Generalization of the Hanrot–Quercia–Zimmermann middle product technique

Determine the extent to which the Hanrot–Quercia–Zimmermann middle product algorithm, which accelerates computation when multiplying a polynomial of length 2N by one of length N by computing only the middle third of the 3N-length product via FFTs of size 2N, can be generalized to other clipped polynomial product scenarios beyond this specific 2N×N case and middle-third range.

Background

The paper studies algorithms for computing specified spans of coefficients in products of integers and polynomials, termed clipped products. For polynomial multiplication, FFT-based methods are considered alongside classical and Karatsuba approaches.

In the specific asymmetric case of multiplying a 2N-length polynomial by an N-length polynomial where only the middle third of the product is needed, Hanrot, Quercia, and Zimmermann showed that FFTs of size 2N (instead of 4N) suffice, yielding a factor-of-2 speedup for certain operations such as Newton iterations for division. The authors note uncertainty about extending this optimization beyond the canonical middle-product setup to broader clipping ranges and operand sizes.