Abstract
We study the following natural question: Which cryptographic primitives (if any) can be realized by functions with constant input locality, namely functions in which every bit of the input influences only a constant number of bits of the output? This continues the study of cryptography in low complexity classes. It was recently shown (Applebaum et al., FOCS 2004) that, under standard cryptographic assumptions, most cryptographic primitives can be realized by functions with constant output locality, namely ones in which every bit of the output is influenced by a constant number of bits from the input.
We (almost) characterize what cryptographic tasks can be performed with constant input locality. On the negative side, we show that primitives which require some form of non-malleability (such as digital signatures, message authentication, or non-malleable encryption) cannot be realized with constant input locality. On the positive side, assuming the intractability of certain problems from the domain of error correcting codes (namely, hardness of decoding a random linear code or the security of the McEliece cryptosystem), we obtain new constructions of one-way functions, pseudorandom generators, commitments, and semantically-secure public-key encryption schemes whose input locality is constant. Moreover, these constructions also enjoy constant output locality. Therefore, they give rise to cryptographic hardware that has constant-depth, constant fan-in and constant fan-out. As a byproduct, we obtain a pseudorandom generator whose output and input locality are both optimal (namely, 3).
Research supported by grant 1310/06 from the Israel Science Foundation.
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
Alekhnovich, M.: More on average case vs approximation complexity. In: Proc. 44th FOCS, pp. 298–307 (2003)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally private randomizing polynomials and their applications. Computional Complexity 15(2), 115–162 (2006)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. SIAM J. Comput. 36(4), 845–888 (2006)
Applebaum, B., Ishai, Y., Kushilevitz, E.: On pseudorandom generators with linear stretch in NC0. In: Proc. 10th Random (2006)
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)
Cook, S.A.: The complexity of theorem-proving procedures. In: STOC 1971. Proceedings of the third annual ACM symposium on Theory of computing, pp. 151–158. ACM Press, New York (1971)
Cryan, M., Miltersen, P.B.: On pseudorandom generators in NC0. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 272–284. Springer, Heidelberg (2001)
Feige, U., Killian, J., Naor, M.: A minimal model for secure computation (extended abstract). In: Proc. of the 26th STOC, pp. 554–563 (1994)
Goldreich, O.: Candidate one-way functions based on expander graphs. Electronic Colloquium on Computational Complexity (ECCC), 7(090) (2000)
Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)
Goldreich, O.: Foundations of Cryptography: Basic Applications. Cambridge University Press, Cambridge (2004)
Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. of the ACM 33, 792–807 (1986)
Goldreich, O., Krawczyk, H., Luby, M.: On the existence of pseudorandom generators. SIAM J. Comput. 22(6), 1163–1175 (1993)
Goldreich, O., Levin, L.: A hard-core predicate for all one-way functions. In: Proc. 21st STOC, pp. 25–32 (1989)
Impagliazzo, R., Levin, L.A., Luby, M.: Pseudorandom generation from one-way functions. In: Proc. 21st STOC, pp. 12–24 (1989)
Ishai, Y., Kushilevitz, E.: Randomizing polynomials: A new representation with applications to round-efficient secure computation. In: Proc. 41st FOCS, pp. 294–304 (2000)
Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002)
Janwa, H., Moreno, O.: Mceliece public key cryptosystems using algebraic-geometric codes. Des. Codes Cryptography 8(3), 293–307 (1996)
Kilian, J.: Founding cryptography on oblivious transfer. In: Proc. 20th STOC, pp. 20–31 (1988)
Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, fourier transform, and learnability. J. ACM 40(3), 607–620 (1993)
McEliece, R.J.: A public-key cryptosystem based on algebraic coding theory. Technical Report DSN PR 42-44, Jet Prop. Lab (1978)
Mossel, E., Shpilka, A., Trevisan, L.: On ε-biased generators in NC0. In: Proc. 44th FOCS, pp. 136–145 (2003)
Naor, M., Reingold, O.: Synthesizers and their application to the parallel construction of pseudo-random functions. J. of Computer and Systems Sciences 58(2), 336–375 (1999)
Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. J. of Computer and Systems Sciences 43, 425–440 (1991)
Sudan, M.: Algorithmic introduction to coding theory - lecture notes (2002), http://theory.csail.mit.edu/~madhu/FT01/
Varshamov, R.: Estimate of the number of signals in error correcting codes. Doklady Akademii Nauk SSSR 117, 739–741 (1957)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Applebaum, B., Ishai, Y., Kushilevitz, E. (2007). Cryptography with Constant Input Locality . In: Menezes, A. (eds) Advances in Cryptology - CRYPTO 2007. CRYPTO 2007. Lecture Notes in Computer Science, vol 4622. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74143-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-74143-5_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74142-8
Online ISBN: 978-3-540-74143-5
eBook Packages: Computer ScienceComputer Science (R0)