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I found my answer from a different question very similar to mine from this(crypto.stackexchange). Hope it helps.


If you are able to compute $m^1 \pmod{N}$, then you have (obviously) recovered the message $m$. So, you should be able to use the extended Euclidean algorithm to express $m^1 \pmod{N}$ in terms of $c_A$ and $c_B$. Hint: It will involve exponentiations and multiplications.


The answer depends on the factorization of $p-1$; for any factor $q$ of $p-1$, the attacker can derive $x \bmod q$ in $\sqrt{q}$ steps (by first computing $(g^x)^q = g^{xq}$, and then searching for the value $y$ with $(g^{xq})^y = 1$ using, say, Pollard Rho. Hence, if the attacker has a computation budget of $n$, and $p-1 = q \cdot r$, where $q$ is $n^2$ ...

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