Two Classes of Quadratic Crooked Functions

Abstract:

It is known that almost perfect nonlinear (APN) functions have many applications in cryptography, and a quadratic function is crooked if and only if it is APN. In this paper, we introduce two infinite classes of quadratic crooked multinomials on fields of order 2^2m. One class of APN functions constructed in [8] is a particular case of the one we construct in Theorem 1. Moreover, we prove that the two classes of crooked functions constructed in this paper are EA inequivalent to power functions and conjecture that CCZ inequivalence between them also holds.

• Periodical: Applied Mechanics and Materials (Volumes 513-517)
• Pages: 2734-2738
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.513-517.2734
• Online since: February 2014
• Authors: Xue Ying Duan*, Yu Long Deng
• Keywords: Almost Perfect Nonlinear, Crooked Functions, Cryptography, Ea Equivalence

References

[1] K. J. Horadam, EA and CCZ equivalence of functions over , volume 5130 of Lecture Notes in Computer Science, pages 134-143, Springer-Verlag, (2008).

DOI: 10.1007/978-3-540-69499-1_12

[2] G. Kyureghyanr, Crooked maps in , Finite Fields Their Applic., 2007, 13: 713-726.

[3] J. Bierbrauer and G. M. Kyureghyanr, Crooked binomials, Des. Codes Cryptogr., 2008, 46: 269-301.

DOI: 10.1007/s10623-007-9157-3

[4] H. Dobbertin, Almost perfect nonlinear power functions over : A new case for n divisible by 5, In Proc. Conf. Finite Fields and Applications FQ5, D. Jungnickel and H. Niederreiter, Eds. Augsburg, Germany: Springer, 2000. 113-121.

DOI: 10.1007/978-3-642-56755-1_11

[5] L. Budaghyan and C. Carlet, Classes of quadratic APN trinomials and Hexanomials and related structures, IEEE Trans. Inf. Theory, 2008, 54(5): 2354-2357.

DOI: 10.1109/tit.2008.920246

[6] L. Budaghyan, C. Carlet, and A. Pott, New classes of almost bent and almost perfect nonlinear functions, IEEE Trans. Inf. Theory, 2006, 52(3): 1141-1152.

DOI: 10.1109/tit.2005.864481

[7] T. Bethand and C. Ding, On almost perfect nonlinear permutations, In Advances in Cryptology - EUROCRYPT 93, volume 765 of Lecture Notes in Computer Science, pages 65-76, Springer-Verlag, (1994).

DOI: 10.1007/3-540-48285-7_7

[8] C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite Fields Their Applic., 2008, 14: 703-714.

DOI: 10.1016/j.ffa.2007.11.002

[9] Q. Wang, J. Peng, H. Kan and X, Xue, Constructions of cryptographically significant Boolean functions using primitive polynomials, IEEE Trans. Inf. Theory, 2010, 56(6): 3048–3053.

DOI: 10.1109/tit.2010.2046195

[10] Q. Wang, T. Johansson and H. Kan, Some results on fast algebraic attacks and higher-order non-linearities, IET Inf. Secur., 2012, 6(1): 41–46.

DOI: 10.1049/iet-ifs.2011.0090

[11] Q. Wang and C. H. Tan, A new method to construct Boolean functions with good cryptographic properties, Inform. Process. Lett., 2013, 113: 567–571.

DOI: 10.1016/j.ipl.2013.04.017

[12] E. R. VanDam and D. Fon-Der-Flaass, Uniformly packed codes and more distance regular graphs from crooked functions, J. Algebraic Combin., 2000, 12(2): 115-121.

DOI: 10.37236/1372

[13] E. R. VanDam and D. Fon-Der-Flaass, Codes, graphs, and schemes from nonlinear functions, European J. Combin., 2003, 24(1): 85-98.

DOI: 10.1016/s0195-6698(02)00116-6