zbMATH — the first resource for mathematics

The affinity of a permutation of a finite vector space. (English) Zbl 1110.11036
Let \(f\) be a permutation of the \(n\)-dimensional vector space \(\mathbb F_q^n\) over a finite field \(\mathbb F_q\), then \(k\)-affinity(\(f\)) denotes the number of \(k\)-dimensional affine subspaces \(X\) of \(\mathbb F_q^n\) such that \(f(X)\) is again a \(k\)-dimensional affine subspace of \(\mathbb F_q^n\). The \(k\)-spectrum(\(n,q\)) is then the set of values \(k\)-affinity(\(f\)) for all permutations \(f\) of \(\mathbb F_q^n\). The authors show that, with few exceptions, \(0 \in k\)-spectrum(\(n,q\)) and present results on the largest values contained in \(k\)-spectrum(\(n,q\)). The paper is a continuation of X. D. Hou [Discrete Appl. Math. 154, 313–325 (2006; Zbl 1089.94020)].
11T99 Finite fields and commutative rings (number-theoretic aspects)
05A05 Permutations, words, matrices
12E10 Special polynomials in general fields
Full Text: DOI arXiv
[1] T. Beth, C. Ding, On almost perfect nonlinear permutations, Eurocrypt ’93, Lofthus, 1993, Lecture Notes in Computer Science, 765, Springer, Berlin, 1994, pp. 65-76.
[2] Carlet, C.; Charpin, P.; Zinoview, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Designs codes cryptography, 15, 125-156, (1998) · Zbl 0938.94011
[3] Dobbertin, H., Almost perfect nonlinear power functions on \(\operatorname{GF}(2^n)\): the niho case, Inform. comput., 151, 57-72, (1999) · Zbl 1072.94513
[4] H. Dobbertin, Almost perfect nonlinear power functions on \(\operatorname{GF}(2^n)\): a new case for n divisible by 5, Finite Fields and Applications, Augsburg, 1999, Springer, Berlin, 2001, pp. 113-121. · Zbl 1010.94550
[5] X. Hou, Affinity of permutations of \(\mathbb{F}_2^n\), Discrete Appl. Math., to appear.
[6] P. Langevin, On the orphans and covering radius of the Reed-Muller codes, Applied Algebra, Algebraic Algorithms and Error-correcting Codes, New Orleans, LA, 1991, Lecture Notes in Computer Science, vol. 539, Springer, Berlin, 1991, pp. 234-240. · Zbl 0767.94008
[7] K. Nyberg, Differentially uniform mappings for cryptography, Eurocrypt ’93, Lofthus, 1993, Lecture Notes in Computer Science, vol. 765, Springer, Berlin, 1994, pp. 55-64. · Zbl 0951.94510
[8] K. Nyberg, S-boxes and round functions with controllable linearity and differential uniformity, Fast Software Encryption, Leuven, 1994, Lecture Notes in Computer Science, vol. 1008, Springer, Berlin, 1995, pp. 111-130. · Zbl 0939.94559
[9] Snapper, E.; Troyer, R.J., Metric affine geometry, (1971), Academic Press New York, London · Zbl 0224.50006
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.