Given a huge composite number with unknown prime factors and a much smaller prime modulus P, is there a way (short of the Herculean task of factorization) to identify the residue classes mod(P) of the factors individually, or to determine whether the factors belong to the same residue class, and if so which? Is the task easier for special choices of the modulus, and if so, how can one find them?
(If you could any of that, you could Chinese Remainder your way to the factors, but you might have to try an awful lot of combinations. A guy can dream, can’t he?)