Secret key generation for pairwise independent networks with curious helpers

We consider the problem of group secret key generation through a multi-hop network modeled by a pair-wise independent network and propose a low complexity scheme which can guarantee strong secrecy. We assume that the legitimate transmitter only has the knowledge of the maximum number of precoded keys known to any helpers, where the precoded keys are the output of a precoder at the transmitter to be propagated through the network. We also assume that nodes in the multi-hop network are curious, i.e., they not only help to relay but may eavesdrop. Confidentiality is required against an eavesdropper outside of the multi-hop network and also against the helpers in the multi-hop network, while both the curious helpers and the external eavesdropper can access a finite rate public channel between the legitimate transmitter-receiver pair. The objective is to generate secret keys shared by a given pair of nodes outside of the multi-hop network at the largest possible rate, with the cooperation of the nodes within the multi-hop network. We propose a low complexity linear transformation-based global key propagation approach to combat curious helpers by the number theoretic transform, under two scenarios: with an additional public or private channel between the pair of nodes aiming to share keys. We analyze the achievable secret key rate and investigate the performances under different network setting by numerical examples.