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A new characterization of semi-bent and bent functions on finite fields. (English) Zbl 1172.11311
Summary: We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a \(\text{GF}(2)\)-linear combination of Gold functions \(\text{Tr}(x^{2i}+1)\) is semi-bent over \(\text{GF}(2^{n})\), \(n\) odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over \(\text{GF}(2^{p})\) when \(p\) belongs to a certain class of primes. Second, we generalize our results to fields \(\text{GF}(p^{n})\) where \(p\) is an odd prime and \(n\) is odd. In that case, we can determine whether a \(\text{GF}(p)\)-linear combination of Gold functions \(\text{Tr}(x^{pi}+1)\) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these \(p\)-ary semi-bent and bent functions are provided.

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
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