Representation Redundancy and Algebraic Complexity in Finite-Field Inversion
- 1Department of Mathematics, Towson University, Towson, MD 21252, USA
Abstract
The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over F2n, with field elements expressed in varying ordered F2-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size n, the correspondence between ordered bases and distinct inversion maps is exactly n-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree n−1 and joint ANF leap 1, the mixed representation formulation has degree 2(n−1) and joint ANF leap 2, and the complete raw formulation has degree at most 3(n−1) and joint ANF leap at least n. Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.
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BibTeX
@article{zhang2026representation,
title = {{Representation Redundancy and Algebraic Complexity in Finite-Field Inversion}},
author = {Zheng Zhang and Na Zhang},
journal = {Mathematics},
year = {2026},
month = sep,
doi = {10.3390/math14193548},
url = {https://www.mdpi.com/2227-7390/14/19/3548},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Representation Redundancy and Algebraic Complexity in Finite-Field Inversion
AU - Zheng Zhang
AU - Na Zhang
PY - 2026
DA - 2026/09/30
JO - Mathematics
DO - 10.3390/math14193548
UR - https://www.mdpi.com/2227-7390/14/19/3548
AB - The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over F2n, with field elements expressed in varying ordered F2-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size n, the correspondence between ordered bases and distinct inversion maps is exactly n-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree n−1 and joint ANF leap 1, the mixed representation formulation has degree 2(n−1) and joint ANF leap 2, and the complete raw formulation has degree at most 3(n−1) and joint ANF leap at least n. Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.
SN - 2227-7390
ER -