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. Fork odd with gcd(n, k) = 1 and a ∈ F
_3n
*, the function f (x) = Tr
₁
ⁿ
(ax
^((3
k
+1)/2)
) is weakly regular bent over F
_3n
, where Tr
₁
ⁿ
(·) is the trace function from F
_3n
to F
₃
. 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 + 5)/2)
^w+1
- ((1 - √5)/2)
^w+1
)/√5 or (((1 +√5)/2)
^n-w+1
- ((1 - √5)/2)
^n-w+1
)/√5 trace terms, where 0 <; w <; n and wk ≡ 1(mod n). Among them, five special cases are especially interesting: 1) for the case of k = n - 1, the number of trace terms is (((1 + 5)/2)
ⁿ
- ((1 - 5)/2)
ⁿ
)/ 5; 2) for the case of k = (n + 1)/2, the number of trace terms is (((1 + 5)/2)
^n-1
- ((1 - 5)/2)
^n-1
)/ 5; 3) for the case of k = (n - 1)/2, the number of trace terms is (((1 + √5)/2)
^n-1
- ((1 - √5)/2)
^n-1
)/√5; 4) for the case of (n, k) = (5t + 4, 4t + 3) or (5t + 1, 4t + 1) with t ≥ 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 ≥ 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.