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Representation Redundancy and Structural Complexity in Finite-Field Inversion

  • Zheng Zhang
  • Na Zhang
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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 nβˆ’1 and joint ANF leap 1, the mixed representation formulation has degree 2(nβˆ’1) and joint ANF leap 2, and the complete raw formulation has degree at most 3(nβˆ’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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Cite

BibTeXRIS
BibTeX
@misc{zhang2026representation,
  title = {{Representation Redundancy and Structural Complexity in Finite-Field Inversion}},
  author = {Zheng Zhang and Na Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04583},
  eprint = {2609.04583},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04583},
}
RIS
TY  - UNPB
TI  - Representation Redundancy and Structural Complexity in Finite-Field Inversion
AU  - Zheng Zhang
AU  - Na Zhang
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04583
UR  - https://arxiv.org/abs/2609.04583
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 𝔽_2^n, with field elements expressed in varying ordered 𝔽_2-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 nβˆ’1 and joint ANF leap 1, the mixed representation formulation has degree 2(nβˆ’1) and joint ANF leap 2, and the complete raw formulation has degree at most 3(nβˆ’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.
ER  -