Can a System of Linear Diophantine Equations be Solved in Strongly Polynomial Time?

We demonstrate that the answer to the question posed in the title is “yes” and “no”: “no” if the set of permissible operations is restricted to {+,−,×,/,mod,<}; “yes” if we are also allowed a gcd-oracle as a permissible operation. It has been shown (see [Sto76, MST91]) that no strongly polynomial algorithm exists for the problem of finding the greatest common divisor (gcd) of two arbitrary integers, a and b. This result precludes the possibility of finding the set of solutions to a system of linear diophantine equations in strongly polynomial time. We show, however, that given an oracle that finds the gcd of two integers a and b and integer multipliers (x0, y0) satisfying ax0 +by0 = gcd(a, b), a system of linear diophantine equations can be solved in strongly polynomial time.