THE HIDDEN NUMBER PROBLEM WITH NON-PRIME MODULUS

We consider a generalization of the Hidden Number Problem for general moduli N, and prove that it can be solved with high probability if roughly 2(log N)(1/2) approximations of quality at least (log N)(1/2) are given, and the multipliers are chosen uniformly at random from Z NZ. We prove a similar result in the case that the multipliers are chosen uniformly at random from (Z/NZ)(x) and N is the product of two distinct primes. The last result holds in the more general case when N is squarefree and a technical condition related to the prime factors of N holds. The condition is related to the distribution of the solutions of a linear equation modulo N. The Hidden Number Problem modulo the product of two primes, with multipliers chosen from (N/NZ)(x), is related to the bit security of the most significant bits of the RSA and Rabin functions. Our solution of the Hidden Number Problem implies that computing roughly (log(N))(1/2) bits of the RSA and Rabin functions is equivalent to computing the entire values.