Abstract
We study the result by Bogdanov and Trevisan (FOCS, 2003), who show that under reasonable assumptions, there is no non-adaptive reduction that bases the average-case hardness of an NP-problem on the worst-case complexity of an NP-complete problem. We replace the hiding and the heavy samples protocol in [BT03] by employing the histogram verification protocol of Haitner, Mahmoody and Xiao (CCC, 2010), which proves to be very useful in this context. Once the histogram is verified, our hiding protocol is directly public-coin, whereas the intuition behind the original protocol inherently relies on private coins.
Chapter PDF
Similar content being viewed by others
Keywords
- Homomorphic Encryption
- Cryptographic Primitive
- Polynomial Hierarchy
- Uniform Random Sample
- Learning With Error
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
Akavia, A., Goldreich, O., Goldwasser, S., Moshkovitz, D.: On basing one-way functions on np-hardness. In: Kleinberg, J.M. (ed.) STOC, pp. 701–710. ACM (2006), See also errata on author’s webpage: http://www.wisdom.weizmann.ac.il/~oded/p_aggm.html
Aiello, W., Håstad, J.: Statistical zero-knowledge languages can be recognized in two rounds. J. Comput. Syst. Sci. 42(3), 327–345 (1991)
Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: Miller, G.L. (ed.) STOC, pp. 99–108. ACM (1996)
Ben-David, S., Chor, B., Goldreich, O., Luby, M.: On the theory of average case complexity. J. Comput. Syst. Sci. 44(2), 193–219 (1992)
Blum, M., Kannan, S.: Designing programs that check their work. J. ACM 42(1), 269–291 (1995)
Bogdanov, A., Lee, C.H.: Limits of provable security for homomorphic encryption. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 111–128. Springer, Heidelberg (2013)
Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) STOC, pp. 575–584. ACM (2013)
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47(3), 549–595 (1993)
Blum, M.: Designing programs to check their work. Technical Report 88-09, ICSI (1988)
Brassard, G.: Relativized cryptography. IEEE Transactions on Information Theory 29(6), 877–893 (1983)
Bogdanov, A., Trevisan, L.: Average-case complexity. Foundations and Trends in Theoretical Computer Science 2(1) (2006)
Bogdanov, A., Trevisan, L.: On worst-case to average-case reductions for NP problems. SIAM J. Comput. 36(4), 1119–1159 (2006)
Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC 2010, June 9-12. IEEE Computer Society (2010)
Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)
Even, S., Yacobi, Y.: Cryptocomplexity and NP-completeness. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 195–207. Springer, Heidelberg (1980)
Feigenbaum, J., Fortnow, L.: Random-self-reducibility of complete sets. SIAM J. Comput. 22(5), 994–1005 (1993)
Goldreich, O., Goldwasser, S.: On the possibility of basing cryptography on the assumption that \(\text{P} \neq \textit{NP}\), Unpublished manuscript (1998)
Goldreich, O.: Notes on levin’s theory of average-case complexity. Electronic Colloquium on Computational Complexity (ECCC) 4(58) (1997)
Goldwasser, S., Sipser, M.: Private coins versus public coins in interactive proof systems. In: Hartmanis, J. (ed.) STOC, pp. 59–68. ACM (1986)
Gutfreund, D., Shaltiel, R., Ta-Shma, A.: If NP languages are hard on the worst-case, then it is easy to find their hard instances. Computational Complexity 16(4), 412–441 (2007)
Gutfreund, D., Ta-Shma, A.: Worst-case to average-case reductions revisited. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX and RANDOM 2007. LNCS, vol. 4627, pp. 569–583. Springer, Heidelberg (2007)
Haitner, I., Mahmoody, M., Xiao, D.: A new sampling protocol and applications to basing cryptographic primitives on the hardness of NP. In: IEEE Conference on Computational Complexity [DBLP10], pp. 76–87
Impagliazzo, R., Levin, L.A.: No better ways to generate hard NP instances than picking uniformly at random. In: FOCS, pp. 812–821. IEEE Computer Society (1990)
Impagliazzo, R.: A personal view of average-case complexity. In: Structure in Complexity Theory Conference, pp. 134–147. IEEE Computer Society (1995)
Impagliazzo, R.: Relativized separations of worst-case and average-case complexities for NP. In: IEEE Conference on Computational Complexity, pp. 104–114. IEEE Computer Society (2011)
Lempel, A.: Cryptology in transition. ACM Comput. Surv. 11(4), 285–303 (1979)
Lyubashevsky, V., Micciancio, D.: On bounded distance decoding, unique shortest vectors, and the minimum distance problem. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 577–594. Springer, Heidelberg (2009)
Micciancio, D.: Almost perfect lattices, the covering radius problem, and applications to Ajtai’s connection factor. SIAM J. Comput. 34(1), 118–169 (2004)
Micciancio, D., Regev, O.: Worst-case to average-case reductions based on gaussian measures. SIAM J. Comput. 37(1), 267–302 (2007)
Mahmoody, M., Xiao, D.: On the power of randomized reductions and the checkability of sat. In: IEEE Conference on Computational Complexity [DBL10], pp. 64–75
Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem: Extended abstract. In: Mitzenmacher, M. (ed.) STOC, pp. 333–342. ACM (2009)
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6) (2009)
Regev, O.: The learning with errors problem (invited survey). In: IEEE Conference on Computational Complexity [DBL10], pp. 191–204 (2010)
Rubinfeld, R.: A mathematical theory of self-checking, self-testing and self-correcting programs. PhD thesis. UC Berkeley (1990)
Sudan, M., Trevisan, L., Vadhan, S.P.: Pseudorandom generators without the xor lemma. J. Comput. Syst. Sci. 62(2), 236–266 (2001)
Watson, T.: Relativized worlds without worst-case to average-case reductions for NP. TOCT 4(3), 8 (2012)
Yap, C.-K.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287–300 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Holenstein, T., Künzler, R. (2014). A New View on Worst-Case to Average-Case Reductions for NP Problems. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-08783-2_18
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08782-5
Online ISBN: 978-3-319-08783-2
eBook Packages: Computer ScienceComputer Science (R0)