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Quaternary cryptographic bent functions and their binary projection

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Abstract

The nonlinearity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, bent functions have been generalized to the alphabet ℤ q  = ℤ/qℤ and also studied by several authors. This generalization maps the classical definition of binary bent functions to generalized bent functions. This paper propose a new approach for generalized bent functions over ℤ4 and their associated derived boolean functions. We present in this paper a new construction of a family of m-variable quaternary (ℤ4 = ℤ/4ℤ = {0,1,2,3}-valued) bent functions over Galois ring and cyclotomic classes using an intern function defined on a particular partition of the Teichmüller set. Using a particular binary projection map we obtain a family of 2m-variable boolean bent functions and a family of 2m + 1-variable boolean plateaued functions of amplitude 2m + 1 with nonlinearity equal to 4m − 2m.

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References

  1. Carlet, C.: Partially-bent functions. In: Proceedings of CRYPTO’ 92, Advances in Cryptology, LNCS vol. 740, pp. 280–291 (1993)

  2. Carlet, C.: One-weight ℤ4-linear codes. In: Proceedings of International Conference on Coding Theory, Cryptography and Related Areas 1998, vol. 8, pp. 57–72. Springer (1998)

  3. Carlet, C., Ding, C.: Highly nonlinear mappings, special issue “Complexity issues in coding and cryptography”. J. Complex. 20, 205–244 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlet, C., Dubuc, S.: On generalized bent and q −ary perfect nonlinear functions. Finite Fields Appl. 1999, 81–94 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carlet, C., Prouff, E.: On plateaued functions and their constructions. In: Fast Software Encryption, LNCS, vol. 2887, pp. 54–73 (2003)

  6. Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Advances in Cryptology, EUROCRYPT 2003, LNCS, vol. 2656, pp. 345–359 (2003)

  7. Courtois, N., Pieprzyk, J.: Cryptanalysis of block ciphers with over-defined systems of equations. In: Advances in Cryptology, ASIACRYPT 2002, LNCS, vol. 2501, pp. 267–287 (2002)

  8. Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The ℤ4 linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)

    Article  MATH  Google Scholar 

  9. Hou, X.: q −ary bent functions constructed from chain rings. Finite Fields Appl. 4, 55–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hou, X.-D.: Bent functions, partial difference sets and quasi-Frobenius rings. Designs Codes Cryptogr. 20, 251–268 (2000)

    Article  MATH  Google Scholar 

  11. Hou, X.: p-ary and q-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl. 10, 566–582 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jadda, Z., Parraud, P.: ℤ4-Nonlinearity of a constructed quaternary cryptographic functions class. In: Proc. 6th SETA, pp. 81–94 (2010)

  13. Kavut, S., Maitra, S., Sarkar, S., Yücel, M.D.: Enumeration of 9 variables rotation symetric boolean functions having non-linearity > 240. In: Advances in Cryptology, INDOCRYPT 2006, LNCS, vol. 4329, pp. 266–279 (2006)

  14. Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theory, Ser A 1(40), 90–107 (1985)

    Article  MathSciNet  Google Scholar 

  15. Kumar, P.V., Hellesth, T., Calderbank, A.R., Hammons, A.R.: Large families of quaternary sequences with low correlation. IEEE Trans. Inf. Theory 42(2), 579–592 (1996)

    Article  MATH  Google Scholar 

  16. Li, N., Qi, W.F.: Construction and analysis of boolean functions of 2t + 1 variables with maximum algebraic immunity. In: Advances in Cryptology, ASIACRYPT 2006, LNCS, vol. 4284, pp. 84–98 (2006)

  17. McDonald, B.R.: Finite Rings with Identity. Marcel Dekker (1974)

  18. Meier, W., Staffelbach, O.: Nonlinearity criteria for cryptographic functions. In: Advances in Cryptology, EUROCRYPT 1989, LNCS, vol. 434, pp. 549–562 (1990)

  19. Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of boolean functions In: Advances in Cryptology, EUROCRYPT 2004, LNCS, vol. 3027, pp. 474–491 (2004)

  20. Rothaus, O.S.: On bent functions. J. Comb. Theory 20, 300–305 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  21. Saber, Z., Uddin, M.F., Youssef, A.: On the existence of (9,3,5,240) resilient functions. IEEE Trans. Inf. Theory 52(5), 2269–2270 (2006)

    Article  MathSciNet  Google Scholar 

  22. Singh, D., Bhaintwal, M., Singh, B.K.: Recent results on generalized q-ary bent functions. In: Cryptology ePrint Archives http://www.eprint.iacr.org/2012/037 (2012)

  23. Solé, P., Tokareva, N.: Connections between quaternary and binary bent functions. In: Cryptology ePrint Archives http://www.eprint.iacr.org/2009/544 (2009)

  24. Tokareva, N.: Generalizations of bent functions, a survey to appear in Journal of Applied and industrial Mathematics. In: Cryptology ePrint Archives http://www.eprint.iacr.org/2011/111 (2011)

  25. Yang, K., Helleseth, T., Kumar, P.V., Shanbhag, A.: On the weighthierarchy of Kerdock codes over ℤ4. IEEE Trans. Inf. Theory 42(5), 1587–1593 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zheng, Y., Zhang, X.M.: Plateaued functions. In: Advances in Cryptology, ICICS99, LNCS, vol. 1726, pp. 284–300 (1999)

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Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments and suggestions that helped improving the paper.

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Authors and Affiliations

  1. ESCC/CREC, UR Macclia, Ministère de la défense - Base de défense de Vannes-Coëtquidan, Guer, France

    Zoubida Jadda & Patrice Parraud

  2. Faculty of des Sciences, Lmca, University of Mohamed V Agdal, Rabat, Morocco

    Soukayna Qarboua

Authors
  1. Zoubida Jadda
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  2. Patrice Parraud
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  3. Soukayna Qarboua
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Correspondence to Patrice Parraud.

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Jadda, Z., Parraud, P. & Qarboua, S. Quaternary cryptographic bent functions and their binary projection. Cryptogr. Commun. 5, 49–65 (2013). https://doi.org/10.1007/s12095-012-0077-3

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  • DOI: https://doi.org/10.1007/s12095-012-0077-3

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