Abstract
We prove complexity lower bounds for the tasks of hardness amplification of one-way functions and construction of pseudo-random generators from one-way functions, which are realized non-adaptively in black-box ways.
First, we consider the task of converting a one-way function \(f : \{0,1\}^n \longrightarrow \{0,1\}^m\) into a harder one-way function \(\overline{f} : \{0,1\}^{\overline{n}} \longrightarrow \{0,1\}^{\overline{m}}\), with \(\overline{n},\overline{m} \leq poly(n)\), in a black-box way. The hardness is measured as the fraction of inputs any polynomial-size circuit must fail to invert. We show that to use a constant-depth circuit to amplify hardness beyond a polynomial factor, its size must exceed 2poly(n), and to amplify hardness beyond a 2o(n) factor, its size must exceed \(2^{2^{o(n)}}\). Moreover, for a constant-depth circuit to amplify hardness beyond an n 1 + o(1) factor in a security preserving way (with \(\overline{n} = O(n)\)), it size must exceed \(2^{n^{o(1)}}\).
Next, we show that if a constant-depth polynomial-size circuit can amplify hardness beyond a polynomial factor in a weakly black-box way, then it must basically embed a hard function in itself. In fact, one can derive from such an amplification procedure a highly parallel one-way function, which is computable by an NC0 circuit (constant-depth polynomial-size circuit with bounded fan-in gates).
Finally, we consider the task of constructing a pseudo-random generator \(G : \{0,1\}^{\overline{n}} \longrightarrow \{0,1\}^{\overline{m}}\) from a strongly one-way function \(f : \{0,1\}^n \longrightarrow \{0,1\}^m\) in a black-box way. We show that any such a construction realized by a constant-depth \(2^{n^{o(1)}}\)-size circuit can only have a sublinear stretch (with \(\overline{m} - \overline{n} = o(\overline{n})\)).
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-3-540-32732-5_32
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References
Alon, N., Babai, L., Håstad, J., Peralta, R.: Some constructions of alomost k-wise independent random variables. Random Structures and Algorithms 3(3), 289–304 (1992)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 166–175 (2004)
Boppana, R.B.: The average sensitivity of bounded-depth circuits. Information Processing Letters 63(5), 257–261 (1997)
Di Crescenzo, G., Impagliazzo, R.: Security-preserving hardnessamplification for any regular one-way function. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pp. 169–178 (1999)
Furst, M.L., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17(1), 13–27 (1984)
Gennaro, R., Trevisan, L.: Lower bounds on the efficiency of generic cryptographic constructions. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, pp. 305–313 (2000)
Goldreich, O., Impagliazzo, R., Levin, L.A., Venkatesan, R., Zuckerman, D.: Security preserving amplification of hardness. In: Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science, pp. 318–326 (1990)
Håstad, J.: Computational limitations for small depth circuits. PhD thesis. MIT Press (1986)
Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM Journal on Computing 28(4), 1364–1396 (1999)
Healy, A., Viola, E.: Constant-depth circuits for arithmetic in finite fields of characteristic two. Electronic Colloquium on Computational Complexity, TR05-087 (2005)
Hesse, W., Allender, E., Barrington, D.A.M.: Uniform constantdepth threshold circuits for division and iterated multiplication. Journal of Computer and System Sciences 65(4), 695–716 (2002)
Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pp. 44–61 (1989)
Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform, and learnability. Journal of the ACM 40(3), 607–620 (1993)
Lin, H., Trevisan, L., Wee, H.: On hardness amplification of oneway functions. In: Proceedings of the 2nd Theory of Cryptography Conference, pp. 34–49 (2005)
Lu, C.-J., Tsai, S.-C., Wu, H.-L.: On the complexity of hardness amplification. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 170–182 (2005)
Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM Journal on Computing 22(4), 838–856 (1993)
Nisan, N.: Pseudorandom bits for constant depth circuits. Combinatorica 11(1), 63–70 (1991)
Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)
Viola, E.: The complexity of constructing pseudorandom generators from hard functions. Computational Complexity 13(3-4), 147–188 (2005)
Viola, E.: On constructing parallel pseudorandom generators from one-way functions. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 183–197 (2005)
Yao, A.C.-C.: Theory and applications of trapdoor functions. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 80–91 (1982)
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Lu, CJ. (2006). On the Complexity of Parallel Hardness Amplification for One-Way Functions. In: Halevi, S., Rabin, T. (eds) Theory of Cryptography. TCC 2006. Lecture Notes in Computer Science, vol 3876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11681878_24
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DOI: https://doi.org/10.1007/11681878_24
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