The complexity of Grigorchuk groups with application to cryptography


The Turing complexity of the word problems of a class of groups introduced by Grigorchuk (1985) is examined. In particular, it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct. A refinement of Savitch's translation theorem as well as a similar result restricted for time complexity follow. New families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time. A new family of public- key cryptosystems based on these word problems is constructed. © 1991.

Publication Title

Theoretical Computer Science