How do I prove that this PRNG is easily distinguished from a random sequence of numbers (modulo m)? I was studying Crypto, and this question came up. Using a PRG like the multiplicative congruence generator:

$b_{i+1} = a \cdot b_i \mod m$

How do I prove that the output of this PRG, or any PRG's whatsoever is easily distinguished from a random sequence of numbers?
Well, this is essentially homework (it wasn't assigned, however you are attempting to learn from it), and so I won't give you an explicit answer; instead, I'll try to point you in a direction where you can figure out the answer yourself.

First of all, how would you solve this if you were given the values $a$ and $m$?

Once you have answered that, let us assume that you were given the output of such a sequence, and also given the value of $m$.  How would you recover the value $a$?

Now comes the hard part, let us assume that you were given the output of such a sequence, but did not know either $a$ or $m$; how would you recover the value $m$?

Hint on the last question: let us assume we were given three consecutive values $b_1, b_2, b_3$ with $b_2 = a b_1 \bmod m$, $b_3 = a^2 b_1 \bmod m$; how can we combine the values $b_1, b_2, b_3$ to come up with some value which is a multiple of $m$?  (Hint: consider $b_2^2$ and $b_1b_3$)?  How can we use that to recover $m$?

Last question: let us assume that the sequence we were given was not generated by such a generator; what happens when we attempt to recover $m$ using the above procedure?