Planar functions and commutative semifields

Lilya Budaghyan; Tor Helleseth

doi:10.2478/v10127-010-0002-0

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Planar functions and commutative semifields

Tatra Mountains Mathematical Publications

Volume 45 (2010): Issue 1 (April 2010)

By: Lilya Budaghyan and Tor Helleseth

Open Access

|Nov 2012

Abstract

This paper gives a short survey on planar functions and commutative semifields and considers a possible extension of CCZ-equivalence which is the most general known equivalence relation of functions preserving the planar property.

References

[1] ALBERT, A. A.: On nonassociative division algebras, Trans. Amer. Math. Soc. 72 (1952), 296-309.10.1090/S0002-9947-1952-0047027-4
Search in Google Scholar
[2] ALBERT, A. A.: Generalized twisted fields, Pacific J. Math. 11 (1961), 1-8.10.2140/pjm.1961.11.1
Search in Google Scholar
[3] BIERBRAUER, J.: New semifields, PN and APN functions, Des. Codes Cryptogr. 54 (2010), 189-200.10.1007/s10623-009-9318-7
Search in Google Scholar
[4] BIHAM, E.-SHAMIR, A.: Differential cryptanalysis of DES-like cryptosystems, J. Cryptology 4 (1991), 3-72.10.1007/BF00630563
Search in Google Scholar
[5] BRACKEN, C.-BYRNE, E.-MARKIN, N.-MCGUIRE, G.: New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite Fields Appl. 14 (2008), 703-714.10.1016/j.ffa.2007.11.002
Search in Google Scholar
[6] BUDAGHYAN, L.: The simplest method for constructing APN polynomials EA-inequivalent to power functions, in: Proc. of 1st Internat. Workshop on Arithmetic of Finite Fields-WAIFI ’07 (C. Carlet et al., eds.), Lecture Notes in Comput. Sci., Vol. 4547, Springer-Verlag, Berlin, 2007, pp. 177-188.
Search in Google Scholar
[7] BUDAGHYAN, L.-CARLET, C.: CCZ-equivalence of single and multi output Boolean functions, “Contemporary Mathematics” of Amer. Math. Soc., 2010 (to appear).10.1090/conm/518/10195
Search in Google Scholar
[8] BUDAGHYAN, L.-CARLET, C.: On CCZ-equivalence and its use in secondary constructions of bent functions, in: Preproc. of Internat. Workshop on Coding and Cryptography-WCC ’09, pp. 19-36, 2009.
Search in Google Scholar
[9] BUDAGHYAN, L.-CARLET, C.-LEANDER, G.: Two classes of quadratic APN binomials inequivalent to power functions, IEEE Trans. Inform. Theory 54, (2008), 4218-4229.10.1109/TIT.2008.928275
Search in Google Scholar
[10] BUDAGHYAN, L.-CARLET, C.-POTT, A.: New classes of almost bent and almost perfect nonlinear functions, IEEE Trans. Inform. Theory 52 (2006), 1141-1152.10.1109/TIT.2005.864481
Search in Google Scholar
[11] BUDAGHYAN, L.-HELLESETH, T.: New perfect nonlinear multinomials over Fp2k for any odd prime p, in: Proc. of Internat. Conference on Sequences and Their Applications-SETA ’08, Lecture Notes in Comput. Sci., Vol. 5203, Springer-Verlag, Berlin, 2008, pp. 401-414.
Search in Google Scholar
[12] BUDAGHYAN, L.-HELLESETH, T.: New commutative semifields defined by new PN multinomials, Cryptography and Communications: Discrete Structures, Boolean Functions and Sequences, 2010 (to appear).10.1007/s12095-010-0022-2
Search in Google Scholar
[13] CARLET, C.-CHARPIN, P.-ZINOVIEV, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr. 15 (1998), 125-156.10.1023/A:1008344232130
Search in Google Scholar
[14] COHEN, S. D.-GANLEY, M. J.: Commutative semifields, two-dimensional over there middle nuclei, J. Algebra 75 (1982), 373-385.10.1016/0021-8693(82)90045-X
Search in Google Scholar
[15] COULTER, R. S.-MATTHEWS, R. W.: Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr. 10 (1997), 167-184.10.1023/A:1008292303803
Search in Google Scholar
[16] COULTER, R. S.-HENDERSON, M.: Commutative presemifields and semifields, Adv. Math. 217 (2008), 282-304.10.1016/j.aim.2007.07.007
Search in Google Scholar
[17] COULTER, R. S.-HENDERSON, M.-KOSICK, P.: Planar polynomials for commutative semifields with specified nuclei, Des. Codes Cryptogr. 44 (2007), 275-286.10.1007/s10623-007-9097-y
Search in Google Scholar
[18] DEMBOWSKI, P.-OSTROM, T.: Planes of order n with collineation groups of order n2, Math. Z. 103 (1968), 239-258.10.1007/BF01111042
Search in Google Scholar
[19] DICKSON, L. E.: On commutative linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc. 7 (1906), 514-522.10.1090/S0002-9947-1906-1500764-6
Search in Google Scholar
[20] DICKSON, L. E.: Linear algebras with associativity not assumed, Duke Math. J. 1 (1935), 113-125.10.1215/S0012-7094-35-00112-0
Search in Google Scholar
[21] DING, C.-YUAN, J.: A new family of skew Paley-Hadamard difference sets, J. Comb. Theory Ser. A 133 (2006), 1526-1535.10.1016/j.jcta.2005.10.006
Search in Google Scholar
[22] EDEL, Y.-POTT, A.: A new almost perfect nonlinear function which is not quadratic, Adv. Math. Commun. 3 (2009), 59-81.10.3934/amc.2009.3.59
Search in Google Scholar
[23] GANLEY, M. J.: Central weak nucleus semifields, European J. Combin. 2 (1981), 339-347.10.1016/S0195-6698(81)80041-8
Search in Google Scholar
[24] HELLESETH, T.-RONG, C.-SANDBERG, D.: New families of almost perfect nonlinear power mappings, IEEE Trans. Inf. Theory 45 (1999), 475-485.10.1109/18.748997
Search in Google Scholar
[25] HELLESETH, T.-SANDBERG, D.: Some power mappings with low differential uniformity, Appl. Algebra Engrg. Comm. Comput. 8 (1997), 363-370.10.1007/s002000050073
Search in Google Scholar
[26] KNUTH, D. E.: Finite semifields and projective planes, J. Algebra 2 (1965), 182-217.10.1016/0021-8693(65)90018-9
Search in Google Scholar
[27] KYUREGHYAN, G.-POTT, A.: Some theorems on planar mappings, in: Proc. of Internat. Workshop on Arithmetic of Finite Fields-WAIFI ’08 (J. von Gathen et al., eds.), Lecture Notes in Comput. Sci., Vol. 5130, Springer-Verlag, Berlin, 2008, pp. 117-122.
Search in Google Scholar
[28] MENICHETTI, G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field, J. Algebra 47 (1977), 400-410.10.1016/0021-8693(77)90231-9
Search in Google Scholar
[29] MINAMI, K.-NAKAGAWA, N.: On planar functions of elementary abelian p-group type (submitted).
Search in Google Scholar
[30] NAKAGAWA, N.: On functions of finite fields, http://www.math.is.tohoku.ac.jp/ taya/sendaiNC/2006/report/nakagawa.pdf.
Search in Google Scholar
[31] NESS, G. J.: Correlation of sequences of different lengths and related topics. PhD Dissertation, University of Bergen, Norway, 2007.
Search in Google Scholar
[32] NYBERG, K.: Perfect nonlinear S-boxes, in: Advances in Cryptography-EUROCRYPT ’91, Lecture Notes in Comput. Sci. 547 (1992), pp. 378-386.
Search in Google Scholar
[33] NYBERG, K.: Differentially uniform mappings for cryptography. in: Advances in Cryptography-EUROCRYPT ’93, Lecture Notes in Comput. Sci., Vol. 765, Springer-Verlag, Berlin, 1994, pp. 55-64.10.1007/3-540-48285-7_6
Search in Google Scholar
[34] PENTTILA, T.-WILLIAMS, B.: Ovoids of parabolic spaces, Geom. Dedicata 82 (2000), 1-19.10.1023/A:1005244202633
Search in Google Scholar
[35] WENG, G.: Private communications, 2007.
Search in Google Scholar
[36] ZHA, Z.-KYUREGHYAN, G.-WANG, X.: Perfect nonlinear binomials and their semifields, Finite Fields Appl. 15(2009), 125-133.10.1016/j.ffa.2008.09.002
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10127-010-0002-0 | Journal eISSN: 1338-9750 | Journal ISSN: 12103195

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Language: English

Page range: 15 - 25

Published on: Nov 12, 2012

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Keywordscommutative semifield,

equivalence of functions,

Related subjects:

© 2012 Lilya Budaghyan, Tor Helleseth, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons License.

Volume 45 (2010): Issue 1 (April 2010)