On the Complexity of Simulating Auxiliary Input

  • Yi-Hsiu ChenEmail author
  • Kai-Min ChungEmail author
  • Jyun-Jie LiaoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10822)


We construct a simulator for the simulating auxiliary input problem with complexity better than all previous results and prove the optimality up to logarithmic factors by establishing a black-box lower bound. Specifically, let \(\ell \) be the length of the auxiliary input and \(\epsilon \) be the indistinguishability parameter. Our simulator is \(\tilde{O}(2^{\ell }\epsilon ^{-2})\) more complicated than the distinguisher family. For the lower bound, we show the relative complexity to the distinguisher of a simulator is at least \(\varOmega (2^{\ell }\epsilon ^{-2})\) assuming the simulator is restricted to use the distinguishers in a black-box way and satisfy a mild restriction.


  1. [AHK12]
    Arora, S., Hazan, E., Kale, S.: The multiplicative weights update method: a meta-algorithm and applications. Theory Comput. 8(1), 121–164 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AS11]
    Artemenko, S., Shaltiel, R.: Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM -2011. LNCS, vol. 6845, pp. 377–388. Springer, Heidelberg (2011). Scholar
  3. [CLP15]
    Chung, K.-M., Lui, E., Pass, R.: From weak to strong zero-knowledge and applications. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9014, pp. 66–92. Springer, Heidelberg (2015). Scholar
  4. [DBL08]
    49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, USA, 25–28 October 2008. IEEE Computer Society (2008)Google Scholar
  5. [DP08]
    Dziembowski, S., Pietrzak, K.: Leakage-resilient cryptography. In: 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, USA, 25–28 October 2008 [DBL08], pp. 293–302Google Scholar
  6. [FK99]
    Frieze, A.M., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19(2), 175–220 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Fre95]
    Freund, Y.: Boosting a weak learning algorithm by majority. Inf. Comput. 121(2), 256–285 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [FS96]
    Freund, Y., Schapire, R.E.: Game theory, on-line prediction and boosting. In: Blum, A., Kearns, M. (eds.) Proceedings of the Ninth Annual Conference on Computational Learning Theory, COLT 1996, Desenzano del Garda, Italy, 28 June–1 July 1996, pp. 325–332. ACM (1996)Google Scholar
  9. [GNW95]
    Goldreich, O., Nisan, N., Wigderson, A.: On Yao’s XOR-lemma. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 2, no. 50 (1995)Google Scholar
  10. [GW11]
    Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Fortnow, L., Vadhan, S.P. (eds.) Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6–8 June 2011, pp. 99–108. ACM (2011)Google Scholar
  11. [HILL99]
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Hol05]
    Holenstein, T.: Key agreement from weak bit agreement. In: Gabow, H.N., Fagin, R. (eds.) Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 22–24 May 2005, pp. 664–673. ACM (2005)Google Scholar
  13. [HS16]
    Hirt, M., Smith, A. (eds.): TCC 2016-B. LNCS, vol. 9985. Springer, Heidelberg (2016). Scholar
  14. [Imp95]
    Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23–25 October 1995, pp. 538–545. IEEE Computer Society (1995)Google Scholar
  15. [JP14]
    Jetchev, D., Pietrzak, K.: How to fake auxiliary input. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 566–590. Springer, Heidelberg (2014). Scholar
  16. [LTW11]
    Lu, C.-J., Tsai, S.-C., Wu, H.-L.: Complexity of hard-core set proofs. Comput. Complex. 20(1), 145–171 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [PS16]
    Pietrzak, K., Skórski, M.: Pseudoentropy: lower-bounds for chain rules and transformations. In: Hirt and Smith [HS16], pp. 183–203Google Scholar
  18. [Rou16]
    Roughgarden, T.: No-Regret Dynamics, pp. 230–246. Cambridge University Press, Cambridge (2016)Google Scholar
  19. [RTTV08]
    Reingold, O., Trevisan, L., Tulsiani, M., Vadhan, S.P.: Dense subsets of pseudorandom sets. In: 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, USA, 25–28 October 2008 [DBL08], pp. 76–85Google Scholar
  20. [Skó16a]
    Skórski, M.: Simulating auxiliary inputs, revisited. In: Hirt and Smith [HS16], pp. 159–179Google Scholar
  21. [Skó16b]
    Skórski, M.: A subgradient algorithm for computational distances and applications to cryptography. IACR Cryptology ePrint Archive, 2016:158 (2016)Google Scholar
  22. [SV10]
    Shaltiel, R., Viola, E.: Hardness amplification proofs require majority. SIAM J. Comput. 39(7), 3122–3154 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [TTV09]
    Trevisan, L., Tulsiani, M., Vadhan, S.P.: Regularity, boosting, and efficiently simulating every high-entropy distribution. In: Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15–18 July 2009, pp. 126–136. IEEE Computer Society (2009)Google Scholar
  24. [VZ13]
    Vadhan, S., Zheng, C.J.: A uniform min-max theorem with applications in cryptography. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 93–110. Springer, Heidelberg (2013). Scholar
  25. [Yao82]
    Yao, A.C.-C.: Theory and applications of trapdoor functions (extended abstract). In 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, 3–5 November 1982, pp. 80–91. IEEE Computer Society (1982)Google Scholar
  26. [Zha11]
    Zhang, J.: On the query complexity for showing dense model. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 18, p. 38 (2011)Google Scholar

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  1. 1.Harvard John A. Paulson School of Engineering and Applied SciencesHarvard UniversityCambridgeUSA
  2. 2.Institute of Information Science, Academia SinicaTaipeiTaiwan

Personalised recommendations