Secrecy Capacity Scaling in Large Cooperative Wireless Networks

We investigate <italic>large</italic> wireless networks subject to security constraints. In contrast to point-to-point, interference-limited communications considered in prior works, we propose active cooperative relaying-based schemes. We consider a network with <inline-formula> <tex-math notation="LaTeX">$n_{l}$ </tex-math></inline-formula> legitimate nodes, <inline-formula> <tex-math notation="LaTeX">$n_{e}$ </tex-math></inline-formula> eavesdroppers, and path loss exponent <inline-formula> <tex-math notation="LaTeX">$\alpha \geq 2$ </tex-math></inline-formula>. As long as <inline-formula> <tex-math notation="LaTeX">$n_{e}^{2}\big (\log (n_{e})\big )^{\gamma }=o(n_{l})$ </tex-math></inline-formula>, for some positive <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>, we show that one can obtain unbounded secure aggregate rate. This means zero-cost secure communication, given fixed total power constraint for the entire network. We achieve this result through: 1) the source using Wyner randomized encoder and a <italic>serial (multi-stage)</italic> block Markov scheme, to cooperate with the relays and 2) the relays acting as a virtual multi-antenna to apply beamforming against the eavesdroppers. Our simpler <italic>parallel (two-stage)</italic> relaying scheme can achieve the same unbounded secure aggregate rate when <inline-formula> <tex-math notation="LaTeX">$n_{e}^{\alpha /{2}+1}\big (\log (n_{e})\big )^{\gamma +\delta ({\alpha }/{2}+1)}=o(n_{l})$ </tex-math></inline-formula> holds, for some positive <inline-formula> <tex-math notation="LaTeX">$\gamma ,\delta $ </tex-math></inline-formula>. Finally, we study the improvement (to the detriment of legitimate nodes) that the eavesdroppers achieve in terms of the information leakage rate in a large <italic>cooperative</italic> network in the case of <italic>collusion</italic>. We show that again the zero-cost secure communication is possible, if <inline-formula> <tex-math notation="LaTeX">$n_{e}^{(2+{2}/{\alpha })}\big (\log n_{e}\big )^{\gamma }=o(n_{l})$ </tex-math></inline-formula> holds, for some positive <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>; that is, in the case of collusion slightly fewer eavesdroppers can be tolerated compared with the non-colluding case.