Abstract
In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field \(\mathbb {F}_{2}\), extending some earlier results of Leander and Rodier.
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References
Aubry, Y., McGuire, G., Rodier, F.: A Few More Functions that are not APN Infinitely Often, Finite Fields: Theory and Applications, 23–31, Contemp Math., vol. 518. Amer. Math. Soc., Providence (2010)
Budaghyan, L.: Construction and Analysis of Cryptographic Functions. Springer, Berlin (2014)
Budaghyan, L., Carlet, C., Helleseth, T., Li, N., Sun, B.: On upper bounds for algebraic degrees of APN functions. IEEE Trans. Inf. Theory 64:6, 4399–4411 (2018)
Budaghyan, L., Carlet, C., Leander, G.: On a construction of quadratic APN functions. In: Proc IEEE Inf. Theory Workshop ITW’09, pp. 374–378 (2009)
Caranti, A., Dalla Volta, F., Sala, M.: On some block ciphers and imprimitive groups. Applic. Algebra Eng. Commun. Comput. 20(5–6), 339–350 (2009)
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models, pp 257–397. Cambridge University Press, Cambridge (2010)
Carlet, C.: Vectorial Boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models, pp 398–472. Cambridge University Press, Cambridge (2010)
Chabaud, F., Vaudenay, S.: Links between differential and linear cryptanalysis. In: Adv. in Crypt.–EUROCRYPT’94, LNCS 950, pp. 356–365 (1995)
Charpin, P., Kyureghyan, G.M.: On sets determining the differential spectrum of mappings. Internat. J. Inf. Coding Theory 4(2–3), 170–184 (2017)
Cusick, T.W., Stanica, P.: Cryptographic Boolean Functions and Applications. 2nd. Academic Press, San Diego (2017)
Férard, E., Oyono, R., Rodier, F.: Some More Functions that are not APN Infinitely Often. The Case of Gold and Kasami Exponents, Arithmetic, Geometry, Cryptography and Coding Theory, 27–36, Contemp Math, vol. 574. Amer. Math. Soc., Providence (2012)
Leander, G., Rodier, F.: Bounds on the degree of APN polynomials: The case of x− 1 + g(x). Des. Codes Cryptogr. 59(1–3), 207–222 (2011)
Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)
Rodier, F.: Functions of degree 4e that are not APN infinitely often. Cryptogr. Commun. 3, 227–240 (2011)
Rodier, F.: Borne sur le degré des polynômes presque parfaitement non-linéaires. Arithmetic, geometry, cryptography and coding theory. In: Lachaud, G., Ritzenthaler, C., Tsfasman, M. (eds.) Contemporary Math. no 487, pp 169–181. AMS, Providence (2009)
Acknowledgements
The authors would like to thank the referees for their thorough reading and useful comments, and the editors for handling our manuscript very efficiently. The paper was started while the fourth named author visited Selmer center at UiB in the Summer of 2018. This author thanks the institution for the excellent working conditions. S.K. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1D1A1B03931912 and No. 2016R1A5A1008055). The research of the first two authors was supported by Trond Mohn foundation.
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Budaghyan, L., Kaleyski, N.S., Kwon, S. et al. Partially APN Boolean functions and classes of functions that are not APN infinitely often. Cryptogr. Commun. 12, 527–545 (2020). https://doi.org/10.1007/s12095-019-00372-8
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DOI: https://doi.org/10.1007/s12095-019-00372-8