I've a programming background and am just about to get into a project where Elliptic Curve Cryptography (ECC) is used. Although our libraries deal with the details I still like to do background reading so started with the ECC chapter of Understanding Cryptography. Everything was fine until I came upon this example of point doubling over $Z_{17}$:
I can't figure out how he gets $s$. For example the $(2\cdot1)^{-1}(3\cdot5^2+2)$ why does that evaluate to $2^{-1} \cdot 9$? From looking at it I would have thought $(3\cdot5^2+2)$ evaluates to $77$ so giving $2^{-1}\cdot77$ or $77\div2$ for the whole expression. Obviously the math doesn't work in the way I expect, is it something to do with the dot product not being normal multiplication? Or something else?
p.s. Sorry about formatting, I'm looking up how to do the Tex for the site now.
All arithmetic is done modulo $17$, so $$(2\cdot 1)^{-1}(3\cdot 5^2+2)\equiv 2^{-1}\cdot 9\bmod 17$$ is a congruence, not an equality in $\mathbb{Q}$.
Moreover, $9\cdot 9\equiv 13 \bmod 17$ because $$9\cdot 9 = 81 = 13 + 68 = 13 + 4\cdot 17 \equiv 13 \bmod 17,$$ or, equivalently, $$9\cdot 9\equiv (-8)\cdot (-8)\equiv 64\equiv 16\cdot 4\equiv -4\equiv 13\bmod 17.$$