Improved routing-based linear algebra for the number field sieve

Efficient custom-hardware designs were proposed for the linear algebra step of the number field sieve integer factoring algorithm. These designs make use of a heuristic mesh routing algorithm, whose performance has been analyzed only experimentally. We show that the routing algorithm may in fact encounter livelocks, i.e., may never terminate. We propose a context-specific solution for this problem. We also show how efficiency can be improved by a factor of ≈ 4 through the use of a parallel tori topology and packet injection during routing.