Some theorems on planar mappings. (English) Zbl 1180.94056

von zur Gathen, Joachim (ed.) et al., Arithmetic of finite fields. 2nd international workshop, WAIFI 2008, Siena, Italy, July 6–9, 2008. Proceedings. Berlin: Springer (ISBN 978-3-540-69498-4/pbk). Lecture Notes in Computer Science 5130, 117-122 (2008).
Summary: A mapping \(f\colon {\mathbb{F}}_p^n\to {\mathbb{F}}_p^n\) is called planar if for every nonzero \(a \in {\mathbb{F}}_p^n\) the difference mapping \(D_{f,a}\colon x \rightarrowtail f(x + a) - f(x)\) is a permutation of \({\mathbb{F}}_p^n\). In this note we prove that two planar functions are Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) exactly when they are extended affine equivalent (EA-equivalent). We give a sharp lower bound on the size of the image set of a planar function. Further we observe that all currently known main examples of planar functions have image sets of that minimal size.
Here two mappings \(f, g\colon {\mathbb{F}}_p^n\to {\mathbb{F}}_p^n\) are called extended affine equivalent (EA-equivalent), if \(g = A_1 \circ f \circ A_2 + A\) for some affine permutations \(A_1, A_2\) and an affine mapping \(A\).
For the entire collection see [Zbl 1141.11003].


94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: DOI


[1] 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 · doi:10.1109/TIT.2005.864481
[2] Carlet, C.; Charpin, P.; Zinoviev, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15, 125-156 (1998) · Zbl 0938.94011 · doi:10.1023/A:1008344232130
[3] 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
[4] 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 · doi:10.1007/s10623-007-9097-y
[5] 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 · doi:10.1023/A:1008292303803
[6] 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
[7] Ding, C.; Yin, J., Signal sets from functions with optimum nonlinearity, IEEE Trans. Communications, 55, 936-940 (2007) · doi:10.1109/TCOMM.2007.894113
[8] Ding, C.; Yuan, J., A family of optimal constant-composition codes, IEEE Trans. Inform. Theory, 51, 3668-3671 (2005) · Zbl 1181.94129 · doi:10.1109/TIT.2005.855609
[9] Ding, C.; Yuan, J., A new family of skew Paley-Hadamard difference sets, J. Comb. Theory Ser. A, 113, 1526-1535 (2006) · Zbl 1106.05016 · doi:10.1016/j.jcta.2005.10.006
[10] Helleseth, T.; Sandberg, D., Some power mappings with low differential uniformity, Applicable Algebra in Engineering, Communications and Computing, 8, 363-370 (1997) · Zbl 0886.11067 · doi:10.1007/s002000050073
[11] Turnwald, G., A new criterion for permutation polynomials, Finite Fields and Appl., 1, 64-82 (1995) · Zbl 0817.11055 · doi:10.1006/ffta.1995.1005
[12] Wan, D., A p-adic lifting lemma and its applications to permutation polynomials, Finite Fields, Coding Theory and Advances in Comm. and Computing, Lect. Notes in Pure and Appl. Math., 141, 209-216 (1993) · Zbl 0792.11049
[13] Weng, G.; Qiu, W.; Wang, Z.; Xiang, Q., Pseudo-Paley graphs and skew Hadamard sets from presemifields, Des. Codes Cryptogr., 44, 49-62 (2007) · Zbl 1126.05026 · doi:10.1007/s10623-007-9057-6
[14] Zha, Z., Kyureghyan, G., Wang, X.: A new family of perfect nonlinear binomials (submitted, 2008)
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