New binomial bent functions over the finite fields of odd characteristic

The p-ary binomial function f(x) mapping GF(p
^4k
) to GF(p) given by f(x) = Tr
_4k
(x
^{p
3k
+p
2k
-p
k}
+1 + x
²
)equations 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. Finally, we characterize the size of a cyclotomic coset containing the exponent of a monomial bent functions (the same result holds in a binomial case) and provide numerical data.