Data rate trading, in which mobile devices trade their real-time data transmission rates to achieve cooperative mobile networks access, not only can meet the increasing data access demands of users but also can reduce the pressure on cellular networks. However, there is no directly available mechanism for data rate trading. In this paper, we propose a truthful online auction mechanism for data rate trading in mobile networks. In the designed auction, the data rate buyers submit their realtime data access requests, including the rate requirement, access time and payment. The auctioneer, which may be the network operator, assigns data rate requests to appropriate sellers who leverage their surplus cellular data plan or other networks to complete the data rate requests and benefit from them. To achieve this model, we first formulate the social welfare maximization problem in data rate trading as an integer linear programming and show its NP-hardness. Then, we resort to the Lagrangian relaxation technique to design an online approximation algorithm to assign data rate requests and compute the corresponding payments in polynomial time. Theoretical analysis and simulation experiments show that the proposed auction mechanism obtains a good competitive ratio and satisfies the desired properties, including individual rationality, truthfulness, and computational efficiency.

Maximum distance separable (MDS) matrices are widely used in the design of block ciphers. However, it is highly nontrival to find MDS matrices which could be used in practice. This paper focuses on the design of efficient MDS matrices for substitution-permutation networks (SPNs). We provide a new method to construct and count these MDS matrices. Moreover, we identified an interesting class of Cauchy matrices (named compact Cauchy matrices) which has the fewest different entries and is thus more favorable for implementation. Finally, we prove that all compact Cauchy matrices could be modified into an involution compact Cauchy matrix, and show how to maximize the occurrences of entry “1” in a compact Cauchy matrix.