understanding the resulted lambda for 2 points on the elliptic curve

Question

I'm following on a tutorial and the resulted lambda for the 2 points p & q is:

$\lambda=\frac{-1}{2} = 11 mod 23$

but I'm not sure how the author got 11?

Answer 1

$\mathbb Z/23 \mathbb Z$ is a field so you can divide by any element that is not zero. (In general any $\mathbb Z/p\mathbb Z$ for $p$ a prime is a field.)

So as your first step, you should think of $1/2 \pmod{23}$ as the multiplicative inverse of $2 \pmod{23}$. To give more details, what multiplied by $2$ gives $1\pmod{23}$? By inspection, the answer is $12$ because $12\times 2=24$ which is $1\pmod{23}$. So $$\frac{1}{2}\equiv 12 \pmod{23}$$

Now you can easily see that $\frac{-1}{2}\equiv -12 \equiv 11 \pmod{23}$