Binary multiplication and parity convolution 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?
Sure, you can reduce it to a factor of 1, but probably at the cost of time. But to answer the question behind the question: multiplication of polynomials mod 2 is easier from a hardware point of view (no need to propagate carry bits), but multiplication of integers is an operation people consider essential, and so has direct support in ALUs and programming languages.