A secure key-exchange protocol with an absence of injective functions

The security of neural cryptography is investigated. A key-exchange protocol over a public channel is studied where the parties exchanging secret messages use multilayer neural networks which are trained by their mutual output bits and synchronize to a time dependent secret key. The weights of the networks have integer values between $\pm L$. Recently an algorithm for an eavesdropper which could break the key was introduced by Shamir et al. [adi]. We show that the synchronization time increases with $L^2$ while the probability to find a successful attacker decreases exponentially with $L$. Hence for large $L$ we find a secure key-exchange protocol which depends neither on number theory nor on injective trapdoor functions used in conventional cryptography.