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 > 2 log_q |S| + log_q (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.