Some optimization problems in error free perfect secrecy systems

Wireless physical layer security has attracted a considerable recent attention. One of the approaches for generating secret keys among different users is by using some physical properties of wireless networks. Since secret keys are valuable resources, how to efficiently use a key to achieve perfect secrecy is as important as how to generate a key. The flexibility to control the size of cipher-text is also important in wireless networks due to the possible asymmetric channel capacities among different links in networks. Recently, Ho et al. have demonstrated the existence of a fundamental trade-off between the expected key consumption and the size of cipher-text. It is important to find the optimal coding scheme which can achieve the trade- off. With the help of an algorithm proposed in this paper, to design the optimal coding scheme is equivalent to solving an optimization problem. Some optimization problems for security systems satisfying different requirements are formulated. This paper also shows a necessary condition for achieving the minimal initial key requirement and the minimal size of cipher-text.