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Bundles, presemifields and nonlinear functions

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Abstract

Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over GF(2n), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order p n, we derive bundles of functions over GF(p n) of the form λ(x)*ρ(x), where λ, ρ are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p 2 and give a representative of each. We compute all bundles of presemifields of orders p n ≤ 27 and in the isotopism class of GF(32) and we measure the differential uniformity of the derived λ(x)*ρ(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2).

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References

  1. Berger T.P., Canteaut A., Charpin P., Laigle-Chapuy Y. (2006) On almost perfect nonlinear functions over \(\mathbb F_{2^n}\) . IEEE Trans. Inform. Theory 52:4160–4170

    Article  MathSciNet  Google Scholar 

  2. Bosma W., Cannon J., Playoust C. (1997) The MAGMA algebra system I: the user language. J. Symbol. Comp. 24:235–265

    Article  MATH  MathSciNet  Google Scholar 

  3. Brinkmann M., Leander G.: On the classification of APN functions up to dimension five. In: Proceedings, International Workshop on Coding and Cryptography, April 16–20, 2007, INRIA-Rocquencourt, France, pp. 39–48 (2007).

  4. Budaghyan L., Carlet C., Leander G.: Another class of quadratic APN binomials over \(\mathbb F_{2^n}\) : the case n divisible by 4. In: Proceedings, International Workshop on Coding and Cryptography, April 16–20, 2007, INRIA-Rocquencourt, France, pp. 49–58 (2007).

  5. Budaghyan L., Carlet C., Felke P., Leander G.: An infinite class of quadratic APN functions which are not equivalent to power mappings, Cryptology ePrint Archive: Report 2005/359 http://eprint.iacr.org/2005/35. In: Proceedings ISIT, July 9–14, 2006, Seattle, USA, IEEE, pp. 2637–2641 (2006).

  6. Budaghyan L., Carlet C., Pott A. (2006) New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inform. Theory 52:1141–1152

    Article  MathSciNet  Google Scholar 

  7. Carlet C.: Boolean functions for cryptography and error-correcting codes; and, Vectorial Boolean functions for cryptography. In: Hammer P., Crama Y. (eds.) Boolean Methods and Models, CUP, Cambridge (to appear).

  8. Carlet C., Charpin P., Zinoviev V. (1998) Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15:125–156

    Article  MATH  MathSciNet  Google Scholar 

  9. Carlet C., Ding C. (2004) Highly nonlinear mappings. J. Complexity 20:205–244

    Article  MATH  MathSciNet  Google Scholar 

  10. Colbourn C.J., Dinitz J.H. (eds) (1996) The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton

    MATH  Google Scholar 

  11. Cordero M., Wene G.P. (1999) A survey of finite semifields. Discrete Math. 208/209:125–137

    Article  MathSciNet  Google Scholar 

  12. Coulter R.S., Matthews R.W. (1997) Planar functions and planes of Lenz-Barlotti Class II. Des. Codes Cryptogr. 10:167–184

    Article  MATH  MathSciNet  Google Scholar 

  13. Dobbertin H. (1999) Almost perfect nonlinear power functions on GF(2n): the Welch case. IEEE Trans. Inform. Theory 45:1271–1275

    Article  MATH  MathSciNet  Google Scholar 

  14. Edel Y., Kyureghyan G., Pott A. (2006) A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52:744–747

    Article  MathSciNet  Google Scholar 

  15. Galati J.C. (2004) A group extensions approach to relative difference sets. J. Combin. Designs 12:279–298

    Article  MATH  MathSciNet  Google Scholar 

  16. Horadam K.J.: Differential uniformity for arrays, cryptography and coding. In: Proceedings of the 9th IMA International Conference, LNCS 2898, pp. 115–124. Springer, Berlin (2003).

  17. Horadam K.J. (2006) A theory of highly nonlinear functions. In: Fossorier M., et al. (eds) AAECC-16, LNCS 3857. Springer, Berlin, pp. 87–100

    Google Scholar 

  18. Horadam K.J. (2007) Hadamard Matrices and Their Applications. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  19. Horadam K.J.: Transversals and graphs: bundles, CCZ and EA equivalence of functions, manuscript in preparation.

  20. Horadam K.J., Farmer D.G.: Bundles, presemifields and nonlinear functions. In: Proceedings, International Workshop on Coding and Cryptography, April 16–20, 2007, INRIA-Rocquencourt, France, pp. 197–206 (2007).

  21. Horadam K.J., Udaya P. (2002) A new construction of central relative (p a, p a, p a, 1)-difference sets. Des. Codes Cryptogr. 27:281–295

    Article  MATH  MathSciNet  Google Scholar 

  22. Knuth D.E. (1965) Finite semifields and projective planes. J. Algebra 2:182–217

    Article  MATH  MathSciNet  Google Scholar 

  23. Kyureghyan G.M. (2007) Crooked maps in \(\mathbb F_{2^n}\) . Finite Field Appl. 13:713–726

    Article  MATH  MathSciNet  Google Scholar 

  24. Leander G., Poschmann A. (2007) On the classification of 4-bit S-boxes. In: Carlet C., Sunar B. (eds) WAIFI 2007, LNCS 4547. Springer, Berlin, pp. 159–176

    Google Scholar 

  25. LeBel A., Horadam K.J.: Direct sums of balanced functions, perfect nonlinear functions and orthogonal cocycles. J. Combin. Designs (2008) to appear.

  26. Nakagawa N., Yoshiara S. (2007) A construction of differentially 4-uniform functions from commutative semifields of characteristic 2. In: Carlet C., Sunar B. (eds). WAIFI 2007, LNCS 4547. Springer, Berlin, pp. 134–146

    Google Scholar 

  27. Perera A.A.I., Horadam K.J. (1998) Cocyclic generalised Hadamard matrices and central relative difference sets. Des. Codes Cryptogr. 15:187–200

    Article  MATH  MathSciNet  Google Scholar 

  28. Pott A. (2004) Nonlinear functions in Abelian groups and relative difference sets. Discrete Appl. Math. 138:177–193

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. RMIT University, Melbourne, VIC, 3001, Australia

    K. J. Horadam & D. G. Farmer

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  1. K. J. Horadam
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  2. D. G. Farmer
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Correspondence to K. J. Horadam.

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This paper is dedicated to Hans Dobbertin, in memory of his inspiring work in nonlinear functions.

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Horadam, K.J., Farmer, D.G. Bundles, presemifields and nonlinear functions. Des. Codes Cryptogr. 49, 79–94 (2008). https://doi.org/10.1007/s10623-008-9172-z

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  • DOI: https://doi.org/10.1007/s10623-008-9172-z

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