Abstract
We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x2i+1) is semi-bent over GF(2n), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pn) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(xpi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided.
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E. Biham A. Shamir (1991) ArticleTitleDifferential cryptanalysis of DES-like cryptosystems Journal of Cryptology 4 3–72 Occurrence Handle10.1007/BF00630563 Occurrence Handle93j:94020
S. Boztas P. V. Kumar (1994) ArticleTitleBinary sequences with Gold-like correlation but larger linear span IEEE Transactions on Information Theory 40 532–537 Occurrence Handle10.1109/18.312181
A. Canteaut C. Carlet P. Charpin C. Fontaine (2001) ArticleTitleOn cryptographic properties of the cosets of R(1,m) IEEE Transactions on Information Theory 47 1494–1513 Occurrence Handle2002h:94048
J. H. Cheon S. Chee (2000) ArticleTitleElliptic curves and resilient functions Lecture Notes in Computer Science 2015 386–397
R. Gold (1968) ArticleTitleMaximal recursive sequences with 3-valued cross correlation functions IEEE Transactions on Information Theory 14 154–156 Occurrence Handle10.1109/TIT.1968.1054106 Occurrence Handle0228.62040
G. Gong K. Khoo (2004) ArticleTitleAdditive autocorrelation of resilient boolean functions Lecture Notes in Computer Science 3006 275–290 Occurrence Handle2005f:06017
T. Helleseth, Correlation of m-sequences and related topics, In Sequences and Their Applications, Springer, Berlin, (1998) pp. 49–66.
T. Helleseth and P. V. Kumar, Sequences with low correlation, In Handbook of Coding Theory, North-Holland, Amsterdam, (1998) pp. 1765–1853
K. Khoo G. Gong (2003) ArticleTitleNew constructions for resilient and highly nonlinear boolean functions Lecture Notes in Computer Science 2727 498–509
K. Khoo, G. Gong and D. R. Stinson, A new family of Gold-like sequences, Proceedings of IEEE International Symposium on Information Theory (2002) p. 181.
Y. S. Kim, J. W. Jang, J. S. No and T. Helleseth, On p-ary bent functions defined on finite fields, In Mathematical Properties of Sequences and Other Combinatorial Structures, Kluwer, Dordrecht, (2002) pp. 65–76.
Y.S. Kim J.S. No (2003) ArticleTitleNew families of binary sequences with low correlation IEEE Transactions on Information Theory 49 3059–3065 Occurrence Handle2004k:94044
Mathworld: Artin’s Conjecture, http://mathworld.wolfram.com/ArtinsConstant.html.
M. Matsui (1994) ArticleTitleLinear cryptanalysis method for DES cipher Lecture Notes in Computer Science 765 386–397 Occurrence Handle0951.94519
R. Lidl H. Niederreiter (1994) Introduction to Finite Fields and Their Applications Cambridge University Press Cambridge
F. J. McWilliams N. J. A. Sloane (1977) Theory of Error-Correcting Codes North-Holland Amsterdam
J. Mykkeltveit (1980) ArticleTitleThe covering radius of the (128,8) Reed Muller code is 56 IEEE Transactions on Information Theory 26 359–362 Occurrence Handle10.1109/TIT.1980.1056187 Occurrence Handle0431.94036 Occurrence Handle81e:94018
K. Nyberg (1992) ArticleTitlePerfect nonlinear S-boxes Lecture Notes in Computer Science 547 378–386 Occurrence Handle1227816
N. J Patterson D. H. Wiedemann (1983) ArticleTitleThe covering radius of the (215,16) Reed-Muller code is at least 16276 IEEE Transactions on Information Theory 29 354–356 Occurrence Handle10.1109/TIT.1983.1056679 Occurrence Handle85h:94031
N. J Patterson D. H. Wiedemann (1990) ArticleTitleCorrection to “The covering radius of the (215,16) Reed-Muller code is at least 16276” IEEE Transactions Information Theory 36 443 Occurrence Handle91b:94046
O.S. Rothaus (1976) ArticleTitleOn bent functions Journal of Combinatorial Theory A 20 300–305 Occurrence Handle0336.12012 Occurrence Handle53 #7797
Song Y. Yan (2000) Number Theory for Computing Springer-Verlag Berlin
Y. Zheng X. M. Zhang (1999) ArticleTitleRelationships between bent functions and complementary plateaued functions Lecture Notes in Computer Science 1787 60–75
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Communicated by: T. Helleseth
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Khoo, K., Gong, G. & Stinson, D.R. A New Characterization of Semi-bent and Bent Functions on Finite Fields*. Des Codes Crypt 38, 279–295 (2006). https://doi.org/10.1007/s10623-005-6345-x
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DOI: https://doi.org/10.1007/s10623-005-6345-x