In this study, we propose an algorithm for computing the network size of communicating agents. The algorithm is distributed: a) it does not require a leader selection; b) it only requires local exchange of information, and; c) its design can be implemented using local information only, without any global information about the network. It is privacy-preserving, namely it does not require to propagate identifying labels. This algorithm is based on system identification, and more precisely on the identification of the order of a suitably-constructed discrete-time linear time-invariant system over some finite field. We provide a probabilistic guarantee for any randomly picked node to correctly compute the number of nodes in the network. Moreover, numerical implementation has been taken into account to make the algorithm applicable to networks of hundreds of nodes, and therefore make the algorithm applicable in real-world sensor or robotic networks. We finally illustrate our results in simulation and conclude the paper with discussions on how our technique differs from a previously-known strategy based on statistical inference.