Abstract
Extractable functions are functions where any adversary that outputs a point in the range of the function is guaranteed to “know” a corresponding preimage. Here, knowledge is captured by the existence of an efficient extractor that recovers the preimage from the internal state of the adversary. Extractability of functions was defined by the authors (ICALP’08) in the context of perfectly one-way functions. It can be regarded as an abstraction from specific knowledge assumptions, such as the Knowledge of Exponent assumption (Hada and Tanaka, Crypto 1998).
We initiate a more general study of extractable functions. We explore two different approaches. The first approach is aimed at understanding the concept of extractability in of itself; in particular we demonstrate that a weak notion of extraction implies a strong one, and make rigorous the intuition that extraction and obfuscation are complementary notions.
In the second approach, we study the possibility of constructing cryptographic primitives from simpler or weaker ones while maintaining extractability. Results are generally positive. Specifically, we show that several cryptographic reductions are either “knowledge-preserving” or can be modified to be so. Examples include reductions from extractable weak one-way functions to extractable strong ones, from extractable pseudorandom generators to extractable pseudorandom functions, and from extractable one-way functions to extractable commitments. Other questions, such as constructing extractable pseudorandom generators from extractable one way functions, remain open.
The original version of the book was revised: The copyright line was incorrect. The Erratum to the book is available at DOI: 10.1007/978-3-642-00457-5_36
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
Barak, B., Goldreich, O., Impagliazzo, R., Rudich, S., Sahai, A., Vadhan, S.P., Yang, K.: On the (Im)possibility of obfuscating programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, p. 1. Springer, Heidelberg (2001)
Bellare, M., Goldreich, O.: On defining proofs of knowledge. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 390–420. Springer, Heidelberg (1993)
Bellare, M., Palacio, A.: The knowledge-of-exponent assumptions and 3-round zero-knowledge protocols. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 273–289. Springer, Heidelberg (2004)
Bellare, M., Palacio, A.: Towards plaintext-aware public-key encryption without random oracles. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 48–62. Springer, Heidelberg (2004)
Bellare, M., Rogaway, P.: Optimal asymmetric encryption. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 92–111. Springer, Heidelberg (1995)
Blum, M.: Coin flipping by phone. In: IEEE Computer conference (1982)
Canetti, R.: Towards realizing random oracles: Hash functions that hide all partial information. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 455–469. Springer, Heidelberg (1997)
Canetti, R., Dakdouk, R.R.: Extractable perfectly one-way functions. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 449–460. Springer, Heidelberg (2008)
Canetti, R., Fischlin, M.: Universally composable commitments. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, p. 19. Springer, Heidelberg (2001)
Di Crescenzo, G.: Equivocable and extractable commitment schemes. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 74–87. Springer, Heidelberg (2003)
Dent, A.W.: The cramer-shoup encryption scheme is plaintext aware in the standard model. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 289–307. Springer, Heidelberg (2006)
Goldreich, O.: Foundations of Cryptography. Cambridge University Press, Cambridge (2001)
Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. Journal of the ACM 33 (1986)
Goldwasser, S., Kalai, Y.T.: On the impossibility of obfuscation with auxiliary input. In: FOCS (2005)
Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof-systems. In: STOC (1985)
Hada, S., Tanaka, T.: On the existence of 3-round zero-knowledge protocols. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, p. 408. Springer, Heidelberg (1998)
Hastad, J., Levin, L., Impagliazzo, R., Luby, M.: Construction of a pseudorandom generator from any one-way function. SIAM Journal on Computing (1999)
Herzog, J.C., Liskov, M., Micali, S.: Plaintext awareness via key registration. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 548–564. Springer, Heidelberg (2003)
Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: FOCS (1995)
Lepinski, M.: On the existence of 3-round zero-knowledge proofs. M.S. Thesis (2002)
Naor, M.: Bit commitments using pseudorandom generators. Journal of Cryptology (1991)
De Santis, A., Di Crescenzo, G., Persiano, G.: Necessary and sufficient assumptions for non-interactive zero-knowledge proofs of knowledge for all NP relations. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, p. 451. Springer, Heidelberg (2000)
Ventre, C., Visconti, I.: Message-aware commitment schemes (unpublished manuscript, 2008)
Yao, A.C.: Theory and application of trapdoor functions. In: FOCS (1982)
Zheng, Y., Seberry, J.: Immunizing public key cryptosystems against chosen ciphertext attacks. Journal on Selected Areas in Communication (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Canetti, R., Dakdouk, R.R. (2009). Towards a Theory of Extractable Functions. In: Reingold, O. (eds) Theory of Cryptography. TCC 2009. Lecture Notes in Computer Science, vol 5444. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00457-5_35
Download citation
DOI: https://doi.org/10.1007/978-3-642-00457-5_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00456-8
Online ISBN: 978-3-642-00457-5
eBook Packages: Computer ScienceComputer Science (R0)