Abstract
We introduce a new approach to constructing extractors. Extractors are algorithms that transform a “weakly random” distribution into an almost uniform distribution. Explicit constructions of extractors have a variety of important applications, and tend to be very difficult to obtain.We demonstrate an unsuspected connection between extractors and pseudorandom generators. In fact, we show that every pseudorandom generator of a certain kind is an extractor.A pseudorandom generator construction due to Impagliazzo and Wigderson, once reinterpreted via our connection, is already an extractor that beats most known constructions and solves an important open question. We also show that, using the simpler Nisan--Wigderson generator and standard error-correcting codes, one can build even better extractors with the additional advantage that both the construction and the analysis are simple and admit a short self-contained description.
- ANDREEV, A. E., CLEMENTI,A.E.F.,AND ROLIM, J. D. P. 1998. Anewgeneral derandomization method. J. ACM 45, 1 (Jan.), 179-213. Google Scholar
Digital Library
- ANDREEV, A., CLEMENTI, A., ROLIM, J., AND TREVISAN, L. 1999. Weak random sources, hitting sets, and BPP simulations. SIAM J. Comput. 28, 6, 2103-2116. Google Scholar
Digital Library
- ARVIND,V.,AND KOBLER, J. 1997. On resource-bounded measure and pseudorandomness. In Proceedings of the 17th Conference on Foundations of Software Technology and Theoretical Computer Science. Lecture Notes in Computer Science, vol. 1346. Springer-Verlag, New York, pp. 235-249. Google Scholar
Digital Library
- BABAI, L., FORTNOW, L., NISAN, N., AND WIGDERSON, A. 1993. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Comput. Compl. 3, 4, 307-318. Google Scholar
Digital Library
- BELLARE, M., GOLDREICH, O., AND SUDAN, M. 1998. Free bits, PCP's and nonapproximability - towards tight results. SIAM J. Comput. 27, 3, 804-915. Google Scholar
Digital Library
- BELLARE, M., AND ROMPEL, J. 1994. Randomness-efficient oblivious sampling. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 276-287.Google Scholar
Digital Library
- BLUM, M., AND MICALI, S. 1984. How to generate cryptographically strong sequences of pseudorandom bits. SIAM J. Comput. 13, 4, 850-864. Google Scholar
Digital Library
- CHOR, B., AND GOLDREICH, O. 1988. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 2 (Apr.), 230-261. Google Scholar
Digital Library
- COHEN, A., AND WIGDERSON, A. 1989. Dispersers, deterministic amplification, and weak random sources. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 14-19.Google Scholar
Digital Library
- GOLDREICH, O. 1999. Modern Cryptography, Probabilistic Proofs and Pseudorandomness. Springer- Verlag, New York. Google Scholar
Digital Library
- GOLDREICH, O., NISAN, N., AND WIGDERSON, A. 1995. On Yao's XOR lemma. Tech. Rep. TR95-50. Electronic Colloquium on Computational Complexity, http://eccc.uni-trier.de/eccc/.Google Scholar
- GOLDREICH, O., AND WIGDERSON, A. 1997. Tiny families of functions with random properties: A quality-size trade-off for hashing. Random Struct. Algor. 11, 4, 315-343. Google Scholar
Digital Library
- GOLDREICH, O., AND ZUCKERMAN, D. 1997. Another proof that BPP ~PH (and more). Tech. Rep. TR97-045. Electronic Colloquium on Computational Complexity, http://eccc.uni-trier.de/eccc/.Google Scholar
- GOLDWASSER, S., AND MICALI, S. 1984. Probabilistic encryption. J. Comput. Syst. Sci. 28, 2, 270- 299.Google Scholar
Cross Ref
- IMPAGLIAZZO, R. 1995. Hard-core distributions for somewhat hard problems. In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 538-545. Google Scholar
Digital Library
- IMPAGLIAZZO, R., AND WIGDERSON, A. 1997. P D BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (El Paso, Tex., May 4-6). ACM, New York, pp. 220-229. Google Scholar
Digital Library
- IMPAGLIAZZO, R., AND WIGDERSON, A. 1998. Randomness versus time: De-randomization under a uniform assumption. In Proceedings of the 39th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 734-743. Google Scholar
Digital Library
- KLIVANS, A., AND VAN MELKEBEEK, D. 1999. Graph nonisomorphism has subexponential size proofs unless the polynomial-time, hierarchy collapses. In Proceedings of the 31st ACM Symposium on Theory of Computing (Atlanta, Ga., May 1-4). ACM, New York, pp. 659-667. Google Scholar
Digital Library
- LEIGHTON, F. 1992. Introduction to Parallel Algorithms and Architectures. Morgan-Kaufmann, San Mateo, Calif. Google Scholar
Digital Library
- MACWILLIAMS,F.,AND SLOANE, N. 1977. The Theory of Error-Correcting Codes. North-Holland, Amsterdam, The Netherlands.Google Scholar
- NISAN, N. 1991. Pseudorandom bits for constant depth circuits. Combinatorica 12, 4, 63-70.Google Scholar
Cross Ref
- NISAN, N. 1996. Extracting randomness: How and why. In Proceedings of the 11th IEEE Conference on Computational Complexity. IEEE Computer Society Press, Los Alamitos, Calif., pp. 44-58. Google Scholar
Digital Library
- NISAN, N., AND TA-SHMA, A. 1999. Extrating randomness : A survey and new constructions. J. Comput. Syst. Sci. 58, 1, 148-173. Google Scholar
Digital Library
- NISAN, N., AND WIGDERSON, A. 1994. Hardness vs randomness. J. Comput. Syst. Sci. 49, 149-167. Google Scholar
Digital Library
- NISAN, N., AND ZUCKERMAN, D. 1993. More deterministic simulation in Logspace. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 235-244. Google Scholar
Digital Library
- RADHAKRISHNAN, J., AND TA-SHMA, A. 1997. Tight bounds for depth-two superconcentrators. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 585-594. Google Scholar
Digital Library
- RAZ, R., REINGOLD, O., AND VADHAN, S. 1999. Extracting all the randomness and reducing the error in Trevisan's extractors. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing (Atlanta, Ga., May 1-4). ACM, New York, pp. 149-158. Google Scholar
Digital Library
- SAKS, M., SRINIVASAN, A., AND ZHOU, S. 1998. Explicit OR-dispersers with polylogarithmic degree. J. ACM 45, 1 (Jan.), 123-154. Google Scholar
Digital Library
- SANTHA, M., AND VAZIRANI, U. 1986. Generating quasi-random sequences from slightly random sources. J. Comput. Syst. Sci. 33, 75-87. Google Scholar
Digital Library
- SRINIVASAN, A., AND ZUCKERMAN, D. 1994. Computing with very weak random sources. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 264-275.Google Scholar
Digital Library
- SUDAN, M., TREVISAN, L., AND VADHAN, S. 1999. Pseudorandom generators without the XOR lemma. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing (Atlanta, Ga., May 1-4). ACM, New York, pp. 537-546. Google Scholar
Digital Library
- TA-SHMA, A. 1996. On extracting randomness from weak random sources. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (Philadelphia, Pa., May 22-24). ACM, New York, pp. 276-285. Google Scholar
Digital Library
- TA-SHMA, A. 1998. Almost optimal dispersers. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (Dallas, Tex., May 23-26). ACM, New York, pp. 196-202. Google Scholar
Digital Library
- VAZIRANI, U., AND VAZIRANI, V. 1985. Random polynomial time is equal to slightly random polynomial time. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 417-428.Google Scholar
Digital Library
- VON NEUMANN, J. 1951. Various techniques used in connection with random digits. NBS, Appl. Math. Seri. 12, 36-38.Google Scholar
- WIGDERSON, A., AND ZUCKERMAN, D. 1993. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 245-251. Google Scholar
Digital Library
- YAO, A. 1982. Theory and applications of trapdoor functions. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 80-91.Google Scholar
Cross Ref
- ZUCKERMAN, D. 1990. General weak random sources. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 534-543.Google Scholar
Digital Library
- ZUCKERMAN, D. 1996a. On unapproximable versions of NP-complete problems. SIAM J. Comput. 25,6, 1293-1304. Google Scholar
Digital Library
- ZUCKERMAN, D. 1996b. Randomness-optimal sampling, extractors and constructive leader election. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (Philadelphia, Pa., May 22-24). ACM, New York, pp. 286-295. Google Scholar
Digital Library
Index Terms
Extractors and pseudorandom generators
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