New Classes of Ternary Bent Functions From the Coulter-Matthews Bent Functions.

It has been an active research issue for many years to construct new bent functions. For k odd with \gcd (n, k)=1 and a\in \mathbb F3^{n}^{*} , the function f(x)=Tr1^{n}(ax^{(({3^{k}+1})/{2})}) is weakly regular bent over \mathbb F3^{n} , where Tr1^{n}(\cdot) is the trace function from \mathbb F3^{n} to \mathbb F3 . This is the well-known Coulter–Matthews bent function. In this paper, we determine the dual function of f(x) completely. As a consequence, we find many classes of ternary bent functions not reported in the literature previously. Such bent functions are not quadratic if k>1 and have (((1+\sqrt 5)/2)^w+1-((1-\sqrt 5)/2)^w+1)/\sqrt 5 or (((1+\sqrt 5)/2)^n-w+1-((1-\sqrt 5)/2)^n-w+1)/\sqrt 5 trace terms, where 0<w<n and wk\equiv 1(\bmod \;n) . Among them, five special cases are especially interesting: 1) for the case of k=n-1 , the number of trace terms is ; 2) for the case of k=(n+1)/2 , the number of trace terms is (((1+\sqrt 5)/2)^n-1-((1-\sqrt 5)/2)^n-1)/\sqrt 5 ; 3) for the case of k=(n-1)/2 , the number of trace terms is (((1+\sqrt 5)/2)^n-1-((1-\sqrt 5)/2)^n-1)/\sqrt 5 ; 4) for the case of $(n, k)=(5t+4, 4t+3)$ or $(5t+1, 4t+1)$ with $t\geq 1$ , the number of trace terms is 8; and 5) for the case of $(n, k)=(7t+6, 6t+5)$ or $(7t+1, 6t+1)$ with $t\geq 1$ , the number of trace terms is 21. As a byproduct, we find new classes of ternary bent functions with only 8 or 21 trace terms. [ABSTRACT FROM PUBLISHER]