Abstract
We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number d of the random input bits. As our main result, we construct a deterministic extractor that, given any d-local source with min-entropy k on n bits, extracts Ω(k2/nd) bits that are 2−nΩ(1)-close to uniform, provided d ≤ o(log n) and k ≥ n2/3+γ (for arbitrarily small constants γ > 0).
Using our result, we also improve a result of Viola [2010] who proved a 1/2 − O(1/log n) statistical distance lower bound for o(log n)-local samplers trying to sample input-output pairs of an explicit boolean function, assuming the samplers use at most n + n1−δ random bits for some constant δ > 0. Using a different function, we simultaneously improve the lower bound to 1/2 − 2−nΩ(1) and eliminate the restriction on the number of random bits.
- Applebaum, B. 2011. Pseudorandom generators with long stretch and low locality from random local one-way functions. Tech. rep. TR11-007, Electronic Colloquium on Computational Complexity.Google Scholar
- Applebaum, B., Ishai, Y., and Kushilevitz, E. 2006. Cryptography in NC0. SIAM J. Comput. 36, 4, 845--888. Google Scholar
Digital Library
- Applebaum, B., Ishai, Y., and Kushilevitz, E. 2008. On pseudorandom generators with linear stretch in NC0. Comput. Complex. 17, 1, 38--69. Google Scholar
Digital Library
- Applebaum, B., Bogdanov, A., and Rosen, A. 2011. A dichotomy for local small-bias generators. Tech. rep. TR11-126, Electronic Colloquium on Computational Complexity.Google Scholar
- Arora, S., Steurer, D., and Wigderson, A. 2009. Towards a study of low-complexity graphs. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming. 119--131. Google Scholar
Digital Library
- Barak, B., Impagliazzo, R., and Wigderson, A. 2006a. Extracting randomness using few independent sources. SIAM J. Comput. 36, 4, 1095--1118. Google Scholar
Digital Library
- Barak, B., Rao, A., Shaltiel, R., and Wigderson, A. 2006b. 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. In Proceedings of the 38th ACM Symposium on Theory of Computing. 671--680. Google Scholar
Digital Library
- Barak, B., Kindler, G., Shaltiel, R., Sudakov, B., and Wigderson, A. 2010. Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors. J. ACM 57, 4. 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. 276--287. Google Scholar
Digital Library
- Ben-Sasson, E. and Gabizon, A. 2011. Extractors for polynomials sources over constant-size fields of small characteristic. Tech. rep. TR11-129, Electronic Colloquium on Computational Complexity.Google Scholar
- Bogdanov, A. and Qiao, Y. 2009. On the security of Goldreich’s one-way function. In Proceedings of the 13th International Workshop on Randomization and Computation. 392--405. Google Scholar
Digital Library
- Bourgain, J. 2005. More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1, 1--32.Google Scholar
Cross Ref
- Bourgain, J. 2007. On the construction of affine-source extractors. Geom. Funct. Anal. 1, 33--57.Google Scholar
Cross Ref
- Chor, B. and Goldreich, O. 1988. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 2, 230--261. Google Scholar
Digital Library
- Chor, B., Friedman, J., Goldreich, O., Håstad, J., Rudich, S., and Smolensky, R. 1985. The bit extraction problem or t-resilient functions. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science. 396--407. Google Scholar
Digital Library
- Cook, J., Etesami, O., Miller, R., and Trevisan, L. 2009. Goldreich’s one-way function candidate and myopic backtracking algorithms. In Proceedings of the 6th Theory of Cryptography Conference. 521--538. Google Scholar
Digital Library
- Cryan, M. and Miltersen, P. B. 2001. On pseudorandom generators in NC0. In Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science. 272--284. Google Scholar
Digital Library
- De, A. and Trevisan, L. 2009. Extractors using hardness amplification. In Proceedings of the 13th International Workshop on Randomization and Computation. 462--475. Google Scholar
Digital Library
- DeVos, M. and Gabizon, A. 2010. Simple affine extractors using dimension expansion. In Proceedings of the 25th IEEE Conference on Computational Complexity. 50--57. Google Scholar
Digital Library
- Dodis, Y., Elbaz, A., Oliveira, R., and Raz, R. 2004. Improved randomness extraction from two independent sources. In Proceedings of the 8th International Workshop on Randomization and Computation. 334--344.Google Scholar
- Dvir, Z. 2009. Extractors for varieties. In Proceedings of the 24th IEEE Conference on Computational Complexity. 102--113. Google Scholar
Digital Library
- Dvir, Z., Gabizon, A., and Wigderson, A. 2009. Extractors and rank extractors for polynomial sources. Comput. Complex. 18, 1, 1--58. Google Scholar
Digital Library
- Dziembowski, S. and Maurer, U. 2004. Optimal randomizer efficiency in the bounded-storage model. J. Cryptol. 17, 1, 5--26. Google Scholar
Digital Library
- Gabizon, A. and Raz, R. 2008. Deterministic extractors for affine sources over large fields. Combinatorica 28, 4, 415--440. Google Scholar
Digital Library
- Gabizon, A., Raz, R., and Shaltiel, R. 2006. Deterministic extractors for bit-fixing sources by obtaining an independent seed. SIAM J. Comput. 36, 4, 1072--1094. Google Scholar
Digital Library
- Goldreich, O. 2011. Candidate one-way functions based on expander graphs. Studies Complex. Cryptograph., 76--87.Google Scholar
- Goldwasser, S., Gutfreund, D., Healy, A., Kaufman, T., and Rothblum, G. 2007. Verifying and decoding in constant depth. In Proceedings of the 39th ACM Symposium on Theory of Computing. 440--449. Google Scholar
Digital Library
- Guruswami, V. and Rudra, A. 2008. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Trans. Inf. Theory 54, 1, 135--150. Google Scholar
Digital Library
- Guruswami, V., Umans, C., and Vadhan, S. 2009. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. J. ACM 56, 4. Google Scholar
Digital Library
- Håstad, J. 1986. Almost optimal lower bounds for small depth circuits. In Proceedings of the 18th ACM Symposium on Theory of Computing. 6--20. Google Scholar
Digital Library
- Håstad, J. 1987. One-way permutations in NC0. Inf. Process. Lett. 26, 3, 153--155. Google Scholar
Digital Library
- Ishai, Y., Kushilevitz, E., Ostrovsky, R., and Sahai, A. 2008. Cryptography with constant computational overhead. In Proceedings of the 40th ACM Symposium on Theory of Computing. 433--442. Google Scholar
Digital Library
- Kamp, J. and Zuckerman, D. 2007. Deterministic extractors for bit-fixing sources and exposure-resilient cryptography. SIAM J. Comput. 36, 5, 1231--1247. Google Scholar
Digital Library
- Kamp, J., Rao, A., Vadhan, S., and Zuckerman, D. 2011. Deterministic extractors for small-space sources. J. Comput. Syst. Sci. 77, 1, 191--220. Google Scholar
Digital Library
- Li, X. 2011a. Improved constructions of three source extractors. In Proceedings of the 26th IEEE Conference on Computational Complexity. 126--136. Google Scholar
Digital Library
- Li, X. 2011b. A new approach to affine extractors and dispersers. In Proceedings of the 26th IEEE Conference on Computational Complexity. 137--147. Google Scholar
Digital Library
- Lovett, S. and Viola, E. 2011. Bounded-depth circuits cannot sample good codes. In Proceedings of the 26th IEEE Conference on Computational Complexity. 243--251. Google Scholar
Digital Library
- Lu, C.-J. 2004. Encryption against storage-bounded adversaries from on-line strong extractors. J. Cryptol. 17, 1, 27--42. Google Scholar
Digital Library
- Mossel, E., Shpilka, A., and Trevisan, L. 2006. On epsilon-biased generators in NC0. Random Struct. Algor. 29, 1, 56--81. Google Scholar
Digital Library
- Nisan, N. and Zuckerman, D. 1996. Randomness is linear in space. J. Comput. Syst. Sci. 52, 1, 43--52. Google Scholar
Digital Library
- Rao, A. 2008. A 2-source almost-extractor for linear entropy. In Proceedings of the 12th International Workshop on Randomization and Computation. 549--556. Google Scholar
Digital Library
- Rao, A. 2009a. Extractors for a constant number of polynomially small min-entropy independent sources. SIAM J. Comput. 39, 1, 168--194. Google Scholar
Digital Library
- Rao, A. 2009b. Extractors for low-weight affine sources. In Proceedings of the 24th IEEE Conference on Computational Complexity. 95--101. Google Scholar
Digital Library
- Rao, A. and Zuckerman, D. 2008. Extractors for three uneven-length sources. In Proceedings of the 12th International Workshop on Randomization and Computation. 557--570. Google Scholar
Digital Library
- Raz, R. 2005. Extractors with weak random seeds. In Proceedings of the 37th ACM Symposium on Theory of Computing. 11--20. Google Scholar
Digital Library
- Raz, R. and Yehudayoff, A. 2011. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. J. Comput. Syst. Sci. 77, 1, 167--190. Google Scholar
Digital Library
- Raz, R., Reingold, O., and Vadhan, S. 1999. Error reduction for extractors. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science. 191--201. Google Scholar
Digital Library
- Raz, R., Reingold, O., and Vadhan, S. 2002. Extracting all the randomness and reducing the error in Trevisan’s extractors. J. Comput. Syst. Sci. 65, 1, 97--128. Google Scholar
Digital Library
- Schmidt, J., Siegel, A., and Srinivasan, A. 1995. Chernoff-Hoeffding bounds for applications with limited independence. SIAM J. Discr. Math. 8, 2, 223--250. Google Scholar
Digital Library
- Shaltiel, R. 2002. Recent developments in explicit constructions of extractors. Bull. Euro. Assoc. Theor. Comput. Sci. 77, 67--95.Google Scholar
- Shaltiel, R. 2008. How to get more mileage from randomness extractors. Random Struct. Algor. 33, 2, 157--186. Google Scholar
Digital Library
- Shaltiel, R. 2011. An introduction to randomness extractors. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming. 21--41. Google Scholar
Digital Library
- Shoup, V. 1988. New algorithms for finding irreducible polynomials over finite fields. In Proceedings of the 29th IEEE Symposium on Foundations of Computer Science. 283--290. Google Scholar
Digital Library
- Sudan, M. 2002. Essential coding theory---Course notes. http://theory.lcs.mit.edu/~madhu/coding/.Google Scholar
- Tauman Kalai, Y., Li, X., and Rao, A. 2009. 2-source extractors under computational assumptions and cryptography with defective randomness. In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science. 617--626. Google Scholar
Digital Library
- Trevisan, L. 2001. Extractors and pseudorandom generators. J. ACM 48, 4, 860--879. Google Scholar
Digital Library
- Trevisan, L. and Vadhan, S. 2000. Extracting randomness from samplable distributions. In Proceedings of the 41st IEEE Symposium on Foundations of Computer Science. 32--42. Google Scholar
Digital Library
- Vadhan, S. 2004. Constructing locally computable extractors and cryptosystems in the bounded-storage model. J. Cryptol. 17, 1, 43--77. Google Scholar
Digital Library
- Vadhan, S. 2011. Pseudorandomness. In Foundations and Trends in Theoretical Computer Science. To appear. Google Scholar
Digital Library
- Viola, E. 2010. The complexity of distributions. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science. 202--211. Google Scholar
Digital Library
- Viola, E. 2011. Extractors for circuit sources. In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science. To appear. Google Scholar
Digital Library
- Yao, A. 1985. Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science. 1--10. Google Scholar
Digital Library
- Yehudayoff, A. 2011. Affine extractors over prime fields. Combinatorica 31, 2, 245--256. Google Scholar
Digital Library
- Zimand, M. 2010. Simple extractors via constructions of cryptographic pseudo-random generators. Theor. Comput. Sci. 411, 10, 1236--1250. Google Scholar
Digital Library
Index Terms
Extractors and Lower Bounds for Locally Samplable Sources
Recommendations
Extractors and lower bounds for locally samplable sources
APPROX'11/RANDOM'11: Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniquesWe consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number d of the random input bits. As our main result, we construct a deterministic ...
Extractors with weak random seeds
STOC '05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computingWe show how to extract random bits from two or more independent weak random sources in cases where only one source is of linear min-entropy and all other sources are of logarithmic min-entropy. Our main results are as follows:
- A long line of research, ...
Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography
We give an efficient deterministic algorithm that extracts $\Omega(n^{2\gamma})$ almost-random bits from sources where $n^{\frac{1}{2}+\gamma}$ of the $n$ bits are uniformly random and the rest are fixed in advance. This improves upon previous ...






Comments