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If now the prime value $P$ gets increased for constant $k$ (order of $g$ still $k$) does this decrease the security? No, for random choice of $g$ among $g$ with large constant order $k$. Why this? If $P \rightarrow \infty$ it would be the normal logarithm .. When $P$ grows, $g$ also grows. So we never reach the threshold where $g^a<P$. Thus we can ...


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(I observe $S=c\cdot P$ for some integer $c$.) Is that always the case? Yes. Proof follows. If $g=P$ then $S=0$. We'll disregard this special case in the following. The set $M$ has $k$ elements, with $k$ the lowest strictly positive integer with $g^k\equiv1\pmod P$. This $k$ is known as the order of $g$ modulo $P$. This $k$ divides $P-1$. $M$ also is $\{g^...


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