Differentially uniform mappings for cryptography
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
On almost perfect nonlinear permutations
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
Almost perfect nonlinear power functions on GF (2n): the Niho case
Information and Computation
Codes, Bent Functions and Permutations Suitable For DES-likeCryptosystems
Designs, Codes and Cryptography
On the Orphans and Covering Radius of the Reed-Muller Codes
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Affinity of permutations of F2n
Discrete Applied Mathematics - Special issue: Coding and cryptography
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For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f), where f runs through all permutations of V. The problem of the complete determination of k-spectrum(n,q) seems very difficult except for small or special values of the parameters. However, we are able to determine (n-1)-spectrum(n,2) and establish that 0@?k-spectrum(n,q) in the following cases: (i) q=3 and 1@?k@?n-1; (ii) q=2, 3@?k@?n-1; (iii) q=2, k=2, n=3 odd. For 1@?k@?n-1 and (q,k)(2,1), the maximum of k-affinity(f) is obtained when f is any semiaffine mapping. We conjecture that the next to largest value of k-affinity(f) occurs when f is a transposition, and we are able to prove it when q=2, k=2, n=3 and when q=3, k=1, n=2.