Aiming at the poor real-time performance of mobile robot path planning in complex indoor environment, a further improvement on A ^* algorithm was proposed by analyzing and comparing Dijkstra algorithm, traditional A ^* algorithm and some improved A ^* algorithms. Firstly, the estimated path cost of the current node and its parent node were weighted in exponentially decreasing way. In this way, when the current code was far away from the target, the improved algorithm could search towards to the target quickly instead of searching around the start node. While the current code was near to the target, the algorithm could search the target carefully to ensure that the target was reachable. Secondly, the generated path was smoothed by quintic polynomia to further shorten the path and facilitate robot control. The simulation results show that compared with the traditional A ^* algorithm, the proposed algorithm can reduce the searching time by 93.8% and reduce the path length by 17.6% and get the path without quarter turning point, so that the robot could get to the destination along the planned path without a break. The proposed algorithm is verified in different scenarios, and the results show that the proposed algorithm can adapt to different environments and has good real-time performance.

By making use of the theory of quadratic residues under the condition of strong prime, a method for studying the algebraic structure of Z*φ(n) of RSA (Rivest-Shamir-Adleman) algorithm was established in this work. A formula to determine the order of element in Z*φ(n) and an expression of maximal order were proposed; in addition, the numbers of quadratic residues and non-residues in the group Z*φ(n) were calculated. This work gave an estimate that the upper bound of maximal order was φ(φ(n))/4 and obtained a necessary and sufficient condition on maximal order being equal to φ(φ(n))/4. Furthermore, a sufficient condition for A1 being cyclic group was presented, where A1 was a subgroup composed of all quadratic residues in Z*φ(n), and a method of the decomposition of Z*φ(n) was also established. Finally, it was proved that the group Z*φ(n) could be generated by seven elements of quadratic non-residues and the quotient group Z*φ(n)/A1 was a Klein group of order 8.