  Group authentication is a method of confirmation that a set of users belong
to a group and of distributing a common key among them. Unlike the standard
authentication schemes where one central authority authenticates users one by
one, group authentication can handle the authentication process at once for all
members of the group. The recently presented group authentication algorithms
mainly exploit Lagrange's polynomial interpolation along with elliptic curve
groups over finite fields. As a fresh approach, this work suggests use of
linear spaces for group authentication and key establishment for a group of any
size. The approach with linear spaces introduces a reduced computation and
communication load to establish a common shared key among the group members.
The advantages of using vector spaces make the proposed method applicable to
energy and resource constrained devices. In addition to providing lightweight
authentication and key agreement, this proposal allows any user in a group to
make a non-member to be a member, which is expected to be useful for autonomous
systems in the future. The scheme is designed in a way that the sponsors of
such members can easily be recognized by anyone in the group. Unlike the other
group authentication schemes based on Lagrange's polynomial interpolation, the
proposed scheme doesn't provide a tool for adversaries to compromise the whole
group secrets by using only a few members' shares as well as it allows to
recognize a non-member easily, which prevents service interruption attacks.
