  The interval discrete logarithm problem is defined as follows: Given some
$g,h$ in a group $G$, and some $N \in \mathbb{N}$ such that $g^z=h$ for some
$z$ where $0 \leq z < N$, find $z$. At the moment, kangaroo methods are the
best low memory algorithm to solve the interval discrete logarithm problem. The
fastest non parallelised kangaroo methods to solve this problem are the three
kangaroo method, and the four kangaroo method. These respectively have expected
average running times of $\big(1.818+o(1)\big)\sqrt{N}$, and $\big(1.714 +
o(1)\big)\sqrt{N}$ group operations. It is currently an open question as to
whether it is possible to improve kangaroo methods by using more than four
kangaroos. Before this dissertation, the fastest kangaroo method that used more
than four kangaroos required at least $2\sqrt{N}$ group operations to solve the
interval discrete logarithm problem. In this thesis, I improve the running time
of methods that use more than four kangaroos significantly, and almost beat the
fastest kangaroo algorithm, by presenting a seven kangaroo method with an
expected average running time of $\big(1.7195 + o(1)\big)\sqrt{N} \pm O(1)$
group operations. The question, 'Are five kangaroos worse than three?' is also
answered in this thesis, as I propose a five kangaroo algorithm that requires
on average $\big(1.737+o(1)\big)\sqrt{N}$ group operations to solve the
interval discrete logarithm problem.
