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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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