Correct the statement and proof of Theorem A.3.2.

Signed-off-by: Daira Hopwood <daira@jacaranda.org>
This commit is contained in:
Daira Hopwood 2018-10-01 11:21:02 +01:00
parent 25b64382e4
commit da7c6fe190
1 changed files with 7 additions and 7 deletions

View File

@ -9791,6 +9791,7 @@ Peter Newell's illustration of the Jubjub bird, from \cite{Carroll1902}.
\item Improved cross-referencing in \crossref{concretepedersenhash}.
\item Clarify the notes concerning domain separation of prefixes in
\crossref{saplingmerklecrh} and \crossref{concretesaplingnotecommit}.
\item Correct the statement and proof of \theoremref{thmconversiontomontnoexcept}.
\end{itemize}
} %sapling
\item Add the QED-it report to the acknowledgements.
@ -11250,16 +11251,15 @@ enumerate all exceptional inputs that may violate the side-conditions.
\vspace{1ex}
\begin{theorem} \label{thmconversiontomontnoexcept}
Let $(u, \varv)$ be an affine point on a complete twisted Edwards curve.
Then the only points with $u \neq 0$ or $\varv \neq 0$
are $(0, 1) = \ZeroJ$; $(0, -1)$ of order $2$; and
$\left(\pm\, 1/\!\ssqrt{\ParamJ{a}}, 0\right)$ of order $4$.
Let $(u, \varv)$ be an affine point on a complete twisted Edwards curve $\Edwards{a,d}$.
Then the only points with $u = 0$ or $1 - \varv = 0$ are $(0, 1) = \ZeroJ$, and
$(0, -1)$ of order $2$.
\end{theorem}
\begin{proof}
Straightforward from the curve equation. (The fact that the points
$\left(\pm\, 1/\!\ssqrt{\ParamJ{a}}, 0\right)$ are of order $4$
can be inferred by applying the doubling formula.)
The curve equation is $a \smult u^2 + \varv^2 = 1 + d \smult u^2 \smult \varv^2$
with $a \neq d$ (see \cite[Definition 2.1]{BBJLP2008}). By substituting $u = 0$ we
obtain $\varv = \pm 1$, and by substituting $\varv = 1$ and using $a \neq d$ we obtain $u = 0$.
\end{proof}
\vspace{0.5ex}