In this paper, we propose a formal notion of network security for ad hoc networks. We adopt a probabilistic security framework, that is, security is defined by a polynomially bounded adversary model, the cost of attack and the cost of defense. In a complex and probabilistic system, we speak of the “infeasibility” of breaking the security system rather than the “impossibility” of breaking the same system. Security is defined on the concept of “negligible”, which is asymptotically sub-polynomial with respect to a pre-defined system parameter x. Intuitively, the parameter x in cryptography is the key length n. We apply the same bounds in ad hoc network security research, but in regard to scalability from now on. We propose an RP (n-runs) complexity class with a global virtual god oracle (GVG) to model a general class of network protocols. In GVG-RP (n-runs) class, the network scale (i.e., number of network members) N replaces the role of key length n in cryptography. From ...