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On the Complexity of Computing Two Nonlinearity Measures

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Computer Science - Theory and Applications (CSR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8476))

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Abstract

We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time 2O(n) given the truth table of length 2n, in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of (2 − ε)n/2. When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit.

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References

  1. Arora, S., Barak, B.: Computational Complexity - A Modern Approach, pp. 1–579. Cambridge University Press (2009)

    Google Scholar 

  2. Boyar, J., Peralta, R., Pochuev, D.: On the multiplicative complexity of Boolean functions over the basis (∧,⊕,1). Theoretical Computer Science 235(1), 43–57 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boyar, J., Damgård, I., Peralta, R.: Short non-interactive cryptographic proofs. J. Cryptology 13(4), 449–472 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boyar, J., Find, M., Peralta, R.: Four measures of nonlinearity. In: Spirakis, P.G., Serna, M. (eds.) CIAC 2013. LNCS, vol. 7878, pp. 61–72. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  5. Boyar, J., Matthews, P., Peralta, R.: Logic minimization techniques with applications to cryptology. J. Cryptology 26(2), 280–312 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, ch. 8, pp. 257–397. Cambridge Univ. Press, Cambridge (2010)

    Chapter  Google Scholar 

  7. Cenk, M., Özbudak, F.: On multiplication in finite fields. J. Complexity 26(2), 172–186 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: Simon, J. (ed.) STOC, pp. 11–19. ACM (1988)

    Google Scholar 

  9. Courtois, N., Bard, G.V., Hulme, D.: A new general-purpose method to multiply 3x3 matrices using only 23 multiplications. CoRR abs/1108.2830 (2011)

    Google Scholar 

  10. Daemen, J., Rijmen, V.: AES proposal: Rijndael (1999), http://csrc.nist.gov/archive/aes/rijndael/Rijndael-ammended.pdf

  11. Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kabanets, V., yi Cai, J.: Circuit minimization problem. In: Yao, F.F., Luks, E.M. (eds.) STOC, pp. 73–79. ACM (2000)

    Google Scholar 

  15. Kolesnikov, V., Schneider, T.: Improved garbled circuit: Free XOR gates and applications. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 486–498. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  16. O’Donnell, R.: Analysis of Boolean Functions. Book draft (2012), http://www.analysisofbooleanfunctions.org

  17. Razborov, A.A., Rudich, S.: Natural proofs. J. Comput. Syst. Sci. 55(1), 24–35 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sergeev, I.S.: A relation between additive and multiplicative complexity of Boolean functions. CoRR abs/1303.4177 (2013)

    Google Scholar 

  19. Sloane, N., MacWilliams, F.J.: The Theory of Error-Correcting Codes. North-Holland Math. Library 16 (1977)

    Google Scholar 

  20. Vaikuntanathan, V.: Computing blindfolded: New developments in fully homomorphic encryption. In: Ostrovsky, R. (ed.) FOCS, pp. 5–16. IEEE (2011)

    Google Scholar 

  21. Wegener, I.: The Complexity of Boolean Functions. Wiley-Teubner (1987)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Denmark

    Magnus Gausdal Find

Authors
  1. Magnus Gausdal Find
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Editor information

Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, 27 Fontanka,, 191023, St. Petersburg, Russia

    Edward A. Hirsch

  2. Higher School of Economics, National Research University, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. CNRS and University Paris Diderot, Paris, France

    Jean-Éric Pin

  4. Moscow State University, 119992, Moscow, Russia

    Nikolay K. Vereshchagin

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Find, M.G. (2014). On the Complexity of Computing Two Nonlinearity Measures. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_13

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  • DOI: https://doi.org/10.1007/978-3-319-06686-8_13

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06685-1

  • Online ISBN: 978-3-319-06686-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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