Chaotic map based key agreement with/out clock synchronization
Introduction
A key agreement/exchange protocol is a protocol which enables two or more parties to agree on a key or exchange some information in such a way that both of them decide on the established session key. If properly done, this precludes undesired third-parties from forcing a key choice on the agreeing/exchanging parties. Protocols which are useful in practice also do not reveal to any eavesdropping party what key has been agreed upon or exchanged.
A secure key agreement protocol can help communication parties to establish a shared secret session key [1], [8], [16]. The communication parties then use the shared session key for subsequent communications. Therefore, building secure key agreement protocols over public channel is one of the primitive goals in information security. Since Diffie and Hellman created the first key agreement protocol [7], some key agreement protocols have been proposed. However, most of the existing key agreement protocols are based on number theory [16].
In recent years, chaos as a universal, random-like and robust phenomenon has received significant concern and research from multidisciplinary areas [2], [3], [6], [9], [14], e.g. from physical science to computer science, from electric engineering to telecommunication, and from mathematics to non-linear science.
One of the successful and significant interdisciplinary applications for chaos is in secure communications [1], [3], [5], [8], [10], [12], [15]. Currently, there have been two main approaches to the use of chaotic systems in designing cryptosystems: the analog one and the digital one. The analog approach uses hardware-based synchronized chaotic circuits where, chaos synchronization is achieved [4], [6], [9]. The digital one is designed for digital computer with chaotic discrete dynamical systems [5], [10].
Kocarev et al. created new encryption system, which was a cryptographic system using chaotic maps, especially Chebyshev chaotic maps [10]. Following Kocarev et al.’s work which applied chaotic map to information security, Xiao et al. proposed a key agreement. Bergamo et al. then developed an attack on the protocol. To improve the security of key agreement based on chaotic maps, Xiao et al. further used chaotic maps to propose a new key agreement protocol [13]. However, their new protocol has been compromised by Han’s method [8]. Subsequently, Chang et al. proposed a new key agreement protocol using chaotic map and passphrase [5]. Chang et al.’s protocol addressed the security issues in previous key agreement using chaotic maps. However, their protocol can only work in clock synchronization environment.
Therefore, in order to enhance the security and extend the flexibility and the usability, in this paper, we will develop two new key agreement protocols which both use chaotic maps. Further, the proposed key agreements will have the following properties: (1) The first one works with clock synchronization. The second one works without clock synchronization; (2) The two protocols can resist guessing attack; (3) The private keys can be not so large numbers as 2048 bit-length; (4) The two communication parties are mutually authenticated; (5) The communication parties use authenticated password for secure communications; (6) If there exists a replaying attacker interfering with the communication, they can still establish a shared session key securely.
Section snippets
Protocols based on chaotic map and their security issues
In this section, two key agreement protocols based on chaotic maps are analyzed. Some issues will be pointed out with them. We first review the Chebyshev map, which has semi-group property.
Mutual authenticated key agreement protocols based on chaotic maps
In this section, we will use two different approaches to develop two key agreement protocols based on chaotic maps. Both of them have the property of mutual authentication. The first approach will create a chaotic map based key agreement with clock synchronization. The second approach will bring out a chaotic map based key agreement without clock synchronization. Therefore, the protocol 1 is suitable for some practical applications where the timeline is critical; while the protocol 2 is
Security resistant to Bergamo et al.’s attack
Bergamo et al.’s security compromise method is based on the condition that an adversary can obtain the related elements x, and . The two new protocols are able to resist Bergamo et al.’s attack. This is because (as well as ) has been enciphered using the hash value h which is only known to the user and the server. As a result, even though an adversary can get X or Y, it cannot work out using Bergamo et al.’s method.
Mutual authentication
The identity authentication in our key agreement
Comparison
We summarize the difference between the new key agreement protocols (i.e. the Protocol 1 (P1) and the Protocol 2 (P2)) in our paper, the XLD protocol in [13], and the CH protocol in [5].
Conclusions
We proposed two secure key agreement protocols based on chaotic maps in this paper. The first protocol works in a situation where clock synchronization is available, while the second one works without clock synchronization. Both of the protocols provide mutual authentication for the server and the user. More importantly, they both can resist the guessing attack and the man-in-the-middle attack. Therefore, on the one hand, the two proposed protocols provided the flexibility and the usability in
Acknowledgement
The authors sincerely thank anonymous reviewers. This work was partially supported by Curtin University of Technology and ARC.
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