Given generator $g$ with prime order $k$ in $\bmod P$. Does increasing $P = 2 \cdot c \cdot k +1$ decrease security? Increasing $g$ increase security? An adversary wants to find $a$ in

$$m \equiv  g^a \bmod P$$

He knows prime $P = 2 \cdot c \cdot k +1$ with it's primes $c,k$, value $m$ and $g$. And he also knows that $g$ only has an order of $k$, so:
$$g^k \equiv 1  \bmod P$$



Question:
If now the prime value $P$ gets increased for constant $k$ (order of $g$ still $k$ only factor $c$ gets increased) does this decrease the security?

Why this? If $P \rightarrow \infty $ it would be the normal logarithm which is easy to compute compared to discrete version.

The smaller the better (for same $k$)? Any boundary for best security?



Discrete Log is hard because it is always reduced to a number from $0$ to $P-1$.
 Is security bound to the number of multiplications of generator $g$ needed until over $P$ again? The less the better?  

Question part 2:
Would be a higher value for $g$ increase security? (best would be $(P-1)/2$?)
Or to be more certain would a higher smallest group element $\not=1$ increase security?
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 $g^a&lt;P$. Thus we can not use the normal logarithm.


  The smaller the better (for same $k$)?


Uh, no, on the contrary! The above says that larger $P$ does not decrease security. Correspondingly, larger $P$ tends to increase security. However, very large $P$ does not much increase security, because the best known algorithms we have for the discrete logarithm for fixed $k$ and large $P$ perform a number of modular multiplications modulo $P$ that is a function¹ of $k$ only, and the time necessary for a multiplication modulo $P$ grows relatively slowly with $P$: in the order of $O(\log^2 P)$ for the simplest and common methods².


  Any boundary for best security?


Only a lower boundary, and that's a nontrivial subject. The short answer might be: at least $2^{4095}$ (that is $P$ of $2^{12}$ bits) with $P$ not chosen close to the power of an integer.

The long answer includes that security against classical computers (not hypothetical quantum computers usable for cryptanalysis per e.g. Shor's algorithm) is believed to be limited by several possible attacks:


the Number Field Sieve, which efficiency is governed mostly by $P$, and which general form the General Number Field Sieve (the only form likely to be applicable to $P$ not chosen close to the power of an integer, a situation where the Special Number Field Sieve shines) is believed to require a $P$ in the order of at least $2^{11}$ to $2^{13}$ bits for security.
Pollard's rho, which efficiency is governed mostly by $k$, requiring a $k$ in the order of at least $2^8$ to $2^9$ bits for security.
Pohlig–Hellman (and possibly other algorithms) depending on the factorization of $k$, that do not work as well as Pollard's rho when $k$ is prime.



  Would be a higher value for $g$ increase security?


Not much. The influence of $g$ on security is usually neglected for the aforementioned three algorithm. I doubt it can exceed $\mathcal O(\log g)$ (which can be approached in Baby-step/Giant-step). Also, I know no way to select a very small $g$ with order $k$ and $\log_2(k)\ll\log_2(P)$, much less for prime $k$, which is desirable for various reasons including blocking Pohlig–Hellman.


  Or to be more certain would a higher smallest group element $\ne 1$ increase security?


Not that we know. Also I know no way to insure that for large parameters.



What we are discussing is a Schnorr group. There are well established methods to generate these, including from a public seed so as to tame suspicion of weak parameters. See FIPS&nbsp;186-4, appendix A (which uses $p,q$ where the question and answer use $P,k$).



¹ That function is at most about $\alpha k^2$ for a constant $\alpha$ in the order of $2$ depending on a variety of details. But that's immaterial to the reasoning in the paragraph with this note.

² The cost can get down to $O(\log^{1.585\ldots}P)$ with Karatsuba which sometime is used; $O(\log P\;\log\log P\;\log\log\log P)$ with Schönhage-Strassen which could be of interest for insanely large $P$, and $O(\log P\;\log\log P)$ with Harvey-Hoeven (a bleeding edge result which seems unlikely to have practical application).