Vtyjkyuk

For the question, we write $ell(a)$ to denote the length of the positive integer $a=(a_1cdots a_k)_2$.

In my cryptography textbook, in the section of binary division, I found the following claim:

Let $ageq bin BbbZ^+$. Then, the binary division $a:b$ requires
$leq ell(b)cdot(ell(a)-ell(b)+1)$ binary bit operations.

Τhen, in order to prove the claim, it continues:

From the division algorithm, $exists a,bin BbbZ^+: a=bq+r, 0leq r<b.$ And from an easy to prove theorem, we know that $ell (q) leqell(a)-ell(b)+1.$

Now, finding the quotient and remainder requires $ell(q)leq ell(a)-ell(b)+1$ subtractions.
subtractions.

Each subtraction requires $ell (b)$ bit operations.

And in this sentence above I faced the problem.

Why do we have $ell(b)$ bit operations instead of $ell(a)$?

For example:

In this example we have $2$ subtractions. In the first $(1101-1001)$ we have $4=ell(b)$ binary bits operations. But, in the second $(10001-1001)$ we have $5>ell(b)=4$ binary bits operations.

Why does this happen? Do I misunderstand something?

Thank you.

You know the result of the subtraction will be less than $b$, so you don't have to carry out the subtraction of the highest-order bit (because you know the result will be $0$).