Remarks on Shapes of Decision Diagrams and Classes of Multiple-Valued Functions Stanislav Stankovi´ c 1 , Radomir S. Stankovi´ c 2 , Jaakko Astola 1 1 Institute of Signal Processing, Tampere University of Technology, FIN-33101 Tampere, Finland 2 Dept. of Computer Science, Faculty of Electronics, 18 000 Niˇ s, Serbia Abstract—The paper studies binary and ternary functions that have decision diagrams of identical shape in the original and spectral (Fourier) domain. These functions are called Fourier- sweet functions. This class of functions involves certain classes of bent functions and quadratic forms in both binary and ternary cases. Bent functions and quadratic forms have applications in cryptography and error-correcting codes. Not all bent functions are Fourier-sweet functions. It follows, that Fourier-sweet func- tions are capable of capturing the differences among the classes of bent functions, and at the same time link them to quadratic forms. Representation by shape invariant decision diagrams in the original and spectral domain might provide some better insight into features of bent functions and quadratic forms. The functions represented by the disjoint quadratic forms in the binary case and diagonal forms in the ternary case are elementary Fourier-sweet functions. In both binary and ternary cases, the application of affine transformations, under certain precisely specified restrictions, to the elementary Fourier-sweet functions produces other Fourier-sweet functions. I. I NTRODUCTION Decision diagrams are a data structure allowing compact representations of large discrete functions by taking advantage of the peculiar properties the function to be represented may possess [15]. Decomposition rules used to assign a function to a decision diagram might capture these features of functions and due to that produce compact decision diagrams. For a given function f , decision diagrams defined with respect to different decomposition rules might express dif- ferent relationships. For example, BDDs and Walsh decision diagrams (WDDs) are used for the classification of switching functions [19], [24]. Furthermore, new classes of functions can be defined by requiring certain relationships between decision diagrams to be fulfilled. In particular, sweet Boolean functions are defined as monotone functions having BDDs and Zero- suppressed decision diagrams (ZDDs) [9], [10] for their prime implicants of identical shapes [6]. In this paper, we define Fourier-sweet switching functions for binary and ternary logic functions by requiring the shapes of BDDs and WDDs to be identical for binary functions, and the same for Multiple-place decision diagrams (MDDs) and Vilenkin-Chrestenson decision diagrams (VCDDs) for ternary functions. II. BACKGROUND THEORY A. Binary and Ternary Logic Functions In this paper, we consider binary and ternary logic functions, that are defined as mappings f : {0, 1} n →{0, 1} and f : {0, 1, 2} n →{0, 1, 2}, respectively, where n is the number of variables. They can be viewed as functions on finite groups C n 2 and C n 3 , where C 2 and C 3 are cyclic groups of order 2 and 3, respectively. In this case, they can be represented and processed by Fourier transforms on these groups. B. Fourier Transforms The Fourier transforms on Abelian groups are defined in terms of group characters. In the case of binary functions, the Fourier transform is the Walsh transform and for ternary functions the Vilenkin-Chrestenson transform, see for instance [5]. In the case of finite groups, these transforms can be simply defined in terms of the corresponding transform matrices. Definition 1: (Walsh transform) For a function f (x 1 ,x 2 ,...,x n ) specified by the function- vector F =[f (0),f (1),...,f (2 n - 1)] T , the Walsh spectrum is a vector S f =[S f (0),S f (1),...,S f (2 n - 1)] T determined as S f = W(n)F, where the Walsh transformation matrix is defined as W(n)= n  i=1 W(1), W(1) =  1 1 1 -1  . When the Walsh transform is applied to binary logic func- tions it is good to use the encoding x i → (-1) xi , i.e., (0, 1) → (1, -1) to make the functions to be processed closer to the basis functions used in the transform, resulting in some useful properties of the Walsh spectrum [2], [3]. Some of these features, will be used in this paper. Example 1: Consider the function f (x 1 ,x 2 ) = x 1 x 2 , x 1 ,x 2 ∈{0, 1}, with multiplication defined modulo 2, i.e., as logic AND. The encoded function vector is F = [1, 1, 1, -1] T and the Walsh spectrum is computed as S f = W(2)F = [2, 2, 2, -2] T . Definition 2: (Vilenkin-Chrestenson transform) For a ternary function f (x 1 ,x 2 ,...,x n ) specified by the function-vector F = [f (0),f (1),...,f (3 n - 1)] T , the Vilenkin-Chrestenson spectrum can be represented by a vector S f =[S f (0),S f (1),...,S f (3 n - 1)] T determined as S f = C ∗ (n)F, where the Vilenkin-Chrestenson transformation matrix is de- fined as the complex-conjugate transpose C ∗ (n) of the 2012 IEEE 42nd International Symposium on Multiple-Valued Logic 0195-623X/12 $26.00 © 2012 IEEE DOI 10.1109/ISMVL.2012.37 134