On the Carlet's Bivariate APN Construction and Representations of Dobbertin Power Functions

Abstract

In this paper we present necessary and sufficient conditions on some families of bivariate and biprojective APN functions. In the first part, we follow the extended methods given by Calderini et al in 2021, and give sufficient and necessary conditions for Carlet's bivariate APN functions. We also give APN-ness conditions for biprojective functions in some special cases. In the second part, we study the representation of the Dobbertin power function following methods given by Budaghyan et al. in 2022 and give new representations for this kind of functions.

Keywords

• APN functions
• bivariate APN construction
• Dobbertin power function

References

1. E. Biham, A. Shamir, Differential cryptanalysis of des-like cryptosystems, Journal of CRYPTOLOGY, 4(1), (1991) 3–72.
2. M. Calderini, L. Budaghyan, C. Carlet, On known constructions of apn and ab functions and their relation to each other, Rad Hrvatske akademije znanosti i umjetnosti: Matematiˇcke znanosti(546= 25), (2021) 79–105.
3. R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1), (1968) 154–156.
4. K. Nyberg, Differentially uniform mappings for cryptography, in: Workshop on the Theory and Application of Cryptographic Techniques, Springer, 1994, pp. 55–64.
5. H. Janwal, R. Wilsonat, Hyperplane sections of fermat varieties in p3 in char. 2 and some applications to cyclic, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 10th International Symposium, AAECC-10, San Juan de Puerto Rico, Puerto Rico, May 10-14, 1993. Proceedings, Vol. 673, Springer Science & Business Media, 1993, p. 180.
6. T. Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary reed-muller codes, Information and Control, 18(4), (1971) 369–394.
7. H. Dobbertin, Almost perfect nonlinear power functions on gf(2n): The welch case, IEEE Transactions on Information Theory, 45(4), (1999) 1271–1275.
8. H. Dobbertin, Almost perfect nonlinear power functions on gf (2n): the niho case, Information and Computation, 151(1-2), (1999) 57–72.

9. T. Beth, C. Ding, On almost perfect nonlinear permutations, in: Workshop on the Theory and Application of Cryptographic Techniques, Springer, 1994, pp. 65–76.
10. H. Dobbertin, Almost perfect nonlinear power functions on gf(2n): A new case for n divisible by 5, in: Finite Fields and Applications, Springer, 2001, pp. 113–121.
11. L. Budaghyan, M. Calderini, C. Carlet, D. Davidova, N. S. Kaleyski, On two fundamental problems on apn power functions, IEEE Transactions on Information Theory, 68(5), (2022) 3389–3403.
12. D. Davidova, On properties of bent and almost perfect nonlinear functions, PhD Dissertation, University of Bergen (2021).
13. F. Chabaud, S. Vaudenay, Links between differential and linear cryptanalysis, in: Workshop on the Theory and Application of Cryptographic Techniques, Springer, 1995, pp. 356–365.
14. S. Yoshiara, Equivalences of quadratic apn functions, Journal of Algebraic Combinatorics, 35(3), (2012) 461–475.
15. S. Yoshiara, Equivalences of power apn functions with power or quadratic apn functions, Journal of Algebraic Combinatorics, 44, (2016) 561–585.
16. L. Budaghyan, C. Carlet, G. Leander, A class of quadratic apn binomials inequivalent to power functions, Cryptology ePrint Archive,.
17. L. Budaghyan, M. Calderini, I. Villa, On relations between ccz-and ea-equivalences, Cryptography and Communications, 12(1), (2020) 85–100.
18. C. Carlet, P. Charpin, V. Zinoviev, Codes, bent functions and permutations suitable for des-like cryptosys-tems, Designs, Codes and Cryptography, 15(2), (1998) 125–156.
19. G. Lachaud, J. Wolfmann, The weights of the orthogonals of the extended quadratic binary goppa codes, IEEE transactions on information theory, 36(3), (1990) 686–692.
20. H. Dobbertin, Almost perfect nonlinear power functions on gf (2 n): a new case for n divisible by 5, in: Finite Fields and Applications: Proceedings of The Fifth International Conference on Finite Fields and Applications F q 5, held at the University of Augsburg, Germany, August 2–6, 1999, Springer, 2001, pp. 113–121.
21. L. Budaghyan, Construction and analysis of cryptographic functions, Springer, Cham, 2015.
22. C. Carlet, Boolean functions for cryptography and coding theory, Cambridge University Press, 2021.
23. C. Carlet, Relating three nonlinearity parameters of vectorial functions and building apn functions from bent functions, Designs, Codes and Cryptography, 59(1), (2011) 89–109.
24. Y. Zhou, A. Pott, A new family of semifields with 2 parameters, Advances in Mathematics, 234, (2013) 43–60.
25. H. Taniguchi, On some quadratic apn functions, Designs, Codes and Cryptography, 87(9), (2019) 1973– 1983.