×

A new family of semifields with 2 parameters. (English) Zbl 1296.12007

In this paper, the authors obtain a new family of commutative semifields defined by two parameters by altering the construction proposed by S. D. Cohen and M. J. Ganley [J. Algebra 75, 373–385 (1982; Zbl 0499.12021)].
Basically, they replace some of the field multiplications which appear in the construction of Cohen-Ganley by the multiplication of Albert’s twisted fields and determine the necessary conditions to have a semifield as a result. The left and the middle nucleus of the resulting semifields are obtained and the isotopism problem is investigated. It is shown that for a particular choice of one of the parameters, the resulting family belongs to the family discovered by L. Budaghyan and T. Helleseth [Sequences and their applications – SETA 2008, Lect. Notes Comput. Sci. 5203, 403–414 (2008; Zbl 1177.94137)] but for the rest of the parameters, the constructed family contains semifields which are not isotopic to any of the known families.
In the last part of the paper, it is shown that one can use this family of semifields to construct APN functions and by choosing the parameters of the semifield appropriately a new APN function on \(\mathbb{F}_{2^8}\) is obtained.

MSC:

12K10 Semifields
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry

Software:

Magma
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albert, A., On nonassociative division algebras, Trans. Amer. Math. Soc., 72, 292-309 (1952) · Zbl 0046.03601
[2] Albert, A., Finite division algebras and finite planes, (Combinatorial Analysis: Proceedings of the 10th Symposium in Applied Mathematics, Symposia in Appl. Math., vol. 10 (1960), American Mathematical Society: American Mathematical Society Providence, RI), 53-70 · Zbl 0096.15003
[3] Ball, S.; Lavrauw, M., Commutative semifields of rank 2 over their middle nucleus, (Finite Fields with Applications to Coding Theory, Cryptography and Related Areas (2002), Springer-Verlag: Springer-Verlag Berlin, New York), 1-21 · Zbl 1102.12300
[4] Bierbrauer, J., New commutative semifields and their nuclei, (Bras-Amorós, M.; Høholdt, T., Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, vol. 5527 (2009), Springer: Springer Berlin, Heidelberg, Tarragona, Spain), 179-185 · Zbl 1273.12006
[5] Bierbrauer, J., Commutative semifields from projection mappings, Des. Codes Cryptogr., 61, 187-196 (2010) · Zbl 1241.12004
[6] Bierbrauer, J., New semifields, PN and APN functions, Des. Codes Cryptogr., 54, 189-200 (2010) · Zbl 1269.12006
[7] J. Bierbrauer, G.M. Kyureghyan, On the projection construction of planar and APN mappings, Talk presented at YACC 2010, Porquerolles Island, France, 2010.
[8] Biliotti, M.; Jha, V.; Johnson, N. L., The collineation groups of generalized twisted field planes, Geom. Dedicata, 76, 97-126 (1999) · Zbl 0936.51003
[9] Bosma, W.; Cannon, J.; Playoust, C., The MAGMA algebra system I: the user language, J. Symbolic. Comput., 24, 235-265 (1997) · Zbl 0898.68039
[10] Budaghyan, L.; Carlet, C.; Pott, A., New classes of almost bent and almost perfect nonlinear polynomials, IEEE Trans. Inform. Theory, 52, 1141-1152 (2006) · Zbl 1177.94136
[11] Budaghyan, L.; Helleseth, T., New perfect nonlinear multinomials over \(F_{p^{2 k}}\) for any odd prime \(p\), (SETA’08: Proceedings of the 5th International Conference on Sequences and their Applications (2008), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 403-414 · Zbl 1177.94137
[12] Budaghyan, L.; Helleseth, T., New commutative semifields defined by new PN multinomials, Cryptogr. Commun., 3, 1-16 (2011) · Zbl 1291.12006
[13] Carlet, C., (Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Encyclopedia of Mathematics and its Applications, vol. 134 (2010), Cambridge University Press), 398-471
[14] Carlet, C., Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions, Des. Codes Cryptogr., 59, 89-109 (2011) · Zbl 1229.94041
[15] Cohen, S.; Ganley, M., Commutative semifields, two-dimensional over their middle nuclei, J. Algebra, 75, 373-385 (1982) · Zbl 0499.12021
[16] Coulter, R. S., Explicit evaluations of some Weil sums, Acta Arith., 83, 241-251 (1998) · Zbl 0924.11098
[17] Coulter, R. S.; Henderson, M., Commutative presemifields and semifields, Adv. Math., 217, 282-304 (2008) · Zbl 1194.12007
[18] Coulter, R. S.; Henderson, M.; Kosick, P., Planar polynomials for commutative semifields with specified nuclei, Des. Codes Cryptogr., 44, 275-286 (2007) · Zbl 1215.12012
[19] Coulter, R. S.; Kosick, P., Commutative semifields of order 243 and 3125, (Finite Fields: Theory and Applications. Finite Fields: Theory and Applications, Contemp. Math., vol. 518 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 129-136 · Zbl 1213.12010
[20] Coulter, R. S.; Matthews, R. W., Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr., 10, 167-184 (1997) · Zbl 0872.51007
[21] Dembowski, P.; Ostrom, T., Planes of order \(n\) with collineation groups of order \(n^2\), Math. Z., 103, 239-258 (1968) · Zbl 0163.42402
[22] Dickson, L., On commutative linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc., 7, 514-522 (1906)
[23] Ding, C.; Yuan, J., A family of skew hadamard difference sets, J. Combin. Theory Ser. A, 113, 1526-1535 (2006) · Zbl 1106.05016
[24] S. Draper, X. Hou, Explicit evaluation of certain exponential sums of quadratic functions over \(\mathbb{F}_{p^n}, p\) odd, 2007. arXiv:0708.3619v1.
[25] Edel, Y.; Pott, A., A new almost perfect nonlinear function which is not quadratic, Adv. Math. Commun., 3, 59-81 (2009) · Zbl 1231.11140
[26] Ganley, M., Central weak nucleus semifields, European J. Combin., 2, 339-347 (1981) · Zbl 0469.51005
[27] Helleseth, T.; Kholosha, A., On the dual of monomial quadratic \(p\)-ary bent functions, (Golomb, S.; Gong, G.; Helleseth, T.; Song, H. Y., Sequences, Subsequences, and Consequences. Sequences, Subsequences, and Consequences, Lecture Notes in Computer Science, vol. 4893 (2007), Springer: Springer Berlin, Heidelberg), 50-61 · Zbl 1154.94395
[28] (Hughes, D.; Piper, F., Projective Planes (1973), Springer: Springer Berlin) · Zbl 0267.50018
[29] D. Knuth, Finite semifields and projective planes, Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1963. · Zbl 0128.25604
[30] Kyureghyan, G. M.; Pott, A., Some theorems on planar mappings, (WAIFI’08: Proceedings of the 2nd International Workshop on Arithmetic of Finite Fields (2008), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 117-122 · Zbl 1180.94056
[31] Lidl, R.; Niederreiter, H., Finite Fields (1997), Cambridge University Press: Cambridge University Press Cambridge, New York
[32] Marino, G.; Polverino, O., On isotopisms and strong isotopisms of commutative presemifields, J. Algebraic Combin., 36, 247-261 (2012) · Zbl 1294.12005
[33] Penttila, T.; Williams, B., Ovoids of parabolic spaces, Geom. Dedicata, 82, 1-19 (2004) · Zbl 0969.51008
[34] Wedderburn, J. H.M., A theorem on finite algebras, Trans. Amer. Math. Soc., 6, 349-352 (1905)
[35] Weng, G.; Zeng, X., Further results on planar DO functions and commutative semifields, Des. Codes Cryptogr., 63, 413-423 (2012) · Zbl 1272.12021
[36] Zha, Z.; Kyureghyan, G. M.; Wang, X., Perfect nonlinear binomials and their semifields, Finite Fields Appl., 15, 125-133 (2009) · Zbl 1194.12003
[37] Zha, Z.; Wang, X., New families of perfect nonlinear polynomial functions, J. Algebra, 322, 3912-3918 (2009) · Zbl 1209.11107
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.