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Two-to-one mappings and involutions without fixed points over \(\mathbb{F}_{2^n}\). (English) Zbl 1484.11218

Summary: In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like criterion for 2-to-1 mappings. Using this criterion, some new constructions of 2-to-1 mappings are proposed and eight classes of 2-to-1 mappings of the form \((x^{2^k}+x+\delta)^s+cx\) are obtained. Finally, a number of classes of involutions without any fixed point are derived from the known 2-to-1 mappings by the relation between them.

MSC:

11T06 Polynomials over finite fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
05A05 Permutations, words, matrices

Software:

PRINCE
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