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Complete mapping polynomials over finite field \(\mathbb F_{16}\). (English) Zbl 1213.11193
Carlet, Claude (ed.) et al., Arithmetic of finite fields. First international workshop, WAIFI 2007, Madrid, Spain, June 21–22, 2007. Proceedings. Berlin: Springer (ISBN 978-3-540-73073-6/pbk). Lecture Notes in Computer Science 4547, 147-158 (2007).
Summary: A polynomial \(f(x)\) over \(\mathbb F_q\), the finite field with \(q\) elements, is called a complete mapping polynomial if the two mappings \(\mathbb F_q \rightarrow \mathbb F_q\) respectively defined by \(f(x)\) and \(f(x) + x\) are one-to-one. In this correspondence, complete mapping polynomials over \(\mathbb F_{16}\) are considered. The nonexistence of the complete mapping polynomial of degree 9 and the existence of the ones of degree 8 and 11 are proved; the result that the reduced degree of complete mapping polynomials over \(\mathbb F_{16}\) are 1, 4, 8, 10, 11, 12, 13 is presented; and by searching with computer, the degree distribution of complete mapping polynomials over the field is given.
For the entire collection see [Zbl 1121.11002].

11T06 Polynomials over finite fields
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