We describe a family of log n protocols for assigning symmetric keys to n processes in a network so that each process can use its assigned keys to communicate securely with every other process. The k-th protocol in our protocol family, where 1 les k les log n, assigns O(k2 kradicn) symmetric keys to each process in the network. (Thus, our (log n)-th protocol assigns O(log2 n) symmetric keys to each process. This is not far from the lower bound of O(log n) symmetric keys which we show is needed for each process to communicate securely with every other process in the network.) The protocols in our protocol family can be used to assign symmetric keys to the processes in a sensor network, or ad-hoc or mobile network, where each process has a small memory to store its assigned keys. We also discuss the vulnerability of our protocols to "collusion". In particular, we show that kradicn colluding processes can compromise the security of the k-th protocol in our protocol family.