Succinct certificates for solutions to binary quadratic Diophantine equations

Abstract

Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. They contain as special cases many examples of natural problems apparantly occupying intermediate stages in the P-NP hierarchy, i.e. problems neither known to be polynomial time or NP-complete. Let L(F) denote the length of the binary encoding of the coefficients of a binary quadratic diophantine equation F(x_1, x_2)=0. This paper shows there is a certificate of length polynomial in L(F) that such an equation has an integer solution (resp. positive integer solution) when one exists. This is interesting because it is known there exist such equations whose minimal nonnegative integer solution is so large that it requires space exponential in L(F) to write it down in binary representation. The certificates are based on the ideas of D. Shank's "infrastructure".