Abstract: The purpose of this article is to realize the 457 bits final modular exponentiation in the Tate pairing Miller algorithm (one kind of bilinear pairings) of binary extension field on the hardware platform which only supports 250 bits modular multiplication. In the procedure of calculation, we adopted an improved Montgomery modular multiplication algorithm and Chinese remainder theorem algorithm. The final modular exponentiation will be realized with definite data, and we will use the mathematical software Sage to verify the improved scheme. So we can prove that this scheme can realize the 457 bits final modular exponentiation through theory analysis and data calculation.

Key words: Montgomery algorithm, Miller algorithm, modular exponentiation, Chinese remainder theorem, modular multiplication