The invention belongs to the technical field of FPGA hardware algorithm acceleration, and discloses an algorithm acceleration method and system for large number modular exponentiation, a medium, a device and application. The system comprises a Montgomery algorithm module, an R-L mode modular exponentiation algorithm module, a top layer module, a serial port communication module, a modular exponentiation state control module, a modular multiplication state control module and a modular multiplication operation module. According to the present invention, various operations are completed through various functional modules and the corresponding design schemes, the form of converting an algorithm into modules is solidified into the functional modules on various circuits, and higher stability and compatibility are achieved; by utilizing the parallelism of an FPGA, the large number modular exponentiation algorithm is greatly accelerated; a special low-complexity Montgomery operational circuit is used for real-time processing, so that the characteristics of high density, high reliability and low power consumption are realized; a IP core is packaged, and the algorithm has the very good transportability and high flexibility.