An Algebraic Approach to Reducing the Number of Variables of Incompletely Defined Discrete Functions

In this paper, we consider incompletely defined discrete functions, i.e., Boolean and multiple-valued functions, f:S→{0,1,...,q-1} where S ⊆ {0,1,...,q-1}n i.e.,the function value is specified only on a certain subset S of the domain of the corresponding completely defined function. We assume the function to be sparse i.e. |S| is 'small' relative to the cardinality of the domain. We show that by embedding the domain {0,1,...,q-1}n, where n is the number of variables and q is a prime power, in a suitable ring structure, the multiplicative structure of the ring can be used to construct a linear function {0,1,...,q-1}n {0,1,...,q-1}m that is injective on S provided that m > 2logq |S| + logq(n - 1). In this way we find a linear transform that reduces the number of variables from n to m, and can be used e.g. in implementation of an incompletely defined discrete function by using linear decomposition.