Advertisement

Quantum Algorithms Related to \(\textit{HN}\)-Transforms of Boolean Functions

  • Sugata GangopadhyayEmail author
  • Subhamoy Maitra
  • Nishant Sinha
  • Pantelimon Stănică
Conference paper
  • 663 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10194)

Abstract

\(\textit{HN}\)-transforms, which have been proposed as generalizations of Hadamard transforms, are constructed by tensoring Hadamard and nega-Hadamard kernels in any order. We show that all the \(2^n\) possible \(\textit{HN}\)-spectra of a Boolean function in n variables, each containing \(2^n\) elements (i.e., in total \(2^{2n}\) values in transformed domain) can be computed in \(O(2^{2n})\) time (more specific with little less than \(2^{2n+1}\) arithmetic operations). We propose a generalization of Deutsch-Jozsa algorithm, by employing \(\textit{HN}\)-transforms, which can be used to distinguish different classes of Boolean functions over and above what is possible by the traditional Deutsch-Jozsa algorithm.

Keywords

Boolean function \(\textit{HN}\)-transform, Deutsch-Jozsa algorithm 

References

  1. 1.
    Aumasson, J.-P., Dinur, I., Meier, W., Shamir, A.: Cube testers and key recovery attacks on reduced-round MD6 and trivium. In: Dunkelman, O. (ed.) FSE 2009. LNCS, vol. 5665, pp. 1–22. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-03317-9_1 CrossRefGoogle Scholar
  2. 2.
    Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego (2017). 1st edn. (2009)Google Scholar
  3. 3.
    Danielsen, L.E., Parker, M.G.: Spectral orbits and peak-to-average power ratio of boolean functions with respect to the \(\mathit{I},\mathit{H},\mathit{N}^{\mathit{n}}\) transform. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 373–388. Springer, Heidelberg (2005). doi: 10.1007/11423461_28 CrossRefGoogle Scholar
  4. 4.
    Danielsen, L.E.: On connections between graphs, codes, quantum states, and Boolean functions. Ph.D. thesis, Department of Informatics, The Selmer Center, University of Bergen, Norway (2008)Google Scholar
  5. 5.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. Roy. Soc. Lond. A439, 553–558 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974)Google Scholar
  7. 7.
    Dinur, I., Shamir, A.: Cube attacks on tweakable black box polynomials. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 278–299. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-01001-9_16. See also: Cube Attacks on Tweakable Black Box Polynomials. http://eprint.iacr.org/2008/385.pdf CrossRefGoogle Scholar
  8. 8.
    Gangopadhyay, S., Pasalic, E., Stănică, P.: A note on generalized bent criteria for boolean functions. IEEE Trans. Inf. Theor. 59(5), 3233–3236 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gangopadhyay, S., Gangopadhyay, A.K., Pollatos, S., Stănică, P.: Cryptographic Boolean functions with biased inputs. Crypt. Commun. Discrete Struct. Seq. 9, 301–314 (2017). doi: 10.1007/s12095-015-0174-1 MathSciNetzbMATHGoogle Scholar
  10. 10.
    Helleseth, T., Kløve, T., Mvkkeltveit, J.: On the covering radius of binary codes. IEEE Trans. Inf. Theor. 24(5), 627–628 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Knudsen, L.R.: Truncated and higher order differentials. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 196–211. Springer, Heidelberg (1995). doi: 10.1007/3-540-60590-8_16 CrossRefGoogle Scholar
  12. 12.
    Litsyn, S., Shpunt, A.: On the distribution of Boolean function nonlinearity. SIAM J. Discrete Math. 23(1), 79–95 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Maitra, S., Mukhopadhyay, P.: Deutsch-Jozsa algorithm revisited in the domain of cryptographically significant boolean functions. Int. J. Quantum Inf. 3(2), 359–370 (2005)CrossRefzbMATHGoogle Scholar
  14. 14.
    Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994). doi: 10.1007/3-540-48285-7_33 Google Scholar
  15. 15.
    Meier, W., Staffelbach, O.: Fast correlation attacks on stream ciphers. In: Barstow, D., Brauer, W., Brinch Hansen, P., Gries, D., Luckham, D., Moler, C., Pnueli, A., Seegmüller, G., Stoer, J., Wirth, N., Günther, C.G. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 301–314. Springer, Heidelberg (1988). doi: 10.1007/3-540-45961-8_28 Google Scholar
  16. 16.
    Parker, M.G.: Generalised \(S\)-box nonlinearity. NESSIE Public Document, 11.02.03: NES/DOC/UIB/WP5/020/AGoogle Scholar
  17. 17.
    Parker, M.G., Pott, A.: On boolean functions which are bent and negabent. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (eds.) SSC 2007. LNCS, vol. 4893, pp. 9–23. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-77404-4_2 CrossRefGoogle Scholar
  18. 18.
    Patterson, N.J., Wiedemann, D.H.: The covering radius of the \((2^{15}, 16)\) Reed-Muller code is at least 16276. IEEE Trans. Inf. Theor. 29(3), 354–356 (1983). See also correction: IEEE Trans. Inf. Theor. 36(2), 443 (1990)CrossRefzbMATHGoogle Scholar
  19. 19.
    Riera, C.: Spectral properties of Boolean functions, graphs and graph states. Ph.D. thesis, University of Bergen (2005)Google Scholar
  20. 20.
    Riera, C., Parker, M.G.: Generalized bent criteria for Boolean functions. IEEE Trans. Inf. Theor. 52(9), 4142–4159 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schmidt, K.-U., Parker, M.G., Pott, A.: Negabent functions in the Maiorana–McFarland class. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 390–402. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-85912-3_34 CrossRefGoogle Scholar
  22. 22.
    Siegenthaler, T.: Decrypting a class of stream ciphers using ciphertext only. IEEE Trans. Comput. 34(1), 81–85 (1985)CrossRefGoogle Scholar
  23. 23.
    Stănică, P., Maitra, S.: Rotation symmetric Boolean functions - count and cryptographic properties. Disc. Appl. Math. 156, 1567–1580 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stănică, P., Gangopadhyay, S., Chaturvedi, A., Kar-Gangopadhyay, A., Maitra, S.: Investigations on bent and negabent functions via the nega-Hadamard transform. IEEE Trans. Inf. Theor. 58(6), 4065–4072 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sugata Gangopadhyay
    • 1
    Email author
  • Subhamoy Maitra
    • 2
  • Nishant Sinha
    • 1
  • Pantelimon Stănică
    • 3
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Applied Statistics UnitIndian Statistical InstituteKolkataIndia
  3. 3.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

Personalised recommendations