This question is about the relationship between normal multiplication of binary numbers and polynomial multiplication mod 2. To make the question concrete, I would ideally like to know if there is a better solution to the question from Knuth vol. 2, 3rd edition, page 420 than that given in the book.
"Can the multiplication of polynomials modulo 2 be facilitated by using the ordinary arithmetic operations on a binary computer, if coefficients are packed into computer words."
Knuth gives a reasonably straightforward reduction which expands the number of bits in the input by a log multiplicative factor in the worst case. Can this log factor be reduced?