Abstract
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.
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References
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
Biham E, Shamir A. Differential cryptanalysis of DES-like cryptosystems. J Cryptography, 1991, 4: 3–72
Bracken C, Byrne E, Markin N, et al. A few more quadratic APN functions. Cryptography Commun, 2011, 3: 43–53
Budaghyan L, Carlet C. Constructing new APN functions from known ones. Finite Fields Appl, 2009, 15: 150–159
Carlet C. Boolean functions for cryptography and error correcting codes. In: Hammer P, Crama Y, eds. Boolean Methods and Models. Cambridge: Cambridge University Press, 2010
Carlet C. Vectorial Boolean functions for cryptography and error correcting codes. In: Hammer P, Crama Y, eds. Boolean methods and Models. Cambridge: Cambridge University Press, 2010
Carlet C. Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des Codes Cryptogr, 2011, 59: 89–109
Courtois N, Pieprzyk J. Cryptanalysis of block ciphers with overdefined systems of equations. In: Advances in Cryptology-ASIACRYPT, vol. 2501. Berlin: Springer-Verlag, 2002, 267–287
Dillon J. APN polynomials: An update. In: 9th International Conference on Finite Fields and Applications of Fq9. Dublin, 2009, http://mathsci.ucd.ie/~gmg/Fq9Talks/Dillon.pdf
Dobbertin H, Mills D, Muller E, et al. APN functions in odd characteristic. Discrete Math, 2003, 267: 95–112
Knudsen L. Truncated and higher order differentials. In: Lecture Notes in Computer Science, vol. 1008. Berlin: Springer, 1995, 196–211
Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv Math Commun, 2009, 3: 59–81
Matsui M. Linear cryptanalysis method for DES cipher. In: Lecture Notes in Computer Science, vol. 765. Berlin: Springer, 1994, 386–397
Nyberg K. Differentially uniform mappings for cryptography. In: Lecture Notes in Computer Science, vol. 765. Berlin: Springer, 1994, 55–64
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
Zhou Y, Pott A. A new family of semifields with 2 parameters. Adv Math, 2013, 234: 43–60
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Qu, L., Li, C., Dai, Q. et al. On the differential uniformities of functions over finite fields. Sci. China Math. 56, 1477–1484 (2013). https://doi.org/10.1007/s11425-013-4658-1
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DOI: https://doi.org/10.1007/s11425-013-4658-1