Abstract
A construction of APN functions using the bent function \(B(x,y)=xy\) is proposed in Carlet (Des Codes Cryptogr 59:89–109, 2011). At this time, two families of APN functions using this construction are known, that is, the family of Carlet (2011) and the family of Zhou and Pott (Adv Math 234:43–60, 2013). In this note, we propose another family of APN functions with this construction, which are not CCZ equivalent to the former two families on \({{\mathbb {F}}}_{2^8}\). We also propose a family of presemifields and determined the middle, left, right nuclei and the center of the associated semifields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albert A.A.: On non associative division algebras. Trans. Am. Math. Soc. 72, 292–309 (1952).
Bartoli D., Bierbrauer J., Kyureghyan G., Giulietti M., Marcugini S., Pambianco F.: A family of semifields in characteristic 2. J. Algebr. Comb. 45, 455–473 (2017).
Bierbrauer J.: Projective polynomials, a projection construction and a family of semifields. Des. Codes Cryptogr. 79, 183–200 (2016).
Bierbrauer J., Bartoli D., Faina G., Marcugini S., Pambianco F.: A family of semifields in odd characteristic. Des. Codes Cryptogr. 86, 611–621 (2018).
Bluher A.: On \(x^{q+1}+ax+b\). Finite Fields Appl. 10, 285–305 (2004).
Bosma W., Cannon J., Playoust C.: The MAGMA algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997).
Bracken C., Tan C., Tan Y.: On a class of quadratic polynomials with no zeros and its application to APN functions. Finite Fields Appl. 25, 26–36 (2014).
Budaghyan L., Helleseth T., Li N., Sun B.: Some results on the known classes of quadratic APN functions, codes, cryptology and information security. In: Hajji S.E., Nitaj A., Souidi E.M. (eds.) Proceedings of Second International Conference, C2SI 2017, Lecture Notes on Computer Science, 10194, 3–16 (2017).
Carlet C.: Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des. Codes Cryptogr. 59, 89–109 (2011).
Dempwolff U.: More translation planes and semifields from Dembowski-Ostrom polynomials. Des. Codes Cryptogr. 68, 81–103 (2013).
Edel Y., Pott A.: A new perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3, 59–81 (2009).
Helleseth T., Kholosha A.: On the equation \(x^{2^l+1}+x+a=0\) over \(GF(2^k)\). Finite Fields Appl. 14, 159–176 (2008).
Lavrauw M., Polverino O.: Finite semifields. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry, pp. 131–160. Nova Science Publishers, Inc, New York (2011).
Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20, 2nd edn. Cambridge University Press, Cambridge (1996).
Zhou Y., Pott A.: A new family of semifields with 2 parameters. Adv. Math. 234, 43–60 (2013).
Acknowledgements
The author thanks the referees for many helpful comments and advices.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Helleseth.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Taniguchi, H. On some quadratic APN functions. Des. Codes Cryptogr. 87, 1973–1983 (2019). https://doi.org/10.1007/s10623-018-00598-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-00598-2