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?
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$2*11=22$ which is $-1 \pmod{23}$.. – Arkady Jul 17 '17 at 20:02
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@Ravi. Thanks, I think whats confusing me here is the minus sign and the fraction. how would you go around that? – adhg Jul 17 '17 at 20:03
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I've posted an answer because it was too long for a comment. Hope this helps. – Arkady Jul 17 '17 at 20:08
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$\mathbb Z/23 \mathbb Z$ 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 $\frac{-1}{2}\equiv -12 \equiv 11 \pmod{23}$
Arkady
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