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

Cryptanalysis of some protocols using matrices over group rings

We address a cryptanalysis of two protocols based on the supposed difficulty of discrete logarithm problem on (semi) groups of matrices over a group ring. We can find the secret key and break entirely the protocols

On the Relations Between Diffie-Hellman and ID-Based Key Agreement from Pairings

This paper studies the relationships between the traditional Diffie-Hellman key agreement protocol and the identity-based (ID-based) key agreement protocol from pairings. For the Sakai-Ohgishi-Kasahara (SOK) ID-based key construction, we show that identical to the Diffie-Hellman protocol, the SOK key agreement protocol also has three variants, namely \emph{ephemeral}, \emph{semi-static} and \emph{static} versions. Upon this, we build solid relations between authenticated Diffie-Hellman (Auth-DH) protocols and ID-based authenticated key agreement (IB-AK) protocols, whereby we present two \emph{substitution rules} for this two types of protocols. The rules enable a conversion between the two types of protocols. In particular, we obtain the \emph{real} ID-based version of the well-known MQV (and HMQV) protocol. Similarly, for the Sakai-Kasahara (SK) key construction, we show that the key transport protocol underlining the SK ID-based encryption scheme (which we call the "SK protocol") has its non-ID counterpart, namely the Hughes protocol. Based on this observation, we establish relations between corresponding ID-based and non-ID-based protocols. In particular, we propose a highly enhanced version of the McCullagh-Barreto protocol

Cryptanalysis of an e_cient three-party password-based key exchange scheme

AbstractIn order to secure communications between two clients with a trusted server's help in public network environments, a three-party password-based authenticated key exchange (3PAKE) scheme is used to provide the transaction confidentiality and e_ciency. In 2010, Lou-Huang proposed a new simple three-party password-based authenticated key exchange (LH-3PAKE) scheme based on elliptic curve cryptography (ECC). By analysis, Lou-Huang claimed that the proposed LH- 3PAKE scheme is not only secure against various attacks, but also more e_cient than previously proposed 3PAKE schemes. However, this paper demonstrates LH-3PAKE scheme is vulnerable to o_-line password guessing attacks by an attacker

Cryptography: Mathematical Advancements on Cyber Security

The origin of cryptography, the study of encoding and decoding messages, dates back to ancient times around 1900 BC. The ancient Egyptians enlisted the use of basic encryption techniques to conceal personal information. Eventually, the realm of cryptography grew to include the concealment of more important information, and cryptography quickly became the backbone of cyber security. Many companies today use encryption to protect online data, and the government even uses encryption to conceal confidential information. Mathematics played a huge role in advancing the methods of cryptography. By looking at the math behind the most basic methods to the newest methods of cryptography, one can learn how cryptography has advanced and will continue to advance

Analysis of two pairing-based three-party password authenticated key exchange protocols

Password-Authenticated Key Exchange (PAKE) protocols allow parties to share secret keys in an authentic manner based on an easily memorizable password. Recently, Nam et al. showed that a provably secure three-party password-based authenticated key exchange protocol using Weil pairing by Wen et al. is vulnerable to a man-in-the-middle attack. In doing so, Nam et al. showed the flaws in the proof of Wen et al. and described how to fix the problem so that their attack no longer works. In this paper, we show that both Wen et al. and Nam et al. variants fall to key compromise impersonation by any adversary. Our results underline the fact that although the provable security approach is necessary to designing PAKEs, gaps still exist between what can be proven and what are really secure in practice