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This thesis deals with the conjugacy problem in groups and its twistedvariants. We analyze recent results by Bogopolski, Martino, Maslakova andVentura on the twisted conjugacy problem in free groups and its implication forthe conjugacy problem in free-by-cyclic groups and some further groupextensions. We also consider the doubly-twisted conjugacy problem in freegroups. Staecker has developed an algorithm for deciding doubly-twistedconjugacy relations in the case where the involved homomorphisms satisfy acertain remnant inequality. We show how a similar condition affects theequalizer subgroup and raise new questions regarding this subgroup. As anapplication we discuss the Shpilrain-Ushakov authentication scheme based on thedoubly-twisted conjugacy search problem in matrix semigroups over truncatedpolynomials over finite fields. Part of this thesis is devoted to theimplementation and testing of some of the previously mentioned concepts in theGAP programming language.

On the Conjugacy Problem in Groups and its Variants