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The modular exponentiation is one of the most important operations in public-key cryptosystems. A modular exponentiation is composed of repetition of modular multiplications. Again, the modular multiplication consists of a multiprecision multiplication and a modular reduction. Therefore, we can reduce the execution time of the modular exponentiation by two methods. One is to reduce the number of modular multiplications and the other is to reduce the execution time of the modular multiplication. In this paper we investigate high-speed mathematical algorithms that focus on the multiplication, reduction and exponentiation operations to reduce the execution time of the modular exponentiation.