This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
We restrict the decoder to be deterministic for simplicity. This restriction does not compromise generality, in the sense that one can transform a randomized decoder to a deterministic one by incorporating the coins of the former in the encoding itself.
- 2.
This can be generally explained by viewing each slice of the padded function \(\hat{f}'\) (i.e., its restriction to inputs of some fixed length) as a perfect randomized encoding of a corresponding slice of \(\hat{f}\).
- 3.
We also note that if CREN≠SREN then there exist two polynomial-time constructible ensembles which are computationally indistinguishable but not statistically close. In [67] it is shown that such ensembles implies the existence of infinitely often OWF, i.e., a polynomial-time computable function which is hard to invert for infinitely many input lengths (see [70, Definition 4.5.4] for a formal definition).
- 4.
The encoding itself should still be computable in (polynomial-time) uniform NC 0.
- 5.
In fact, this is a specific instance of the statistical difference problem which was shown to be complete for the class SZK [129].
References
Aiello, W., Hastad, J.: Statistical zero-knowledge languages can be recognized in two rounds. J. Comput. Syst. Sci. 42, 327–345 (1991)
Applebaum, B.: Randomly encoding functions: a new cryptographic paradigm (invited talk). In: ICITS, pp. 25–31 (2011)
Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: Proc. of 22nd STOC, pp. 503–513 (1990)
Boppana, R.B., Håstad, J., Zachos, S.: Does co-NP have short interactive proofs? Inf. Process. Lett. 25, 127–132 (1987)
Fortnow, L.: The complexity of perfect zero-knowledge (extended abstract). In: Proc. of 19th STOC, pp. 204–209 (1987)
Goldreich, O.: A note on computational indistinguishability. Inf. Process. Lett. 34(6), 277–281 (1990)
Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)
Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989). Preliminary version in STOC 1985
Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: Proc. of 41st FOCS, pp. 294–304 (2000)
Sahai, A., Vadhan, S.: A complete problem for statistical zero knowledge. J. ACM 50(2), 196–249 (2003). Preliminary version in FOCS 1997
Yap, C.-K.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287–300 (1983)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Applebaum, B. (2014). Randomized Encoding of Functions. In: Cryptography in Constant Parallel Time. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17367-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-17367-7_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17366-0
Online ISBN: 978-3-642-17367-7
eBook Packages: Computer ScienceComputer Science (R0)