Accelerating cross twisted Ate pairing with ordinary pairing friendly curve of composite order that has two large prime factors

Recently, pairing-based cryptographic applications such as ID-based cryptography have received much attention. On the other hand, RSA cryptography has been widely used and is defined over a certain composite order as the modulus. In detail, it generally needs to be a product of two large prime numbers. In order to apply RSA-based techniques to pairing-based cryptography, the authors have proposed a method for generating ordinary pairing-friendly curves of such a composite order especially when the embedding degree k is equal to 3 and the order of curve is given as a polynomial of degree 2 with an integer variable. Then, as the next problem, its pairing calculation needs to be efficiently carried out. This paper shows the implementation of cross twisted Ate pairing using the obtained composite order curve and some experimental results.