TCC 2011: Theory of Cryptography pp 1-18

# Input Locality and Hardness Amplification

• Andrej Bogdanov
• Alon Rosen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6597)

## Abstract

We establish new hardness amplification results for one-way functions in which each input bit influences only a small number of output bits (a.k.a. input-local functions). Our transformations differ from previous ones in that they approximately preserve input locality and at the same time retain the input size of the original function.

Let f : {0, 1}n → {0, 1}m be a one-way function with input locality d, and suppose that f cannot be inverted in time $$\exp(\tilde{O}(\sqrt{n}\cdot d))$$ on an ε-fraction of inputs. Our main results can be summarized as follows:

• If f is injective then it is equally hard to invert f on a (1 − ε)-fraction of inputs.

• If f is regular then there is a function g: {0, 1}n → {0, 1}m + O(n) that is d + O(log3n) input local and is equally hard to invert on a (1 − ε)-fraction of inputs.

A natural candidate for a function with small input locality and for which no sub-exponential time attacks are known is Goldreich’s one-way function. To make our results applicable to this function, we prove that when its input locality is set to be d = O(logn) certain variants of the function are (almost) regular with high probability.

In some cases, our techniques are applicable even when the input locality is not small. We demonstrate this by extending our first main result to one-way functions of the “parity with noise” type.

### Keywords

one-way function input locality hardness amplification parity with noise

### References

1. 1.
Alon, N., Spencer, J.: The Probabilistic Method. John Wiley, Chichester (1992)
2. 2.
Applebaum, B., Barak, B., Wigderson, A.: Public-key cryptography from different assumptions. In: STOC 2010, pp. 171–180 (2010)Google Scholar
3. 3.
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. In: FOCS 2004, pp. 166–175 (2004)Google Scholar
4. 4.
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography with Constant Input Locality. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 92–110. Springer, Heidelberg (2007); Journal version in J. Cryptology 22(4), 429-469 (2009)
5. 5.
Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. In: STOC 2000, pp. 435–440 (2000)Google Scholar
6. 6.
Bogdanov, A., Qiao, Y.: On the security of goldreich’s one-way function. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 392–405. Springer, Heidelberg (2009)
7. 7.
Cook, J., Etesami, O., Miller, R., Trevisan, L.: Goldreich’s One-Way Function Candidate and Myopic Backtracking Algorithms. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 521–538. Springer, Heidelberg (2009)
8. 8.
Goldreich, O.: Candidate One-Way Functions Based on Expander Graphs. Electronic Colloquium on Computational Complexity (ECCC) 7(90) (2000)Google Scholar
9. 9.
Goldreich, O., Impagliazzo, R., Levin, L.A., Venkatesan, R., Zuckerman, D.: Security Preserving Amplification of Hardness. In: FOCS 1990, pp. 318–326 (1990)Google Scholar
10. 10.
Goldreich, O., Krawczyk, H., Luby, M.: On the Existence of Pseudorandom Generators. SIAM J. Comput. 22(6), 1163–1175 (1993)
11. 11.
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)
12. 12.
Impagliazzo, R., Levin, L., Luby, M.: Pseudo-random Generation from One-way Functions. In: STOC 1989, pp. 12–24 (1989)Google Scholar
13. 13.
Impagliazzo, R., Luby, M.: One-way Functions are Essential for Complexity Based Cryptography. In: FOCS 1989, pp. 230–235 (1989)Google Scholar
14. 14.
Mansour, Y., Nisan, N., Tiwari, P.: The Computational Complexity of Universal Hashing. Theor. Comput. Sci. 107(1), 121–133 (1993)
15. 15.
Lin, H.C., Trevisan, L., Wee, H.: On Hardness Amplification of One-Way Functions. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 34–49. Springer, Heidelberg (2005)
16. 16.
Yao, A.C.-C.: Theory and Applications of Trapdoor Functions (Extended Abstract). In: FOCS 1982, pp. 80–91 (1982)Google Scholar

© International Association for Cryptologic Research 2011

## Authors and Affiliations

• Andrej Bogdanov
• 1
• 2
• Alon Rosen
• 3
1. 1.Dept. of CSEChinese Univ. of Hong KongHong Kong
2. 2.ITCSCChinese Univ. of Hong KongHong Kong
3. 3.Efi Arazi School of Computer ScienceIDC HerzliyaIsrael