Given a bent function f (x) of n variables, its max-weight and min-weight functions are introduced as the Boolean functions f + (x) and f − (x) whose supports are the sets {a ∈ Fn2 | w( f ⊕la) = 2n−1+2 n 2 −1} and {a ∈ Fn2 | w( f ⊕la) = 2n−1−2 n 2 −1} respectively, where w( f ⊕ la) denotes the Hamming weight of the Boolean function f (x) ⊕ la(x) and la(x) is the linear function defined by a ∈ Fn2 . f + (x) and f − (x) are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple ( f0(x), f1(x), f2(x), f3(x)) of bent functions of n variables such that f0(x) ⊕ f1(x) ⊕ f2(x) ⊕ f3(x) = 1, a bent function of n + 2 variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.
展开▼
机译：给定n个变量的折弯函数f（x），将其最大权重和最小权重函数作为布尔函数f +（x）和f-（x）引入，其布尔函数为{a∈Fn2 | w（f⊕la）= 2n-1 + 2 n 2 -1}和{a∈Fn2 | w（f⊕la）= 2n-1-2 n 2 -1}，其中w（f⊕la）表示布尔函数f（x）⊕la（x）的汉明权重，而la（x）是由∈Fn2定义的线性函数。 f +（x）和f-（x）被证明是弯曲函数。此外，将2个变量的4个最小项与n个弯曲函数的4元组（f0（x），f1（x），f2（x），f3（x））的最大权重或最小权重函数组合变量使得f0（x）⊕f1（x）⊕f2（x）⊕f3（x）= 1，得到n + 2个变量的弯曲函数。介绍了满足上述条件的四元族弯曲函数族，最后，得到了可以用本文介绍的方法构造的弯曲函数数。此外，我们的结构与其他弯曲功能的结构进行了比较。
展开▼
