On the image of an affine subspace under the inverse function within a finite field
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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 proven that the image of an affine subspace L, |L| > 2, is an affine subspace if and only if L = q 𝔽_p^k, where q ∈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 q. This generalizes the results that were obtained for linear invariant subspaces in 2006. As a consequence, 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 condition works for S-box of AES. 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.
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