One of the possible generalizations of the discrete logarithm problem to
arbitrary groups is the so-called conjugacy search problem (sometimes
erroneously called just the conjugacy problem): given two elements a, b of a
group G and the information that a^x=b for some x \in G, find at least one
particular element x like that. Here a^x stands for xax^{-1}. The computational
difficulty of this problem in some particular groups has been used in several
group based cryptosystems. Recently, a few preprints have been in circulation
that suggested various "neighbourhood search" type heuristic attacks on the
conjugacy search problem. The goal of the present survey is to stress a
(probably well known) fact that these heuristic attacks alone are not a threat
to the security of a cryptosystem, and, more importantly, to suggest a more
credible approach to assessing security of group based cryptosystems. Such an
approach should be necessarily based on the concept of the average case
complexity (or expected running time) of an algorithm.
These arguments support the following conclusion: although it is generally
feasible to base the security of a cryptosystem on the difficulty of the
conjugacy search problem, the group G itself (the "platform") has to be chosen
very carefully. In particular, experimental as well as theoretical evidence
collected so far makes it appear likely that braid groups are not a good choice
for the platform. We also reflect on possible replacements.Comment: 10 page