Ew, I wouldn't have chosen linked lists to implement a BigNumber library...

I wrote my large number library using char* to represent numbers, reversed so that units were always str[0], tens str[1], 10^n's str[n]. e.g. 375 was stored as "573", 27 as "72".

Multiplication worked along the lines of long multiplication that I did at school on paper:
375
27x
===
2625 <- this line is 375*7
7500 <- this line is 375*2*10
===
10125 <- this line is the sum of the above two lines

So to implement this you'll need functions to add (which you'll probably already have cos it's easy to add) and to multiply by a single digit. It shouldn't be impossible to do this with linked lists, if you really want to put yourself through that pain.