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Crooked maps in $$\mathbb F_{2^n}$$. (English) Zbl 1170.94009
Almost perfect nonlinear (APN) maps provide the best resistance against the differential cryptanalysis. A special class of APN maps are called $$crooked$$. Crooked maps can be used to construct many interesting combinatorial objects – codes, graphs, schemes, etc. The only known crooked maps are polynomials with exponents of binary weight 2. In this paper, the authors study the question whether other crooked maps exist. Using combinatorics in the cyclic group of order $$n$$, the authors show that in a class of maps including power maps only the ones with exponents of binary weight 2 can be crooked.

##### MSC:
 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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