Bilinear Pairing Computation Based On Several Type Algebraic Curves

In the early days,the bilinear pairing on the algebraic curves was an important tool for analyzing cryptographic algorithms.The main idea was to transform the discrete logarithm problems on algebraic curves into the discrete logarithm problems over finite fields.Because of their high security and powerful practicability,the bilinear pairing can be used to construct the complex,secure and efficient cryptographic protocols or schemes.For example,the Weil pairing can be used to construct the signature scheme,the Tate pairing can be used to construct the multi-party key agreement scheme,and the bilinear pairing can also be used to construct the ID-based encryption scheme by combining with Diffie-Hellman.The most common bilinear pairing is Tate pairing,while the Eta pairing and Ate pairing are the improvements of Tate pairing.Eta pairing and Ate pairing are more efficient than Tate pairing in computationThis thesis mainly studies the bilinear pairing on several type algebraic curves and some specific computation algorithms for the bilinear pairing are also given.Our main results are as follows(1)For a class of elliptic curves y2+y=x3+x+b over field F2,with b?F2,in this thesis,the computation algorithms of the bilinear pairing are given in four cases,including Tate pairing,Eta pairing and improved Eta pairing.The efficiency of these algorithms is analyzed at last,and the results of the analysis show that Eta pairing algorithms are about 30%faster than Tate pairing algorithms.The improved Eta pairing algorithms are about 50%faster than Eta pairing algorithms,and they are about 65%faster than Tate pairing algorithms.(2)For a class of hyperelliptic curves y2=xp-ax-b over field Fpn with a,b? Fp,in this thesis,the computation algorithms of Ate pairing based on these curves are proposed,and then the efficiency about the Ate pairing algorithms and Tate pairing algorithms is analyzed.Our analysis show that Ate pairing algorithms are about 42%,59%,or 72%faster than Tate pairing algorithms when p is 5,7,or 9 respectively.(3)For a class of hyperelliptic curves y2+y=x5+ax+b over field F2n with a,b?F2,the computation algorithms of Tate pairing and Ate pairing based on the curves are proposed.The efficiency of these two algorithms is analyzed,and the analysis shows that the proposed Ate pairing algorithms are about 83%faster than Tate pairing algorithms.(4)Based on discrete logarithm problem and inverse computation problem,a signature scheme using bilinear pairing is proposed in this thesis,the scheme not only can ensure the security of the private key and the unforgeability of the signature,but also can ensure the simplicity and efficiency of the bilinear pairing.