( ) 1 1985 Koblitz Miller 2 2.1 $p>3$ $\mathbb{F}_{p}$ $E$ : $y^{2}=x^{3}+ax+b$ $(a, b\in \mathbb{F}_{p})$ ( $O$ ) : (i) $P+O=O+P=P.$ (ii) $P=(x, y)$ $-P=(x, -y)$. (iii) $P_{1}=(x_{1}, y_{1}),$ $P_{2}=(x_{2}, y_{2})$ $P_{1}+P_{2}=(x_{3}, y_{3})$ $P_{1}\neq\pm P_{2}$ $\{\begin{array}{l}x_{3}=y_{3}=\end{array}\}x_{2}-x_{1}x2-x_{1\{\begin{array}{l}2-x_{l}-x_{2}(x_{1}-x_{3})-y_{1}\end{array}}$ (iv) $P