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 appear to be new
to group theory, and they open many interesting research - enues within
group theory. Then, we employ complexity theory, notably generic case
complexity of algorithms, for cryptanalysisof various
cryptographicprotocols based on in?nite groups. We also use the ideas
and machinery from the theory of generic case complexity to study
asymptotically dominant properties of some in?nite groups that have been
used in public key cryptography so far. It turns out that for a relevant
cryptographic scheme to be secure, it is essential that keys are
selected from a "very small" (relative to the whole group, say) subset
rather than from the whole group. Detecting these subsets ("black
holes") for a part- ular cryptographic scheme is usually a very
challenging problem, but it holds the keyto
creatingsecurecryptographicprimitives basedonin?nite non-commutative
groups. The book isbased onlecture notesfor the Advanced Courseon
Group-Based CryptographyheldattheCRM, BarcelonainMay2007.
Itisagreatpleasureforus to thank Manuel Castellet, the HonoraryDirector
of the CRM, for supporting the idea of this Advanced Course. We are also
grateful to the current CRM Director, JoaquimBruna, and to the friendly
CRM sta?, especially Mrs. N. PortetandMrs. N. Hern´ andez, for their
help in running the Advanced Course and in preparing the lecture notes.
