Related discussion: https://codereview.stackexchange.com/questions/181556/multiplying-binary-numbers-digit-by-digit-and-carry-method-in-python

carryover.py

def is_zero(n):
    return not n[1:] and not n[0]

def carry_over(first_num, second_num):
    stack = []
    for i, j in enumerate(second_num):
        if j:
            # basicly same as map(sum, zip(stack, a)) but with zero appends
            stack = [(stack.pop(0) + n) if stack else n \
                     for n in ([0] * i + first_num)]
    overflow = 0
    for i, j in enumerate(stack):
        overflow, stack[i] = divmod(j + overflow, 2)
    while overflow > 0:
        stack.append(overflow % 2)
        overflow //= 2
    return stack

def multiply(first_num, second_num):
    if is_zero(first_num) or is_zero(second_num):
        return [0]
    else:
        return carry_over(first_num, second_num)