A note on multiplicative-inverse chaining in finite fields

2025/07/29 by Mohan, Divyarthi, Ravindraraj, R.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.22142

Abstract

We consider chaining multiplicative-inverse operations in finite fields under alternating polynomial bases. When using two distinct polynomial bases to alternate the inverse operation we obtain a partition of \mathbb F_pⁿ∖ \mathbb F_p into disjoint cycles of even length. This allows a natural interpretation of the cycles as permutation cycles. Finally, we explore chaining under more than two polynomial bases.

Citations

• How to write a permutation as a product of involutions (and why you might care)

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• Permutations from an arithmetic setting
• Construction of Permutation Polynomials of Certain Specific Cycle Structure over Finite Fields
• On the inverses of some classes of permutations of finite fields
• Some permutation polynomials via linear translators