Succinct certificates for the solvability of binary quadratic Diophantine equations

Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. This class of equations includes as special cases many of the known examples of natural problems apparently occupying intermediate stages in the P − NP hierarchy, i.e., problems not known to be solvable in polynomial time nor to be NP-complete, for example the problem of factoring integers. Let L(F) denote the length of the binary encoding of the binary quadratic Diophantine equation F given by ax 2 +bx1x2+cx 2 +dx1+ex2+f = 0. Suppose F is such an equation having a nonnegative integer solution. This paper shows that there is a proof (i.e., “certificate”) that F has such a solution which can be checked in O(L(F) 5 logL(F)log logL(F)) bit operations. A corollary of this result is that the set � = {F : F has a nonnegative integer solution} is in