We describe an enhanced version of the TWINKLE factoring device and analyse to what extent it can be expected to speed up the sieving step of the quadratic Sieve and number field Sieve factoring algorithms. The bottom line of our analysis is that the TWINKLE-assisted factorization of 768 bit numbers is difficult but doable in about 9 months (including the sieving and matrix parts) by a large organization which can use 80000 standard Pentium II PC's and 5000 TWINKLE devices