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Crooked binomials. (English) Zbl 1196.11162
Summary: A function $$f: \text{GF}(2^r) \rightarrow \text{GF}(2^r)$$ is called crooked if the sets $$\{f(x) + f(x + a)\mid x \in \text{GF}(2^r)\}$$ is an affine hyperplane for any nonzero $$a \in \text{GF}(2^r)$$. We prove that a crooked binomial function $$f(x) = x^d + ux^e$$ defined on $$\text{GF}(2^r)$$ satisfies that both exponents $$d, e$$ have 2-weights at most 2.

##### MSC:
 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94A60 Cryptography 11T06 Polynomials over finite fields
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