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

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.

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