Fast interpolation and multiplication of unbalanced polynomials

We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Let f is an element of Z[x] be an unknown polynomial and s,D be bounds on its total bit-length and degree, our new interpolation algorithm returns.. with high probability using (O) over tilde (s log D.) bit operations and O (s log D log s) black box evaluation. For polynomial multiplication, assuming the bit-length s of the product is not given, our algorithm has an expected running time of (O) over tilde (s log D), whereas previous methods for (resp.) dense or sparse arithmetic have at least (O) over tilde (sD) or (O) over tilde (s(2)) bit complexity.