A Relation between Group Order of Elliptic Curve and Extension Degree of Definition Field

Recent innovative public key cryptographic applications such as ID-based cryptography are based on pairing cryptography. They efficiently use some torsion group structures constructed on certain elliptic curves defined over finite fields. For this purpose, this paper shows that a relation between group order of elliptic curve and extension degree of definition field especially from the viewpoint of torsion structure for pairing- based cryptographic use. In detail, it is shown that the order of elliptic curve over r
ⁱ
-th extension field denoted by #E(F
_q
r
ⁱ
) is divisible by r
²ⁱ
and it has the torsion structure denoted by Z
_r
i ⊕ Z
_r
i when the base order of elliptic curve denoted by #E(F
_q
) is divisible by r
ⁱ
and the order of the multiplicative group of the definition field is also divisible by r
ⁱ
, where r denotes the order of one cyclic group in the torsion structure.