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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.

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