Suppose we are aiming to solve the discrete logarithm problem $\alpha^x=\beta$ in some cyclic group $G=<\alpha>$. Then we are looking for a (uniformly) random sequence of elements of the form $\alpha^{a_{i}}\beta^{b_{i}}$, $i=0,1,2,...$ so that we can invoke the 'Birthday Paradox'. Pollard suggested a function which apparently works fairly well but can be improved on.
My question concerns the concrete implementation of Pollard's Rho: if we generated sequences of integers $\{a_{i}\}$, $\{b_{i}\}$ with a random number generator (from any given programming language), would this yield an acceptable 'level of randomness'?
The reason I am asking this is because I stumbled upon articles on 'random walks' for Pollard's Rho, which all seemed rather extensive; this led me to believe that achieving an desired level of randomness is quite tricky, and that there is a flaw in my idea of generating random integers. However, I can't seem to find it.