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(Fq^(r^i) ) is divisible by r^(2i) and it has the torsion structure denoted by Zri＋Zri when the base order of elliptic curve denoted by #E(Fq) 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.
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