| Abstrakt: |
We consider the function x - 1 that inverses a finite field element x ∈ F p n (p is prime, 0 - 1 = 0 ) and affine F p -subspaces of F p n such that their images are affine subspaces as well. It is proved that the image of an affine subspace L, | L | > 2 , is an affine subspace if and only if L = s F p k , where s ∈ F p n ∗ and k ∣ n . In other words, it is either a subfield of F p n or a subspace consisting of all elements of a subfield multiplied by s . This generalizes the results that were obtained for linear invariant subspaces in 2006. As a consequence, the function x - 1 maps the minimum number of affine subspaces to affine subspaces among all invertible power functions. In addition, we propose a sufficient condition providing that a function A (x - 1) + b has no invariant affine subspaces U of cardinality 2 < | U | < p n for an invertible linear transformation A : F p n → F p n and b ∈ F p n ∗ . As an example, it is shown that the S-box of the AES satisfies the condition. Also, we demonstrate that some functions of the form α x - 1 + b have no invariant affine subspaces except for F p n , where α , b ∈ F p n ∗ and n is arbitrary. [ABSTRACT FROM AUTHOR] |
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