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In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have strongly unified arithmetic. In some operations (see the last sections), it is close in speed to Edwards curves. ==Definition== Let ''K'' be a field. According to〔(【引用サイトリンク】title=Twisted Hessian curves )〕 twisted Hessian curves were introduced by Bernstein, Lange, and Kohel. The twisted Hessian form in ''affine coordinates'' is given by: and in ''projective coordinates'': where and and ''a'', ''d'' in ''K'' Note that these curves are birationally equivalent to Hessian curves. The Hessian curve is just a special case of Twisted Hessian curve, with a=1. Considering the equation ''a'' · ''x''3 + ''y''3 + 1 = ''d'' · ''x'' · ''y'', note that: if ''a'' has a cube root in ''K'', there exists a unique ''b'' such that ''a'' = ''b''3.Otherwise, it is necessary to consider an extension field of ''K'' (e.g., ''K''(''a''1/3)). Then, since ''b''3 · ''x''3 = ''bx''3, defining ''t'' = ''b'' · ''x'', the following equation is needed (in Hessian form) to do the transformation: . This means that Twisted Hessian curves are birationally equivalent to elliptic curve in Weierstrass form. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Twisted Hessian curves」の詳細全文を読む スポンサード リンク
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