The Residues Modulo M Of Products Of Random Integers

For two (possibly stochastically dependent) random variables X and Y taking values in {O, ... , m 1} we study the distribution of the random residue U = XY mod m. In the case of independent and uniformly distributed X and Y we provide an exact solution in terms of generating functions that are computed via p-adic analysis. We show also that in the uniform case it is stochastically smaller than (and very close to) the uniform distribution. For general dependent X and Y we prove an inequality for the distance sUPxE[O,l] lFu(x) xl.