Rigorous, subexponential algorithms for discrete logarithms over finite fields

There are numerous cryptosystems whose security is based on the difficulty of solving the discrete logarithm problem. This dissertation shows that there exist rigorous probabilistic algorithms for computing discrete logarithms in the multiplicative group of finite fields $GF(q)$ where $q=p\sp{n}$ for p a prime and $n=2$ or log $p < n\sp{.98},$ which have a subexponential expected running time. The discrete logarithm problem is attacked from two viewpoints. The first looks at the problem with an analytic viewpoint and succeeds in extending the work of Odlyzko in the case $GF(2\sp{n}),$ to the case $GF(p\sp{n})$ where log $p < n\sp{.98}.$ The second involves mainly algebraic number theory; I rigorously handle the case of $GF(p\sp2).$ To do this we must use the idea of "multipliers". Letting $L\sb{q}\lbrack b,a\rbrack$ = exp$(a(\log q)\sp{b}(\log\log q)\sp{1-b}),$ our algorithms are shown to have a rigorously provable expected running time at most $L\sb{q}\lbrack 1/2,3/2 + o(1)\rbrack$ bit operations for $GF(p\sp2)$ and $L\sb{p}\lbrack 1/2, \sqrt2 + o(1)\rbrack$ bit operations for $GF(p\sp{n})$ as $q\to\infty.$