The distribution of solutions to xy = N mod a with an application to factoring integers

We consider the uniform distribution of solutions $(x,y)$ to $xy=N \mod a$, and obtain a bound on the second moment of the number of solutions in squares of length approximately $a^{1/2}$. We use this to study a new factoring algorithm that factors $N=UV$ provably in $O(N^{1/3+\epsilon})$ time, and discuss the potential for improving the runtime to sub-exponential.