Reducing many small products to one large multiplication

Develop a reduction that, given integers u1,…,un and v1,…,vn of p bits each, computes all n products ui·vi using a single integer multiplication of size O(np) (or O(1) such multiplications) on a multitape Turing machine.

Background

The authors speculate that if one could reduce n independent p-bit products to a single O(np)-bit multiplication, many results in the paper could be strengthened and simplified. In particular, the term accounting for many small multiplications (e.g., by twiddle and Bluestein factors) could be absorbed into one large multiplication step, potentially replacing Minc(m)-type bounds with Mcost(m) directly and improving the strength of the main implications.