  It is currently known from the work of Shoup and Nechaev that a generic
algorithm to solve the discrete logarithm problem in a group of prime order
must have complexity at least $k\sqrt{N}$ where $N$ is the order of the group.
In many collision search algorithms this complexity is achieved. So with
generic algorithms one can only hope to make the $k$ smaller. This $k$ depends
on the complexity of the iterative step in the generic algorithms. The
$\sqrt{N}$ comes from the fact there is about $\sqrt{N}$ iterations before a
collision. So if we can find ways that can reduce the amount of work in one
iteration then that is of great interest and probably the only possible
modification of a generic algorithm. The modified $r$-adding walk allegedly
does just that. It claims to reduce the amount of work done in one iteration of
the original $r$-adding walk. In this paper we study this modified $r$-adding
walk, we critically analyze it and we compare it with the original $r$-adding
walk.
