Abstract
Let \(g\) be a function over \(\mathbb {F}_q\). If there exist a function \(f\) and \(a\in \mathbb {F}_q^*\) such that \(g(x)=f(x+a)-f(x)\), then we call \(g\) a differential function and call \(f\) a differential-inverse of \(g\). We present two criteria to decide whether a given \(g\) is a differential function. The set of the degrees of all differential functions over \({\mathbb {F}}_q\) is determined. Then we give a lower bound and an upper bound on the number of differential functions over \({\mathbb {F}}_q\). Besides, we show how to construct differential inverses of a given differential function. At last, some applications of our results are introduced.
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References
Berger, T.P., Canteaut, A., Charpin, P., Laigle, C.Y.: On almost perfect nonlinear functions over \(\mathbb{F}_2^n\). IEEE Trans. Inf. Theory 52(9), 4160–4170 (2006)
Bierbrauer, J.: Introduction to Coding Theory. Chapman & Hall/CRC, London (2005)
Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptogr. 4(1), 3–72 (1991)
Carlet, C.: Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications. IEEE Trans. Inf. Theory 54(3), 1262–1272 (2008)
Carlet, C., Ding, C., Yuan, J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2012 (2005)
Carlet, C., Charpin, P., Zinoviev, V.: Codes, Bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)
Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009)
Golomb, S.W., Gong, G.: Signal Design for Good Correlation. Cambridge University Press, New York (2005)
Granville, A.: Arithmetic properties of binomial coefficients I: binomial coefficients modulo prime powers. Can. Math. Soc. Conf. Proc. 20, 253–275 (1997)
Hasse, H.: Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik. J. Reine. Ang. Math. 175, 50–54 (1936)
Lee, S., Park, T.: Efficient interpolation architecture for soft-decision list decoding Reed–Solomon codes. In: ICWMC, pp. 25–30 (2012)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, New York (2003)
Massey, J.L., Seemann, N., Schoeller, P.A.: Hasse derivatives and repeated-root cyclic codes. In: IEEE International Symposium on Information Theory, ISIT (1986)
Nyberg, K., Differentially uniform mappings for cryptography. In: EUROCRYPT 1993, LNCS 765, pp. 55–64. Springer, New York (1994)
Olive, H.M., Souza, R.M.C.: Introducing an Analysis in Finite Fields. http://www2.ee.ufpe.br/codec/Analise_Gf
Qu, L., Tan, Y., Tan, C., Li, C.: Constructing differentially 4-uniform permutations over \(\mathbb{F}_{2^{2k}}\) via the switching method. IEEE Trans. Inf. Theory 59(7), 4675–4686 (2013)
Rocha, V.C.: Digital sequence and the Hasse derivative. Commun. Coding Signal Process. 3, 256–268 (1997)
van Lint, J.H.: Introduction to Coding Theory, 3rd edn. Springer, Berlin (2003)
Zhou, Y., Pott, A.: A new family of semifields with 2 parameters. Adv. Math. 234, 43–60 (2013)
Acknowledgments
The authors would like to thank the anonymous referees for the helpful comments, which improve much both on the technical quality and the editorial quality of the paper. The work was supported by the the Research Project of National University of Defense Technology (No. CJ 13-02-01), the open research fund of Science and Technology on Information Assurance Laboratory (Grant No. KJ-12-02), the Hunan Provincial Innovation Foundation for Postgraduate (No. CX2013B007) and the Innovation Foundation of NUDT (No. B130201).
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Xiong, H., Qu, L., Li, C. et al. Some results on the differential functions over finite fields. AAECC 25, 189–195 (2014). https://doi.org/10.1007/s00200-014-0220-9
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DOI: https://doi.org/10.1007/s00200-014-0220-9