This paper proposes a decomposition method for symmetric multiple-valued functions. It decomposes a given symmetric multiple-valued function into three parts. By using suitable decision diagrams for the three parts, we can represent symmetric multiple-valued functions compactly. By deriving theorems on sizes of the decision diagrams, this paper shows that space complexity of the proposed representation is low. This paper also presents algorithms to construct the decision diagrams for symmetric multiple-valued functions with low time complexity. Experimental results show that the proposed method represents randomly generated symmetric multiple-valued functions more compactly than the conventional representation method using standard multiple-valued decision diagrams. Symmetric multiple-valued functions are a basic class of functions, and thus, their compact representation benefits many applications where they appear.

Multiple-valued bent functions are functions with highest nonlinearity which makes them interesting for multiple-valued cryptography. Since the general structure of bent functions is still unknown, methods for construction of bent functions are often based on some deterministic criteria. For practical applications, it is often necessary to be able to construct a bent function that does not belong to any specific class of functions. Thus, the criteria for constructions are combined with exhaustive search over all possible functions which can be very CPU time consuming. A solution is to restrict the search space by some conditions that should be satisfied by the produced bent functions. In this paper, we proposed the construction method based on spectral subsets of multiple-valued bent functions satisfying certain appropriately formulated restrictions in Galois field (GF) and Reed-Muller-Fourier (RMF) domains. Experimental results show that the proposed method efficiently constructs ternary and quaternary bent functions by using these restrictions.