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given some examples $k_{n_i},k_{m_i}$ out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}=G_n$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. The factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)

given some examples $k_{n_i},k_{m_i}$ out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}=G_n$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. For a given $k$ the factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)

given some examples out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}=G_n$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. The factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)

given some examples $k_{n_i},k_{m_i}$ out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}=G_n$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. The factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)

given some examples out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. The factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)

given some examples out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}=G_n$

Each set has size of $S$ which is a prime and known. Value $P$ is also a prime with $P = 2 \cdot S \cdot f+1$. Factor $f$ is (product of) prime(s) which is known as well. The Generator $g$ is known too. The factors $n,m$ and related exponent $a$ is unknown.

As shown here for each $k$ multiple value pairs $(n,a)$ can be computed very fast (pick an $a$ and compute $n=kg^{-a} \mod P$). That means those sets can be equal with $n\not=m$.
Now is there a way to check if they generate the same sets (without computing all combinations?)