### Abelian -> commutative, homogeneous -> projective

parent dda26b57
 ... ... @@ -25,7 +25,7 @@ Theorem RFC_Correct: forall (n p : list Z) We first review the work of Bartzia and Strub \cite{BartziaS14} (\ref{subsec:ECC-Weierstrass}). We extend it to support Montgomery curves (\ref{subsec:ECC-Montgomery}) with homogeneous coordinates (\ref{subsec:ECC-projective}) and prove the with projective coordinates (\ref{subsec:ECC-projective}) and prove the correctness of the ladder (\ref{subsec:ECC-ladder}). We discuss the twist of Curve25519 (\ref{subsec:Zmodp}) and explain how we deal with it in the proofs (\ref{subsec:curvep2}). ... ... @@ -40,7 +40,7 @@ the green tiles represent major lemmas and theorems. The plan is as follows. We consider the field $\K$ and formalize the Montgomery curves ($M_{a,b}(\K)$). Then, by using the equivalent Weierstra{\ss} form ($E_{a',b'}(\K)$) from the library of Bartzia and Strub, we prove that $M_{a,b}(\K)$ forms an abelian group. we prove that $M_{a,b}(\K)$ forms an commutative group. Under the hypothesis that $a^2 - 4$ is not a square in $\K$, we prove the correctness of the ladder (\tref{thm:montgomery-ladder-correct}). ... ... @@ -96,7 +96,7 @@ Definition oncurve (p : point) := Inductive ec : Type := EC p of oncurve p. \end{lstlisting} Points on an elliptic curve form an abelian group when equipped with the following structure. Points on an elliptic curve form an commutative group when equipped with the following structure. \begin{itemize} \item The negation of a point $P = (x,y)$ is defined by reflection over the $x$-axis, \ie $-P = (x, -y)$. \item The addition of two points $P, Q \in E_{a,b}(\K) \setminus \{\Oinf\}$ with $P \neq Q$ and $P \neq -Q$ ... ... @@ -134,7 +134,7 @@ Definition addec (p1 p2 : ec) : ec := \subsubsection{Montgomery curves} \label{subsec:ECC-Montgomery} Speedups can be obtained by switching to homogeneous coordinates and other forms Speedups can be obtained by switching to projective coordinates and other forms than the Weierstra{\ss} form. We consider the Montgomery form \cite{MontgomerySpeeding}. \begin{dfn} ... ... @@ -222,7 +222,7 @@ After we have verified the group properties, it follows that the bijection is a % Lemma ec_of_mc_bij : bijective ec_of_mc. % \end{lstlisting} \subsubsection{Homogeneous coordinates} \subsubsection{Projective coordinates} \label{subsec:ECC-projective} In a projective plane, points are represented by the triples $(X:Y:Z)$ excluding $(0:0:0)$. ... ... @@ -263,7 +263,7 @@ We define $\chi$ and $\chi_0$ to return the \xcoord of points on a curve. \end{align*} \end{dfn} Using homogeneous coordinates we prove the formula for differential addition.% (\lref{lemma:xADD}). Using projective coordinates we prove the formula for differential addition.% (\lref{lemma:xADD}). \begin{lemma} \label{lemma:xADD} Let $M_{a,b}$ be a Montgomery curve such that $a^2-4$ is not a square in \K, and ... ... @@ -296,7 +296,7 @@ Similarly, we also prove the formula for point doubling.% (\lref{lemma:xDBL}). \end{lemma} With \lref{lemma:xADD} and \lref{lemma:xDBL}, we are able to compute efficiently differential additions and point doublings using homogeneous coordinates. differential additions and point doublings using projective coordinates. \subsubsection{Scalar multiplication algorithms} \label{subsec:ECC-ladder} ... ...
 ... ... @@ -32,7 +32,7 @@ are isomorphic via $(x,y) \mapsto (x, \sqrt{b/b'} \cdot y)$. a curve that is isomorphic over $\F{p^2}$~\cite{cryptoeprint:2017:212}. \end{dfn} Points in $M_{a,b}(\K)$ can be equipped with a structure of an abelian group Points in $M_{a,b}(\K)$ can be equipped with a structure of an commutative group with the addition operation $+$ and with neutral element the point at infinity $\Oinf$. For a point $P \in M_{a,b}(\K)$ and a positive integer $n$ we obtain the scalar product $$n\cdot P = \underbrace{P + \cdots + P}_{n\text{ times}}.$$ ... ...
 ... ... @@ -20,7 +20,7 @@ % M is a finite assoc group \begin{scope}[yshift=0 cm,xshift=3 cm] \draw [fill=green!20] (0,0) -- (3.25,0) -- (3.25,-0.75) -- (0, -0.75) -- cycle; \draw (1.675,-0.375) node[textstyle, anchor=center] {$M_{a,b}(\K)$ is an abelian group}; \draw (1.675,-0.375) node[textstyle, anchor=center] {$M_{a,b}(\K)$ is an commutative group}; \end{scope} % Hypothesis x square is not 2 ... ...
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