Although Wolf Pack Algorithm (WPA) is a novel optimal algorithm with good performance, there is still room for improvement with respect to its convergence. In order to speed up its convergence and strengthen the search ability, we improve WPA with the Differential Evolution (DE) elite set strategy. The new proposed algorithm is called the WPADEES for short. WPADEES is faster than WPA in convergence, and it has a more feasible adaptability for various optimizations. Six standard benchmark functions are applied to verify the effects of these improvements. Our experiments show that the performance of WPADEES is superior to the standard WPA and other intelligence optimal algorithms, such as GA, DE, PSO, and ABC, in several situations.

Electronic payment protocols provide secure service for electronic commerce transactions and protect private information from malicious entities in a network. Formal methods have been introduced to verify the security of electronic payment protocols; however, these methods concentrate on the accountability and fairness of the protocols, without considering the impact caused by timeliness. To make up for this deficiency, we present a formal method to analyze the security properties of electronic payment protocols, namely, accountability, fairness and timeliness. We add a concise time expression to an existing logical reasoning method to represent the event time and extend the time characteristics of the logical inference rules. Then, the Netbill protocol is analyzed with our formal method, and we find that the fairness of the protocol is not satisfied due to the timeliness problem. The results illustrate that our formal method can analyze the key properties of electronic payment protocols. Furthermore, it can be used to verify the time properties of other security protocols.