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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].

##### MSC:
 11T06 Polynomials over finite fields
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