Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

Public-key cryptography and invariant theory

Public-key cryptosystems are suggested based on invariants of groups. We give also an overview of the known cryptosystems which involve groups.Comment: 10 pages, LaTe

Length-Based Attacks for Certain Group Based Encryption Rewriting Systems

In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group

Аналіз складності реалізацій криптосистем на групах

This paper presents comparative analysis of cryptographic realyzations on groups. It is shown that the construction of cryptosystems in groups requires efficient algorithm for the mapping of number to the group and feedback mapping with computationally simple operation group. To date, there is only one known implementation of a cryptosystem MST3, built on the base of the abelian center of Suzuki group.Представлений порівняльний аналіз реалізацій криптосистем на групах. Показано, що побудова криптосистем на групах вимагає ефективного алгоритму для відображень числа на групу і зворотного відображення з обчислювально простою груповою опера­цією. До теперішнього часу відома тільки одна реалізація крипто­системи MST3, побудованої за Абелевим центром групи Судзук