Constructing Polynomials for Functions over Residue Rings Modulo a Composite Number in Linear Time

摘要

We show how to cheek in linear time if a function f : Z_k~N → Z_K, where k = p~m p is a prime number, and m ≥ 2. Specified by its values, can be represented by a polinomial in the ring Z_k[x_1. … ,x_n]. If so, our algorithm also constructs (in linear time) its canonical polynomial representation. We also show how to extend our techniques (with linear time) to the cases of an arbitrary composite number k. More precisely, we prove that the circuit-size complexity of solving the problem, if a given function f : Z_k~N → Z_K,. Where k is a fixed composite number, specified by its values, is represented by a polynomial in the ring Z_k[x_1…,x_n] and, if so, finding its polynomial, is linear.