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We generalize Boneh-Rubin-Silverberg method [3] to construct ordinary elliptic curves with embedding degree one, which provides composite order groups for cryptographic protocols based on such bilinear groups. Our construction is more efficient and almost optimal for parameter setting. In addition, we analyze the non-degeneracy of symmetric pairing derived from the reduced Tate pairing on such curves, and prove that its non-degeneracy only relies on the existence of distortion maps. Based on this observation, we propose a new method for computing the reduced Tate pairing on ordinary curves with embedding degree one. Compared with previous methods, our formulae provide faster computation of the reduced Tate pairing on such curves, which also implies that the reduced Tate pairing may be preferred to use as symmetric pairing instead of the modified Weil pairing in certain cases.

Generation and Tate Pairing Computation of Ordinary Elliptic Curves with Embedding Degree One