The MMO problem

We consider a two polynomials analogue of the polynomial interpolation problem. Namely, we consider the Mixing Modular Operations (MMO) problem of recovering two polynomials f ∈ Z_p[x] and g ∈ Z_q[x] of known degree, where p and q are two (un)known positive integers, from the values of f(t) mod p+g(t) mod q at polynomially many points t ∈ Z. We show that if p and q are known, the MMO problem can be reduced to computing a close vector in a lattice with respect to the infinity norm. Using the Gaussian heuristic we also implemented in the SAGE system a polynomial-time algorithm. If p and q are kept secret, we do not know how to solve this problem. This problem is motivated by several potential cryptographic applications.