A new characterization of semi-bent and bent functions on finite fields
We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x(2i+1)) is semi-bent over F(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 GF(2(p)) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(p(n)) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions 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.
bent functions finite fields semi-bent functions cyclic matrix cyclic codes sophie-german princes