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Clarify that Theorem 5.4.2 depends on the parameters of the Jubjub curve.
Signed-off-by: Daira Hopwood <daira@jacaranda.org>
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@ -6235,9 +6235,10 @@ We show that there are no points in $\GroupJ$ with affine-ctEdwards $u$-coordina
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Suppose for a contradiction that $(u, \varv) \in \GroupJ$ for $u = 1$ and some
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Suppose for a contradiction that $(u, \varv) \in \GroupJ$ for $u = 1$ and some
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$\varv \typecolon \GF{\ParamS{r}}$. By writing the curve equation as
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$\varv \typecolon \GF{\ParamS{r}}$. By writing the curve equation as
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$\varv^2 = (1 - \ParamJ{a} \smult u^2) / (1 - \ParamJ{d} \smult u^2)$, and noting that
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$\varv^2 = (1 - \ParamJ{a} \smult u^2) / (1 - \ParamJ{d} \smult u^2)$, and noting that
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$1 - \ParamJ{d} \smult u^2 \neq 0$, we have $\varv^2 = (1 - \ParamJ{a}) / (1 - \ParamJ{d})$.
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$1 - \ParamJ{d} \smult u^2 \neq 0$ because $\ParamJ{d}$ is nonsquare,
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The right-hand-side is a nonsquare in $\GF{\ParamS{r}}$, so there are no solutions for $\varv$
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we have $\varv^2 = (1 - \ParamJ{a}) / (1 - \ParamJ{d})$.
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(contradiction).
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The right-hand-side is a nonsquare in $\GF{\ParamS{r}}$ (for the \jubjubCurve parameters),
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so there are no solutions for $\varv$ (contradiction).
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\end{proof}
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\end{proof}
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} %sapling
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} %sapling
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@ -9999,6 +10000,8 @@ Peter Newell's illustration of the Jubjub bird, from \cite{Carroll1902}.
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\begin{itemize}
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\begin{itemize}
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\item Give a definition for \completeTwistedEdwardsEllipticCurves in \crossref{jubjub}.
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\item Give a definition for \completeTwistedEdwardsEllipticCurves in \crossref{jubjub}.
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\item Clarify that \theoremref{thmnohashtouncommittedsapling} depends on the
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parameters of the \jubjubCurve.
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\item Ensure that this document builds correctly and without missing
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\item Ensure that this document builds correctly and without missing
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characters on recent versions of \TeX Live.
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characters on recent versions of \TeX Live.
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\item Update the \texttt{Makefile} to use Ghostscript for PDF optimization.
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\item Update the \texttt{Makefile} to use Ghostscript for PDF optimization.
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