A Multiplicative Extension for Discrete Logarithms on Ordinary Pairing-Friendly Curves of Embedding Degree

This paper deals with r-torsion rational points on ordinary pairing-friendly curves such that the embedding degree k divides r + 1, where r is the order of rational points. Especially, its group structure is focused on. In this case, the twisted characteristic polynomial f´(Π̃̂d) becomes irreducible over F_r, where Π̃̂d is the skew Frobenius map with twist degree d such as quadratic, cubic, quartic, and sextic. Then, using the irreducible polynomial f´(Π̃̂d), this paper considers a multiplicative representation of r-torsion rational points.