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Short answer The distinguisher is only given the output of the generator on a uniformly chosen seed of appropriate length, along with a truly random string of the same length as the output of the generator. So, no, the distinguisher is not given $a$, $b$ and $m$. However, as you note we can still consider an algorithm in which those values are hardcoded, ...


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You're right: If you know the setup, calculating the next output from any given $x$ is fully deterministic and you know everything already. Even if you don't know $a$ and $b$, those are easy to calculate from three consecutive $x_i$. If $n$ is not known, its calculation is still pretty easy, given a few consecutive $x_i$. Anyway, LCGs are very unsuitable ...


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The goal of a random number generator is to generate a uniform distribution. Taken the LCG computational indistinguishability means that every sequence generated $\mod m$ is a uniform distribution over the values $0, ..., m-1$. The function parameters are not given. Instead you would have a sequnce $x_0, x_1, ...$ given and have to determine if all values ...



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