomially small error. The best previous extractor, by Bourgain, required each source to have min-entropy .499n. A key ingredient in our construction is an explicit construction of a monotone, almost-balanced Boolean functions that are resilient to coalitions. In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on n bits, where some unknown n-q bits are chosen almost polylogarithmic-wise independently, and the remaining q bits are chosen by an adversary as an arbitrary function of the n-q bits. The best previous construction, by Viola, achieved q quadratically smaller than our result. Our explicit two-source extractor directly implies improved constructions of a K-Ramsey graph over N vertices, improving bounds obtained by Barak et al. and matching independent work by Cohen.