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Non-degenerate bilinear pairings have played a key role in public-key cryptography since they have been used to construct identity-based encryption schemes and one-round three-way key exchange protocols. Because the performance of pairing-based cryptosystems relies heavily on the efficiency of pairing computation, the development of efficient pairings has been an important mathematical issue in cryptographic research areas. The desired pairings are obtained from the Weil and Tate pairings defined on the rational points on elliptic curves over finite fields. Especially the most widely used pairing is the Tate pairing.
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