# zbMATH — the first resource for mathematics

New commutative semifields defined by new PN multinomials. (English) Zbl 1291.12006
Summary: We introduce two infinite classes of quadratic PN multinomials over $$\mathbb F_{p^{2k}}$$ 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 L. E. Dickson [Trans. Am. Math. Soc. 7, 514–522 (1906; JFM 37.0112.01)] and A. A. Albert [Trans. Am. Math. Soc. 72, 296–309 (1952; Zbl 0046.03601)], 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.

##### MSC:
 12K10 Semifields 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94A60 Cryptography 51E99 Finite geometry and special incidence structures
Full Text:
##### References:
 [1] Albert, A.A.: On nonassociative division algebras. Trans. Am. Math. Soc. 72, 296–309 (1952) · Zbl 0046.03601 · doi:10.1090/S0002-9947-1952-0047027-4 [2] Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991) · Zbl 0729.68017 · doi:10.1007/BF00630563 [3] 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) · Zbl 1153.11058 · doi:10.1016/j.ffa.2007.11.002 [4] 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) · Zbl 1177.94136 · doi:10.1109/TIT.2005.864481 [5] Budaghyan, L., Helleseth, T.: New perfect nonlinear multinomials over $$$\backslash$$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) · Zbl 1177.94137 [6] Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998) · Zbl 0938.94011 · doi:10.1023/A:1008344232130 [7] Coulter, R.S., Matthews R.W.: Planar functions and planes of Lenz–Barlotti class II. Des. Codes Cryptogr. 10, 67–184 (1997) · Zbl 0872.51007 [8] Coulter, R.S., Henderson, M.: Commutative presemifields and semifields. Adv. Math. 217, 282–304 (2008) · Zbl 1194.12007 · doi:10.1016/j.aim.2007.07.007 [9] Dembowski, P., Ostrom, T.: Planes of order n with collineation groups of order n 2. Math. Z. 103, 239–258 (1968) · Zbl 0163.42402 · doi:10.1007/BF01111042 [10] Dickson, L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc 7, 514–522, (1906) · JFM 37.0112.01 · doi:10.1090/S0002-9947-1906-1500764-6 [11] Dickson, L.E.: Linear algebras with associativity not assumed. Duke Math. J. 1, 113–125 (1935) · Zbl 0012.14801 · doi:10.1215/S0012-7094-35-00112-0 [12] Helleseth, T., Rong, C., Sandberg, D.: New families of almost perfect nonlinear power mappings. IEEE Trans. Inf. Theory 45, 475–485 (1999) · Zbl 0960.11051 · doi:10.1109/18.748997 [13] Helleseth, T., Sandberg, D.: Some power mappings with low differential uniformity. Appl. Algebra Eng. Commun. Comput. 8, 363–370 (1997) · Zbl 0886.11067 · doi:10.1007/s002000050073 [14] 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) · Zbl 1180.94056 [15] Minami, K., Nakagawa, N.: On planar functions of elementary abelian p-group type. Hokkaido Math. J. 37, 531–544 · Zbl 1161.51003 [16] Ness, G.J.: Correlation of sequences of different lengths and related topics. Ph.D. dissertation, University of Bergen, Norway (2007) · Zbl 1318.94060 [17] Nyberg, K.: Differentially uniform mappings for cryptography. In: Advances in Cryptography, EUROCRYPT’93. LNCS, vol. 765, pp. 55–64 (1994) · Zbl 0951.94510 [18] Zha, Z., Kyureghyan, G., Wang, X.: Perfect nonlinear binomials and their semifields. Finite Fields Their Appl. 15(2), 125–133 (2009) · Zbl 1194.12003 · doi:10.1016/j.ffa.2008.09.002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.