Clarify that Theorem 5.4.2 depends on the parameters of the Jubjub curve.

Signed-off-by: Daira Hopwood <daira@jacaranda.org>
This commit is contained in:
Daira Hopwood 2019-06-18 22:58:14 +01:00
parent f4f4682d57
commit 847a002eff
1 changed files with 6 additions and 3 deletions

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