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Here's a cyclic group of any order $q \ge 1$: Identity: $0$. Generator: $1$. Group operation: $a \cdot b$ is (a + b) % q.


Let $\mathbb{Z}, +$ be the group integers, $\mathbb{Z}/n\mathbb{Z}, \times$ the multiplicative group of integers modulo $n$, and $\varphi(n)$ its order. Then $\varphi(n)\mathbb{Z}, +$, the additive group of multiples of $\varphi(n)$ is a subgroup of $\mathbb{Z}$. The function $f : \mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z} : x \mapsto a^x \mod n$ for ...


As a add on to the above answer, it would be insecure to use $g^{x+y}$ as key , because both $g^x$ and $g^y$ will be transmitted publicly and by simply eavesdropping one can easily find the required key i.e $g^{x+y}$.


You may find it useful to play around with a toy example, such as the integers modulo a Fermat prime, like $p = 257$. Since $g$ is a generator of the Group, $h \equiv g^x$ for some unknown exponent $x$. In other words, $\log_gh = x$, and for Groups of order $2^k$, this discrete log is easily computed like so: Interpret $x$ as a $k$ bit number, i.e. $x = ...

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