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This study deals with opportunistic relay selection in cooperative networks with secrecy constraints, where an eavesdropper node tries to overhead the source message. Previously reported relay selection techniques are optimised for non-eavesdropper environments and cannot ensure security. Two new opportunistic relay selection techniques, which incorporate the quality of the relay-eavesdropper links and take into account secrecy rate issues, are investigated. The first scheme assumes an instantaneous knowledge of the eavesdropper channels and maximises the achievable secrecy rate. The second one assumes an average knowledge of the eavesdropper channels and is a suboptimal selection solution appropriate for practical applications. Both schemes are analysed in terms of secrecy outage probability and their enhancements against conventional opportunistic selection policies are validated via numerical and theoretical results.