This book is about relations between three di?erent areas of mathematics and theoreticalcomputer science: combinatorialgroup theory, cryptography, and c- plexity theory. We explorehownon-commutative(in?nite) groups, which arety- callystudiedincombinatorialgrouptheory, canbeusedinpublickeycryptography. We also show that there is a remarkable feedback from cryptography to com- natorial group theory because some of the problems motivated by cryptography...