Abstract
A randomized encoding of a function f(x) is a randomized function \(\hat{f}(x,r)\), such that the “encoding” \(\hat{f}(x,r)\) reveals f(x) and essentially no additional information about x. Randomized encodings of functions have found many applications in different areas of cryptography, including secure multiparty computation, efficient parallel cryptography, and verifiable computation.
We initiate a complexity-theoretic study of the class \(\mathsf {SRE} \) of languages (or boolean functions) that admit an efficient statistical randomized encoding. That is, \(\hat{f}(x,r)\) can be computed in time poly(|x|), and its output distribution on input x can be sampled in time poly(|x|) given f(x), up to a small statistical distance.
We obtain the following main results.
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Separating \(\mathsf {SRE} \) from efficient computation: We give the first examples of promise problems and languages in \(\mathsf {SRE} \) that are widely conjectured to lie outside \(\mathsf {P/poly}\). Our candidate promise problems and languages are based on the standard Learning with Errors (LWE) assumption, a non-standard variant of the Decisional Diffie Hellman (DDH) assumption and the “Abelian Subgroup Membership problem” (which generalizes Quadratic-Residuosity and a variant of DDH).
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Separating \(\mathsf {SZK} \) from \(\mathsf {SRE} \) : We explore the relationship of \(\mathsf {SRE} \) with the class \(\mathsf {SZK} \) of problems possessing statistical zero knowledge proofs. It is known that \(\mathsf {SRE} \subseteq \mathsf {SZK} \). We present an oracle separation which demonstrates that a containment of \(\mathsf {SZK} \) in \(\mathsf {SRE} \) cannot be proved via relativizing techniques.
Y. Ishai–Research supported by the European Union’s Tenth Framework Programme (FP10/2010-2016) under grant agreement no. 259426 ERC-CaC, ISF grants 1361/10 and 1709/14 and BSF grant 2012378.
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References
Aiello, W., Håstad, J.: Relativized perfect zero knowledge is not BPP. Inf. Comput. (1991)
Applebaum, B.: Cryptography in Constant Parallel Time. Ph.D. thesis, Technion (2007)
Applebaum, B.: Randomly encoding functions: a new cryptographic paradigm. In: Fehr, S. (ed.) ICITS 2011. LNCS, vol. 6673, pp. 25–31. Springer, Heidelberg (2011)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally private randomizing polynomials and their applications. In: IEEE Conference on Computational Complexity, pp. 260–274. IEEE Computer Society (2005)
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. SIAM J. Comput. 36(4), 845–888 (2006)
Applebaum, B., Ishai, Y., Kushilevitz, E.: From secrecy to soundness: efficient verification via secure computation. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 152–163. Springer, Heidelberg (2010)
Baker, T.P., Gill, J., Solovay, R.: Relativizatons of the P =? NP question. SIAM J. Comput. 4(4), 431–442 (1975)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: STOC. ACM (1988)
Chaum, D., Crépeau, C., Damgard, I.: Multiparty unconditionally secure protocols. In: STOC, pp. 11–19. ACM, New York (1988)
Dvir, Z., Gutfreund, D., Rothblum, G.N., Vadhan, S.: On approximating the entropy of polynomial mappings. In: ICS, pp. 460–475 (2011)
Feige, U., Killian, J., Naor, M.: A minimal model for secure computation (extended abstract). In: STOC, pp. 554–563 (1994)
Galbraith, S.D., Rotger, V.: Easy decision-diffie-hellman groups (2004)
Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: FOCS, pp. 294–304. IEEE Computer Society (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)
Ishai, Y., Kushilevitz, E., Paskin-Cherniavsky, A.: From randomizing polynomials to parallel algorithms. In: ITCS. ACM, New York (2012)
Kilian, J.: Founding crytpography on oblivious transfer. In: STOC, pp. 20–31. ACM, New York (1988)
Naor, M., Reingold, O.: Number-theoretic constructions of efficient pseudo-random functions. J. ACM 51(2), Mar 2004
Sahai, A., Vadhan, S.: A complete problem for statistical zero knowledge. J. ACM 50(2), 196–249 (2003). http://doi.acm.org/10.1145/636865.636868
Yao, A.C.C.: How to generate and exchange secrets (extended abstract). In: FOCS, pp. 162–167 (1986)
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Agrawal, S., Ishai, Y., Khurana, D., Paskin-Cherniavsky, A. (2015). Statistical Randomized Encodings: A Complexity Theoretic View. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_1
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DOI: https://doi.org/10.1007/978-3-662-47672-7_1
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