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8

A safe prime is a prime number $p$ for which $(p-1)/2$ is also prime. The order of an element $g$ of the group $\mathbf{Z}^*_p$ (the integers modulo $p$, excluding 0) is the smallest integer $n$ such that $g^n\equiv 1\pmod{p}$; this is always a factor of $p-1$. The orders of the subgroups of the group generated by $g$ are the factors of the order of $g$; ...

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Assume we are working in a cyclic group $G$ with generator $g$ and let $A$ denote the public key in use. From the definition of ElGamal encryption, we have $R_i = g^{r_i}$ and $c_i=A^{r_i}\cdot m_i$, where $r_i$ is some random number, for $i\in\{1,2\}$. Hence, with $R:=R_1\cdot R_2$ and $c:=c_1\cdot c_2$ (where $\cdot$ denotes $G$'s operation), we have ...

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Why is diffie-hellman defined on a cyclic group[0]? Doesn't it work for any commutative operation which the inverse is hard to find? No, you need associativity as well; once you have that, your idea would work fine, once we find a semigroup (that's what we call sets with an operator that is associative) with the appropriate properties. That's the ...

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Diffie-Hellman operates in a cyclic group by definition: the elements $g, g^a, g^b, g^{ab}$ are in the cyclic group generated by $g$. Technically, a monoid is sufficient, but since cryptography mostly operates in finite structures, you get a group anyway. In your example, you operate in the cyclic group $c\mathbf{Z}$, and as you were told in the comments, ...

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