On constructions and properties of (n,m)-functions with maximal number of bent components

05/25/2019
by   Lijing Zheng, et al.
0

For any positive integers n=2k and m such that m≥ k, in this paper we show the maximal number of bent components of any (n,m)-functions is equal to 2^m-2^m-k, and for those attaining the equality, their algebraic degree is at most k. It is easily seen that all (n,m)-functions of the form G(x)=(F(x),0) with F(x) being any vectorial bent (n,k)-function, have the maximum number of bent components. Those simple functions G are called trivial in this paper. We show that for a power (n,n)-function, it has such large number of bent components if and only if it is trivial under a mild condition. We also consider the (n,n)-function of the form F^i(x)=x^2^ih( Tr^n_e(x)), where h: F_2^e→F_2^e, and show that F^i has such large number if and only if e=k, and h is a permutation over F_2^k. It proves that all the previously known nontrivial such functions are subclasses of the functions F^i. Based on the Maiorana-McFarland class, we present constructions of large numbers of (n,m)-functions with maximal number of bent components for any integer m in bivariate representation. We also determine the differential spectrum and Walsh spectrum of the constructed functions. It is found that our constructions can also provide new plateaued vectorial functions.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset