Does a different exponent and base but same key help to resolve discrete logarithm? E.g.:

$k = N^a \mod P$

The attacker knows the prime $P$ and $N$, which is also a prime and

(1.) prime root of $P$ or
(2.) has a cycle size of $s$, so $1 = N^s \mod P$, (and for $\forall s'&lt;s$, $1 &lt; N^{s'} \mod P$

The attacker want to resolve variable $a$ with some discrete log. algorithm for a certain $k$.

About the $k$ he knows:

$k = M^b \mod P$

With $M \neq N$ but $M$ has same properties as case (1.) or case (2.) (same cycle size as $N$). For this equation he knows all variables $k,b,M,P$. Does the attacker help this knowledge about $k$ (in case (1.) or (2.)?

(3.) Somehow the attacker got knowledge of all variables ($a,b,N,M,P$) for one $k$. Does it help him to resolve $a'$ for other $k'$ (with same $N,M,P$), in case (1.)/(2.)?



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side node:

The sets of generated $k$ values are equal for $N$,$M$ in case (2.):

$\{N^c, \forall c \in \mathbb{N}\}=\{M^d, \forall d \in \mathbb{N}\}=S$

with $|S|=s$. The numbers in it have just a different order. (In case (1.) $s=P-1$)

(4.)side question:

bottleneck of this would be the size of $s$, right? No matter how big $P$ is, it would not help security with same $s$.
Does the attacker help this knowledge about $k$ (in case (1.) or (2.)?


No; suppose you had an Oracle that could recover $a$ given $N, P, k, b, M$, then you could recover $a$ just given $N, P, k$ (that is, solve the Discrete Log problem).  Because we believe that the Discrete Log problem (given appropriate choices for $N, P$) is hard, we can conclude that constructing such an Oracle is also hard.

Here is how we would use such an Oracle; we already have been given $N, P, k$, and so all we need to do is construct $b, M$.  This can be done by selecting a $b$ relatively prime to $P-1$ and random otherwise; we can then compute $M = k^{b^{-1} \bmod P-1} \bmod P$; such a $b, M$ pair satisfies $k = M^b \mod P$, and so satisfies the Oracle's preconditions.  We then supply $N, P, k, b, M$ to the Oracle, and recover $a$

A similar method can be constructed for your question (3)