On the differential uniformities of functions over finite fields. (English) Zbl 1303.94100
Summary: In this paper, the possible value of the differential uniformity of a function over finite fields is discussed. It is proved that the differential uniformity of a function over \(\mathbb F_q\) can be any even integer between 2 and \(q\) when \(q\) is even; and it can be any integer between 1 and \(q\) except \(q-1\) when \(q\) is odd. Moreover, for any possible differential uniformity \(t\), an explicit construction of a differentially \(t\)-uniform function is given.
References:
| [1] | Berger T, Canteaut A, Charpin P, et al. On almost perfect nonlinear functions over \[\mathbb{F}_2^n\]. IEEE Trans Inform Theory, 2006, 52: 4160-4170 · Zbl 1184.94224 · doi:10.1109/TIT.2006.880036 |
| [2] | Biham E, Shamir A. Differential cryptanalysis of DES-like cryptosystems. J Cryptography, 1991, 4: 3-72 · Zbl 0729.68017 |
| [3] | Bracken C, Byrne E, Markin N, et al. A few more quadratic APN functions. Cryptography Commun, 2011, 3: 43-53 · Zbl 1282.11162 · doi:10.1007/s12095-010-0038-7 |
| [4] | Budaghyan L, Carlet C. Constructing new APN functions from known ones. Finite Fields Appl, 2009, 15: 150-159 · Zbl 1184.94228 · doi:10.1016/j.ffa.2008.10.001 |
| [5] | Carlet, C.; Hammer, P. (ed.); Crama, Y. (ed.), Boolean functions for cryptography and error correcting codes (2010), Cambridge · Zbl 1209.94035 |
| [6] | Carlet, C.; Hammer, P. (ed.); Crama, Y. (ed.), Vectorial Boolean functions for cryptography and error correcting codes (2010), Cambridge |
| [7] | Carlet C. Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des Codes Cryptogr, 2011, 59: 89-109 · Zbl 1229.94041 · doi:10.1007/s10623-010-9468-7 |
| [8] | Courtois, N.; Pieprzyk, J., Cryptanalysis of block ciphers with overdefined systems of equations, 267-287 (2002), Berlin · Zbl 1065.94543 |
| [9] | Dillon, J., APN polynomials: An update (2009) |
| [10] | Dobbertin H, Mills D, Muller E, et al. APN functions in odd characteristic. Discrete Math, 2003, 267: 95-112 · Zbl 1028.11076 · doi:10.1016/S0012-365X(02)00606-4 |
| [11] | Knudsen, L., Truncated and higher order differentials, 196-211 (1995), Berlin · Zbl 0939.94556 |
| [12] | Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv Math Commun, 2009, 3: 59-81 · Zbl 1231.11140 · doi:10.3934/amc.2009.3.59 |
| [13] | Matsui, M., Linear cryptanalysis method for DES cipher, 386-397 (1994), Berlin · Zbl 0951.94519 |
| [14] | Nyberg, K., Differentially uniform mappings for cryptography, 55-64 (1994), Berlin · Zbl 0951.94510 |
| [15] | Qu L, Tan Y, Tan C, et al. Constructing differentially 4-uniform permutations over \[\mathbb{F}_{2^k }\] via the switching method. IEEE Trans Inform Theory, 2013, 59: 4675-4686 · Zbl 1364.94565 · doi:10.1109/TIT.2013.2252420 |
| [16] | Zhou Y, Pott A. A new family of semifields with 2 parameters. Adv Math, 2013, 234: 43-60 · Zbl 1296.12007 · doi:10.1016/j.aim.2012.10.014 |
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