Computing Sparse Multiples of Polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f??Fx] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h??Fx] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=? and t is constant, we give an algorithm which requires polynomial-time in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.