Abstract
We revisit the problem of basing pseudorandom generators on regular one-way functions, and present the following constructions:
-
For any known-regular one-way function (on n-bit inputs) that is known to be ε-hard to invert, we give a neat (and tighter) proof for the folklore construction of pseudorandom generator of seed length Θ(n) by making a single call to the underlying one-way function.
-
For any unknown-regular one-way function with known ε-hardness, we give a new construction with seed length Θ(n) and O(n/log(1/ε)) calls. Here the number of calls is also optimal by matching the lower bounds of Holenstein and Sinha (FOCS 2012).
Both constructions require the knowledge about ε, but the dependency can be removed while keeping nearly the same parameters. In the latter case, we get a construction of pseudo-random generator from any unknown-regular one-way function using seed length \(\tilde{O}(n)\) and \(\tilde{O}(n/\log{n})\) calls, where \(\tilde{O}\) omits a factor that can be made arbitrarily close to constant (e.g. logloglogn or even less). This improves the randomized iterate approach by Haitner, Harnik and Reingold (CRYPTO 2006) which requires seed length O(n·logn) and O(n/logn) calls.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Workshop on leakage, tampering and viruses (June 2013), http://www.crypto.edu.pl/events/workshop2013
Barak, B., Shaltiel, R., Wigderson, A.: Computational analogues of entropy. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 200–215. Springer, Heidelberg (2003)
Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo random bits. In: Proceedings of the 23rd IEEE Symposium on Foundation of Computer Science, pp. 112–117 (1982)
Carter, J.L., Wegman, M.N.: Universal classes of hash functions. Journal of Computer and System Sciences 18, 143–154 (1979)
Dedić, N., Harnik, D., Reyzin, L.: Saving private randomness in one-way functions and pseudorandom generators. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 607–625. Springer, Heidelberg (2008)
Dodis, Y., Elbaz, A., Oliveira, R., Raz, R.: Improved randomness extraction from two independent sources. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 334–344. Springer, Heidelberg (2004)
Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press (2001)
Goldreich, O.: Three XOR-lemmas — an exposition. In: Goldreich, O. (ed.) Studies in Complexity and Cryptography. LNCS, vol. 6650, pp. 248–272. Springer, Heidelberg (2011)
Goldreich, O., Krawczyk, H., Luby, M.: On the existence of pseudorandom generators. SIAM Journal on Computing 22(6), 1163–1175 (1993)
Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: Johnson, D.S. (ed.) Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, Seattle, Washington, May 15-17, pp. 25–32 (1989)
Haitner, I., Harnik, D., Reingold, O.: On the power of the randomized iterate. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 22–40. Springer, Heidelberg (2006)
Haitner, I., Harnik, D., Reingold, O.: On the power of the randomized iterate. SIAM Journal on Computing 40(6), 1486–1528 (2011)
Haitner, I., Reingold, O., Vadhan, S.P.: Efficiency improvements in constructing pseudorandom generators from one-way functions. In: Proceedings of the 42nd ACM Symposium on the Theory of Computing, pp. 437–446 (2010)
Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: Construction of pseudorandom generator from any one-way function. SIAM Journal on Computing 28(4), 1364–1396 (1999)
Holenstein, T.: Pseudorandom generators from one-way functions: A simple construction for any hardness. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 443–461. Springer, Heidelberg (2006)
Holenstein, T., Sinha, M.: Constructing a pseudorandom generator requires an almost linear Number of calls. In: Proceedings of the 53rd IEEE Symposium on Foundation of Computer Science, pp. 698–707 (2012)
Hsiao, C.-Y., Lu, C.-J., Reyzin, L.: Conditional computational entropy, or toward separating pseudoentropy from compressibility. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 169–186. Springer, Heidelberg (2007)
Lee, C.-J., Lu, C.-J., Tsai, S.-C., Tzeng, W.-G.: Extracting randomness from multiple independent sources. IEEE Transactions on Information Theory 51(6), 2224–2227 (2005)
Levin, L.A.: One-way functions and pseudorandom generators. Combinatorica 7(4), 357–363 (1987)
Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)
Nisan, N., Zuckerman, D.: Randomness is linear in space. Journal of Computer and System Sciences 52(1), 43–53 (1996)
Stinson, D.R.: Universal hash families and the leftover hash lemma, and applications to cryptography and computing. Journal of Combinatorial Mathematics and Combinatorial Computing 42, 3–31 (2002), http://www.cacr.math.uwaterloo.ca/~dstinson/publist.html
Vadhan, S.P., Zheng, C.J.: Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In: Proceedings of the 44th ACM Symposium on the Theory of Computing, pp. 817–836 (2012)
Vadhan, S.P., Zheng, C.J.: A uniform min-max theorem with applications in cryptography. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 93–110. Springer, Heidelberg (2013)
Vazirani, U.V., Vazirani, V.V.: Efficient and secure pseudo-random number generation (extended abstract). In: Proceedings of the 25th IEEE Symposium on Foundation of Computer Science, pp. 458–463 (1984)
Yao, A.C.-C.: Theory and applications of trapdoor functions (extended abstract). In: Proceedings of the 23rd IEEE Symposium on Foundation of Computer Science, pp. 80–91 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yu, Y., Li, X., Weng, J. (2013). Pseudorandom Generators from Regular One-Way Functions: New Constructions with Improved Parameters. In: Sako, K., Sarkar, P. (eds) Advances in Cryptology - ASIACRYPT 2013. ASIACRYPT 2013. Lecture Notes in Computer Science, vol 8270. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-42045-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-42045-0_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-42044-3
Online ISBN: 978-3-642-42045-0
eBook Packages: Computer ScienceComputer Science (R0)