Abstract
We consider the question of constructing cryptographic pseudorandom generators (PRGs) in NC0, namely ones in which each bit of the output depends on just a constant number of input bits. Previous constructions of such PRGs were limited to stretching a seed of n bits to n + o(n) bits. This leaves open the existence of a PRG with a linear (let alone superlinear) stretch in NC0. In this work we study this question and obtain the following main results:
1. We show that the existence of a linear-stretch PRG in NC0 implies non-trivial hardness of approximation results without relying on PCP machinery. In particular, that Max 3SAT is hard to approximate to within some constant.
2. We construct a linear-stretch PRG in NC0 under a specific intractability assumption related to the hardness of decoding “sparsely generated” linear codes. Such an assumption was previously conjectured by Alekhnovich [1].
We note that Alekhnovich directly obtains hardness of approximation results from the latter assumption. Thus, we do not prove hardness of approximation under new concrete assumptions. However, our first result is motivated by the hope to prove hardness of approximation under more general or standard cryptographic assumptions, and the second result is independently motivated by cryptographic applications.
Research supported by grant 36/03 from the Israel Science Foundation.
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References
Alekhnovich, M.: More on average case vs approximation complexity. In: Proc. 44th FOCS, pp. 298–307 (2003)
Alon, N., Roichman, Y.: Random cayley graphs and expanders. Random Struct. Algorithms 5(2), 271–285 (1994)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. SIAM J. Comput. (to appear); Preliminary version in FOCS 2004
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and hardness of approximation problems. J. of the ACM 45(3), 501–555 (1998)
Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of np. J. of the ACM 45(1), 70–122 (1998)
Ben-Sasson, E., Sudan, M., Vadhan, S., Wigderson, A.: Randomness-efficient low-degree tests and short pcps via epsilon-biased sets. In: Proc. 35th STOC, pp. 612–621 (2003)
Blum, A., Furst, M., Kearns, M., Lipton, R.J.: Cryptographic primitives based on hard learning problems. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 278–291. Springer, Heidelberg (1994)
Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput. 13, 850–864 (1984)
Capalbo, M., Reingold, O., Vadhan, S., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: Proc. 34th STOC, pp. 659–668 (2002)
Cryan, M., Miltersen, P.B.: On pseudorandom generators in NC0. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136. Springer, Heidelberg (2001)
Dodis, Y., Smith, A.: Correcting errors without leaking partial information. In: Proc. 37th STOC, pp. 654–663 (2005)
Feige, U.: Relations between average case complexity and approximation complexity. In: Proc. of 34th STOC, pp. 534–543 (2002)
Goldreich, O.: Candidate one-way functions based on expander graphs. ECCC 7(090) (2000)
Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)
Goldreich, O., Krawczyk, H., Luby, M.: On the existence of pseudorandom generators. SIAM J. Comput. 22(6), 1163–1175 (1993)
Goldreich, O., Wigderson, A.: Tiny families of functions with random properties: A quality-size trade-off for hashing. Random Struct. Random Struct. Algorithms 11(4), 315–343 (1997)
Mossel, E., Shpilka, A., Trevisan, L.: On ε-biased generators in NC0. In: Proc. 44th FOCS, pp. 136–145 (2003)
Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993)
Naor, M.: Bit commitment using pseudorandomness. J. of Cryptology 4, 151–158 (1991)
Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. J. of Computer and Systems Sciences 43, 425–440 (1991)
Radhakrishnan, J., Ta-Shma, A.: Tight bounds for depth-two superconcentrators. SIAM J. Discrete Math. 13(1), 2–24 (2000)
Vazirani, U.: Randomness, Adversaries and Computation. Ph.d. thesis, UC Berkeley (1986)
Viola, E.: On constructing parallel pseudorandom generators from one-way functions. In: Proc. 20th CCC, pp. 183–197 (2005)
Yao, A.C.: Theory and application of trapdoor functions. In: Proc. 23rd FOCS, pp. 80–91 (1982)
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Applebaum, B., Ishai, Y., Kushilevitz, E. (2006). On Pseudorandom Generators with Linear Stretch in NC0 . In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 2006. Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_25
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DOI: https://doi.org/10.1007/11830924_25
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