Computing Sparse Multiples of Polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f F[x] of degree d, and a desired sparsity t, our goal is to determine if there exists a multiple h F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F = Q and t is constant, we give a polynomial-time algorithm 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.

• Type: Journal
• Year: 2010
• Where: ISAAC
• Authors: Mark Giesbrecht, Daniel S. Roche, Hrushikesh Tilak
• Keywords: Algorithms, ISAAC 2010, Non-zero Terms, Polynomial-time Algorithm, Sparse Multiple