# Light-Weighted Password-Based Multi-Group Authenticated Key Agreement for Wireless Sensor Networks
## 1 Introduction  
A security association which manages security in a network layer is an important matter and it involves the establishment of a shared security key between two end points to support secure associations [1]. Wireless sensor networks (WSNs) consist of a large number of sensor nodes, which cannot support heavy computations, extensive communications or extensive storage and have limited bandwidth. They can be applied in many environments, such as medical monitors, military reconnaissance and communication, and others. WSNs are deployed to allow a legitimated user to login to the network and access data. The sensor node authentication has become one of the important security issues [2]. Group authenticated key agreements for WSNs enable a group of sensor nodes to authenticate each other and to establish a common key for securely communicating over public sensor networks. Group authenticated key agreement protocols typically fall into two categories, which are group key agreement protocols without public keys and group key agreement protocol using public keys. The former realize authentication and negotiates a group key using shared weak passwords or a shared long-term secret key [3; 4; 5; 6], while the latter realize authentication and negotiate a common group key using public key systems [7; 8; 9]. Most group key agreement protocols that use public keys have higher security than those without. However, they depend on time-consuming modular exponential computations and scalar multiplications on an elliptic curve, and thus are not suitable for sensor networks. Recently, several group-authenticated key agreement approaches have been presented. Unfortunately, most of these protocols were developed for two communication entities (two-party) or three communication entities (three-party), and can only be extended to a group key agreement protocol with difficulty. Thus, most authenticated key agreement protocols are difficult to extend to multi-group authenticated key agreement for WSNs. A multi-group key agreement protocol for WSNs allows communicating entities (sensor nodes) to belong to multiple groups, and enables each group to establish an independent group session key. A key hypergraph [9; 10; 11] is a graph where each vertex represents a party and each hyper-edge represents a relation among parties who to share a key. For instance, group members \(SN_{1}\), \(SN_{2}\), \(SN_{3}\), \(SN_{4}\) and \(SN_{5}\) involve groups \(\{SN_{1}\), \(SN_{2}\), \(SN_{3}\)}, \(\{SN_{3}\), \(SN_{4}\), \(SN_{5}\}\), and \(\{SN_{1}\), \(SN_{2}\), \(SN_{3}\), \(SN_{4}\), \(SN_{5}\}\) and establish group keys used for secure communication. Then its key hypergraph can be denoted as \(G=\{V,E\}\), where \(V=\{SN_{1}\), \(SN_{2}\), \(SN_{3}\), \(SN_{4}\), \(SN_{5}\}\) is a finite set of vertices and \(E=\{e_{1}=\{SN_{1}\), \(SN_{2}\), \(SN_{3}\}\), \(e_{2}=\{SN_{3}\), \(SN_{4}\), \(SN_{5}\}\), \(e_{3}=\{SN_{1}\), \(SN_{2}\), \(SN_{3}\), \(SN_{4}\), \(SN_{5}\}\}\) is a set of subsets of \(V\), as presented in Figure 1.  
Key management issues are also considered to have a major impact on the security scale of WSNs [12]. Recently, several group-authenticated key agreement approaches have been presented for WSNs. For example, in 2007, [PERSON] and [PERSON] [9] proposed a group-authenticated key agreement protocol that uses a public key system to build a session key; these group key approaches can be extended for hypergraphs and are suitable for use with multiple groups [9; 10; 11]. Users also need extra storage, such as radio frequency identification (RFID) tags, flash drives, smart cards and so on, to store public/private key pairs. In 2006, [PERSON] et al. [3] developed a password-based group-authenticated key exchange that can be executed in a constant number of rounds. In the same year, [PERSON] and [PERSON] [13] proposed a password-based encrypted group-authenticated key agreement protocol.