Generating deterministic sequences of pseudo-random group elements (awaiting review)

In order to generate labels for divisions and conjugacy classes, we use a method for picking pseudo-random group elements.

We use an exponential pseudo-random number generator given by \( a_j = a_0\, b^j \bmod p\) where $$ \begin{aligned} a_0 &= 123 \\ b&=25096281518912105342191851917838718629 \\ p &= 340282366920938463463374607431768211297. \end{aligned} $$ (Here $p$ is the largest prime less than $2^{128}$.)

To produce a pseudo-random sequence of group elements we follow a standard method. Starting with $k$ generators, $g_1, \ldots, g_k$ we initialize the process by forming the $(k+5)$-tuple $v$ with $v_i=g_i$ (taking subscripts modulo $k$). We then have $20$ rounds of picking distinct indices $i$ and $j$ and replace $v_i$ with $v_i\cdot v_j$. After this initialization step, we generate the sequence by again picking random subscripts $i$ and $j$ according to our pseudo-random sequence of integers, replacing $v_i$ with $v_i\cdot v_j$ and returning the new $v_i$.