TY - JOUR

T1 - On the image of an affine subspace under the inverse function within a finite field

AU - Kolomeec, Nikolay

AU - Bykov, Denis

N1 - The work is supported by the Mathematical Center in Akademgorodok under the Agreement No. 075–15–2022–282 with the Ministry of Science and Higher Education of the Russian Federation. The authors would like to thank the anonymous reviewers for their valuable comments.

PY - 2023

Y1 - 2023

N2 - We consider the function x- 1 that inverses a finite field element x∈Fpn (p is prime, 0 - 1= 0) and affine Fp -subspaces of Fpn 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=sFpk , where s∈Fpn∗ and k∣ n . In other words, it is either a subfield of Fpn 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| < pn for an invertible linear transformation A:Fpn→Fpn and b∈Fpn∗ . 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 αx- 1+ b have no invariant affine subspaces except for Fpn , where α,b∈Fpn∗ and n is arbitrary.

AB - We consider the function x- 1 that inverses a finite field element x∈Fpn (p is prime, 0 - 1= 0) and affine Fp -subspaces of Fpn 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=sFpk , where s∈Fpn∗ and k∣ n . In other words, it is either a subfield of Fpn 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| < pn for an invertible linear transformation A:Fpn→Fpn and b∈Fpn∗ . 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 αx- 1+ b have no invariant affine subspaces except for Fpn , where α,b∈Fpn∗ and n is arbitrary.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85176729425&origin=inward&txGid=1419973f9ce0cb346a846f7e188a61e9

UR - https://www.mendeley.com/catalogue/86a6a1cb-8da5-3821-9251-1fe2c4531f4e/

U2 - 10.1007/s10623-023-01316-3

DO - 10.1007/s10623-023-01316-3

M3 - Article

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

ER -