New Binomial Bent Functions Over the Finite Fields of Odd Characteristic

摘要

The $p$-ary function $f(x)$ mapping $ {rm GF}(p^{4k})$ to $ {rm GF}(p)$ and given by $f(x)={rm Tr}_{4k}big (x^{p^{3k}+p^{2k}-p^{k}+1}+x^{2}big )$ is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems.