p-Ary and q-ary versions of certain results about bent functions and resilient functions

Abstract

Using the Teichmu¨ller character and Gauss sums, we obtain the following results concerning p-ary bent functions and q-ary resilient functions: (1) a characterization of certain q-ary resilient functions in terms of their coefficients; (2) stronger upper bounds for the degree of p-ary bent functions; (3) determination of all bent functions on F p ; (4) a characterization of ternary weakly regular bent functions in terms of their coefficients.

Monomial and quadratic bent functions over the finite fields of odd characteristic

“…These identities completely agree with (2) and prove that f is a bent function. The sign of S f (b) does not depend on b which means that f is a (weakly) regular bent function.…”

“…Thus, regular bent functions can only be found for even n and for odd n with p ≡ 1 (mod 4). Here we note a minor inaccuracy found in [3,Theorem 9] that contains a statement equivalent to (2). In the correct version e there should be replaced with n and ± ω k should also stand for the case of even n.…”

“…However, this technique does not help in telling if the function is (weakly) regular. Citing the recent work [2] by Hou, "it appears that all known p-ary bent functions are weakly regular". In the current paper we not only describe new classes of bent functions but also prove a stronger property that they are (weakly) regular.…”

Some results onq-ary bent functions

“…It has been proved in [6] that generalized bent functions exist for every value of q and n, except when n is odd and q = 2 mod 4, whereas Boolean bent functions exist only for even n. Kumar et al [6] have provided an analogue of classical Maiorana-McFarland class of bent functions in the generalized setup and discussed several properties of these functions. For more results on q-ary bent functions we refer to [1][2][3][4]. Generalized bent functions are widely applicable in Code-Division Multiple-Access (CDMA) communications systems [10].…”

A Family of p-ary Binomial Bent Functions

“…Highly nonlinear functions including Bent functions have been extensively applied to cryptography, sequences and coding theory [1,2,10]. The concept of Bent function is also generalized to more general notations such as generalized Bent functions [2,6,9,12,13,19]. People have paid lots of attention to this topic, however, the complete classification of Bent functions is still hopeless.…”