I am trying to understand Schnorr signature scheme, and in the text book there is the following computetion:
$$g^s\cdot y^{-r}\stackrel{?}{=}I$$
The context is cyclic groups, and $g$ is a generator of the group, and $y$ is some element $y = g^x$ for some $x$ chosen at random.
My question is:
What is the definition of $y^{-r}$?
How do you compute that given $y$ and $r$.
Thank you.
Let $G$ be a group and $g\in G$. For $r\in \Bbb N_0$ we define $g^r\in G$ recursively via $g^{k+1}:=g\cdot g^k$ with base case $g^0:=1\in G$. We extend this to $r\in \Bbb Z$ by taking inverses, i.e., $g^{-r}=(g^{-1})^r$.
In more fancy language: For every group $G$ and $g\in G$, there exists a unique homomorphism $\phi_g\colon \Bbb Z\to G$ such that $\phi_g(1)=g$. We write $g^r$ for $\phi_g(r)$. Note that $\phi_g(-1)$ is the inverse of $g$, so ambiguity with the notation $g^{-1}$ for the inverse of $g$ does not arise. The usual power laws $g^{r+s}=g^r\cdot g^s$ and $g^{rs}=(g^r)^s$ follow readily from the homomorphism property.