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I'm solving some task about Diffie-Hellman Key Exchange and came down to the equation :

$5^b \pmod {11} = 10^a \pmod {11} = S $ (Secret key, which is not important right now). is there any way I can find $a$ and $b$?

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As you are working mod 11 you always have 10 = -1. Thus 10^a (mod 11) equals 10 (=-1) if a is odd and it equals 1 if a is even. So let's make a case distinction here. Case 1: a is even. Then b has to be such that 5^b = 1 (mod 11). This means that b = 5x for a $x \in \mathbb{Z}$. If you compute the second case you get a contradiction, thus the set of all solutions is: a even, b divisible by 5. Now you might have some bounds on a and b that might help you to turn this into a unique solution.

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    now when I think I actually don't need to know the value of $a$ and $b$, I only need this $S$, so what you are saying is that, $S$ in my equation would always be $1$, right? – 2017-01-25