Lightweight Noninteractive Membership Authentication and Group Key Establishment for WSNs

Algorithm 1

Membership authentication and group key establishment.

	Token generation
	For n users the MRC selects a random asymmetric polynomial, where is degree in and degree (i.e., ). We will prove this condition in Theorem 1 in , , and is a prime integer with The MRC computes a pair of shares, and for each user, where is the public information associated with each user, The MRC sends each pair of shares as ’s token, to user through the secure channel.
	Membership authentication
	We assume that (i.e., ) users, for example want to engage in a group key establishment in WSNs.
	Step 1. Each member broadcasts a random integer, to all other members, where .
	Step 2. Assume that the value with is used as the pairwise shared key between the shareholders and Each member uses one of shares of his token, or to compute pairwise shared keys, between any other users, where is the secret key shared between users, and
	Step 3. Each member computes authentication responses, where is a one-way hash output with and as inputs. Each is sent to member publicly for authentication.
	Step 4. After receiving from member the member uses his computed pairwise shared key, in Step 2 to compute and check whether If the checking is successful, member has been authenticated; otherwise, member has not been authenticated. Repeat this process for all other members
	Group key establishment and authentication
	Let us assume that at the end of membership authentication, all m members, have been successfully authenticated. Then, members follow an XOR operation algorithm to complete the group key establishment process. However, all exchange information among members is encrypted under the pairwise shared keys, , in the Step 2 of membership authentication.
	Step 1. Each member needs to select a secret input and broadcasts a random integer, to all other members, where .
	Step 2. Each member uses his pairwise shared keys with other members to compute
	Step 3. Each member uses his computed pairwise shared keys, in the Step 2 of membership authentication to encrypt as Member sends each to member
	Step 4. After receiving from other member, member uses his computed pairwise shared key, in the Step 2 of membership authentication to decrypt as Repeat this process for all
	Step 5. After obtaining from all other members, member computes , where is the XOR operation.
	Step 6. Each member computes and broadcasts and then checks if where and is a one-way hash output with and as input If the checking is successful, the group key has been authenticated, is the secret group communication key; otherwise, the group key has not been authenticated. Repeat this process for all group members