# Pairings for Beginners: Pohlig–Hellman attack time complexity

I'm reading Pairings for beginners by Craig Costello.

I'm trying to understand this example of (what I think) is the Pohlig–Hellman algorithim (on page 31 of the book).

Consider $$E/\mathbb{F}_{1021}\,:\,y^2=x^3+905x+100$$ with group order $$\#E(\mathbb{F}_q)=966=2\cdot3\cdot7\cdot23$$ and generator $$P = (1006,416)$$. We are given $$Q = (612,827)$$ and we seek to find $$k$$ such that $$[k]P = Q$$. Rather than seeking $$i$$ in the full group $$(2 \leq i \leq 965)$$, we can map the instance into each prime order subgroup by multiplying by the appropriate cofactor, and then solve for $$k_j = k\, \text{mod}\,j, j \in \{2,3,7,23\}$$. For $$j =2$$, we have $$P_j = P_2 = [966/2]P = [483](1006,416) = (174, 0)$$ and $$Q_j = Q_2 = [483](612,827) = (174, 0)$$ so $$Q_2 = [k_2]P_2$$ gives $$k_2 = 1$$.

He then gives the values for $$k_2$$, $$k_3$$, etc.

For $$k_{23}$$, he says

For $$Q_{23} = [k_{23}]P_{23}$$ we exhaust $$k_{23} \in \{1, ..., 22\}$$ to see that $$k_{23} = 20$$.

I'm not sure if this is just a typo, or if I'm mis-understanding something more fundamental. If $$k_{23} = 20$$ then, he did not exhaust $$\{1, ..., 22\}$$, he exhausted $$\{1, ....,20\}$$. He repeats this same thing elsewhere, so I figured it wasn't a typo and I've been left feeling a little confused.

Anyone have an explanation?

I'm not sure if this is just a typo, or if I'm mis-understanding something more fundamental. If $$k_{23} = 20$$ then, he did not exhaust $$\{1, ..., 22\}$$, he exhausted $$\{1, ....,20\}$$. He repeats this same thing elsewhere, so I figured it wasn't a typo and I've been left feeling a little confused.
What he means is, trying all the elements in the set $$\{1, ..., 22\}$$ exhaustively is guaranteed to find $$k_{23}$$. It happens that he found it after 20, and didn't have to test 21 or 22 (assuming he started testing from 1). In another universe, he could have found $$k_{23} = 5$$ and stopped there. But regardless of the true value, the approach is still to exhaust the set $$\{1, ..., 22\}$$.