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\begin{document}
\headers{Math373}{Homework 12}{Due Wed, Apr 19, 2023}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item Let $E$ be the elliptic curve given by $y^2 = x^3 + 17$ over the real numbers.  Let $P=(-1,4)$ and $Q=(2,5)$.
\begin{enumerate}
	\item Compute $PQ$ and $\frac{P}{Q}$.  Hint: What does it mean to divide by a point in an elliptic curve group?
	\item Compute $P^2$ and $Q^2$.
\end{enumerate}

\item Let $E$ be the elliptic curve given by $y^2 = x^3 + 5x + 1$ over $\F_{19}$.  Compute the following.
\begin{enumerate}
    \item $(4,3)\O$.
    \item $(4,3)^{-1}$.
    \item $(4,3)(10,-5)$.
    \item $(4,3)^2$.
    \item $(4,3)^4$.
    \item $(4,3)^8$.
\end{enumerate}

\item Let $E$ be the elliptic curve given by $y^2 = x^3 + 5x + 1$ over $\F_{19}$ and let $g=(4,3)$.  Find the following.
\begin{enumerate}
    \item $\log_g(\O)$
    \item $\log_g((4,-3))$
    \item $\log_g((11,0))$
\end{enumerate}

\end{enumerate}
\end{document}
