Abstract
We introduce two infinite classes of quadratic PN multinomials over \(\textbf{F}_{p^{2k}}\) where p is any odd prime. We prove that for k odd one of these classes defines a new family of commutative semifields (in part by studying the nuclei of these semifields). After the works of Dickson (Trans Am Math Soc 7:514–522, 1906) and Albert (Trans Am Math Soc 72:296–309, 1952), this is the firstly found infinite family of commutative semifields which is defined for all odd primes p. These results also imply that these PN functions are CCZ-inequivalent to all previously known PN mappings.
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 nonassociative division algebras. Trans. Am. Math. Soc. 72, 296–309 (1952)
Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)
Bracken, C., Byrne, E., Markin, N., McGuire, G.: New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields Their Appl. 14(3), 703–714 (2008)
Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear functions. IEEE Trans. Inf. Theory 52(3), 1141–1152 (2006)
Budaghyan, L., Helleseth, T.: New perfect nonlinear multinomials over \(\textbf{F}_{p^{2k}}\) for any odd prime p. In: Proceedings of International Conference on Sequences and Their Applications SETA 2008. Lecture Notes in Computer Science, vol. 5203, pp. 401–414 (2008)
Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)
Coulter, R.S., Matthews R.W.: Planar functions and planes of Lenz–Barlotti class II. Des. Codes Cryptogr. 10, 67–184 (1997)
Coulter, R.S., Henderson, M.: Commutative presemifields and semifields. Adv. Math. 217, 282–304 (2008)
Dembowski, P., Ostrom, T.: Planes of order n with collineation groups of order n 2. Math. Z. 103, 239–258 (1968)
Dickson, L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc 7, 514–522, (1906)
Dickson, L.E.: Linear algebras with associativity not assumed. Duke Math. J. 1, 113–125 (1935)
Helleseth, T., Rong, C., Sandberg, D.: New families of almost perfect nonlinear power mappings. IEEE Trans. Inf. Theory 45, 475–485 (1999)
Helleseth, T., Sandberg, D.: Some power mappings with low differential uniformity. Appl. Algebra Eng. Commun. Comput. 8, 363–370 (1997)
Kyureghyan, G., Pott, A.: Some theorems on planar mappings. In: Proceedings of WAIFI 2008. Lecture Notes in Computer Science, vol. 5130, pp. 115–122 (2008)
Minami, K., Nakagawa, N.: On planar functions of elementary abelian p-group type. Hokkaido Math. J. 37, 531–544
Ness, G.J.: Correlation of sequences of different lengths and related topics. Ph.D. dissertation, University of Bergen, Norway (2007)
Nyberg, K.: Differentially uniform mappings for cryptography. In: Advances in Cryptography, EUROCRYPT’93. LNCS, vol. 765, pp. 55–64 (1994)
Zha, Z., Kyureghyan, G., Wang, X.: Perfect nonlinear binomials and their semifields. Finite Fields Their Appl. 15(2), 125–133 (2009)
Acknowledgements
This work was supported by Norwegian Research Council and partly by the grant NIL-I-004 from Iceland, Liechtenstein and Norway through the EEA and Norwegian Financial Mechanisms.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Budaghyan, L., Helleseth, T. New commutative semifields defined by new PN multinomials. Cryptogr. Commun. 3, 1–16 (2011). https://doi.org/10.1007/s12095-010-0022-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-010-0022-2