Constructing elliptic curves with given group order over large finite fields

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Abstract

A procedure is developed for constructing elliptic curves with given group order over large finite fields. The generality of the construction allows an arbitrary choice of the parameters involved. For instance, it is possible to specify the finite field, the group order or the class number of the endomorphism ring of the elliptic curve. This is important for various applications in computational number theory and cryptography. Moreover, we give a method that yields all representations of a given integer as a norm in an imaginary quadratic field.