Some small changes to chapter V.

parent 7d8e02a7
 ... ... @@ -35,8 +35,10 @@ We consider elliptic curves over a field $\K$. We assume that the characteristic of $\K$ is neither 2 or 3. \begin{dfn} Let a field $\K$, using an appropriate choice of coordinates, an elliptic curve $E$ is a plane cubic algebraic curve $E(x,y)$ defined by an equation of the form: Given a field $\K$, using an appropriate choice of coordinates, an elliptic curve $E$ is a plane cubic algebraic curve defined by an equation $E(x,y)$ of the form: $$E : y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$$ where the $a_i$'s are in \K\ and the curve has no singular point (\ie no cusps or self-intersections). The set of points, written $E(\K)$, is formed by the ... ... @@ -53,7 +55,7 @@ This equation $E(x,y)$ can be reduced into its short Weierstra{\ss} form. Let $a \in \K$, and $b \in \K$ such that $$\Delta(a,b) = -16(4a^3 + 27b^2) \neq 0.$$ The \textit{elliptic curve} $E_{a,b}(\K)$ is the set of all points $(x,y) \in \K^2$ satisfying the equation: $$y^2 = x^3 + ax + b,$$ along with an additional formal point $\Oinf$, at infinity''. Such a curve does not present any singularity. along with an additional formal point $\Oinf$, at infinity''. Such a curve does not have any singularity. \end{dfn} In this setting, Bartzia and Strub defined the parametric type \texttt{ec} which ... ... @@ -75,14 +77,14 @@ Definition oncurve (p : point) := Inductive ec : Type := EC p of oncurve p. \end{lstlisting} Points of an elliptic curve are equipped with a structure of an abelian group. Points of an elliptic curve are equipped with the structure of an abelian group. \begin{itemize} \item The negation of a point $P = (x,y)$ by taking the symmetric with respect to the x axis $-P = (x, -y)$. \item The addition of two points $P$ and $Q$ is defined by the negation of the third intersection of the line passing by $P$ and $Q$ or tangent to $P$ if $P = Q$. \item The negation of a point $P = (x,y)$ is defined by reflecting in the $x$ axis $-P = (x, -y)$. \item The addition of two points $P$ and $Q$ is defined as the negation of the third intersection point of the line passing through $P$ and $Q$ or tangent to $P$ if $P = Q$. \item $\Oinf$ is the neutral element under this law: if 3 points are collinear, their sum is equal to $\Oinf$. \end{itemize} These operations are defined in Coq as follow: These operations are defined in Coq as follows (where we omit the code for the tangent case): \begin{lstlisting}[language=Coq] Definition neg (p : point) := if p is (| x, y |) then (| x, -y |) else EC_Inf. ... ... @@ -98,7 +100,7 @@ Definition add (p1 p2 : point) := (| xs, - s * (xs - x1 ) - y1 |) end. \end{lstlisting} And are proven internal to the curve (with coercion): The value of \texttt{add} is proven to be on the curve (with coercion): \begin{lstlisting}[language=Coq] Lemma addO (p q : ec): oncurve (add p q). ... ... @@ -119,8 +121,9 @@ than the Weierstra{\ss} form. We consider the Montgomery form \cite{MontgomerySp $$by^2 = x^3 + ax^2 + x,$$ along with an additional formal point $\Oinf$, at infinity''. \end{dfn} Using a similar representation, we defined the parametric type \texttt{mc} which represents the points on a specific Montgomery curve. It is parameterized by Similar to the definition of \texttt{ec}, we defined the parametric type \texttt{mc} which represents the points on a specific Montgomery curve. It is parameterized by a \texttt{K : ecuFieldType} -- the type of fields which characteristic is not 2 or 3 -- and \texttt{M : mcuType} -- a record that packs the curve parameters $a$ and $b$ along with the proofs that $b \neq 0$ and $a^2 \neq 4$. ... ... @@ -135,8 +138,8 @@ Inductive mc : Type := MC p of oncurve p. Lemma oncurve_mc: forall p : mc, oncurve p. \end{lstlisting} We define the addition on Montgomery curves the same way as it is in the Weierstra{\ss} form, however the actual computations will be slightly different. We define the addition on Montgomery curves in the similar way as in the Weierstra{\ss} form. %, however the actual computations will be slightly different. \begin{lstlisting}[language=Coq] Definition add (p1 p2 : point K) := match p1, p2 with ... ... @@ -152,14 +155,16 @@ Definition add (p1 p2 : point K) := (| xs, - s * (xs - x1) - y1 |) end. \end{lstlisting} But we prove it is internal to the curve (again with coercion): And again we prove the result is on the curve (again with coercion): \begin{lstlisting}[language=Coq] Lemma addO (p q : mc) : oncurve (add p q). Definition addmc (p1 p2 : mc) : mc := MC p1 p2 (addO p1 p2) \end{lstlisting} We then prove a bijection between a Montgomery curve and its Weierstra{\ss} equation. We then define a bijection between a Montgomery curve and its Weierstra{\ss} form. In this way we get associativity of addition on Montgomery curves from the corresponding property for Weierstra{\ss} curves. \begin{lemma} Let $M_{a,b}(\K)$ be a Montgomery curve, define $$a' = \frac{3-a^2}{3b^2} \text{\ \ \ \ and\ \ \ \ } b' = \frac{2a^3 - 9a}{27b^3}.$$ ... ... @@ -193,12 +198,12 @@ Lemma ec_of_mc_bij : bijective ec_of_mc. \label{subsec:ECC-projective} On a projective plane, points are represented with a triple $(X:Y:Z)$. With the exception of $(0:0:0)$, any points can be projected. With the exception of $(0:0:0)$, any point can be projected. Scalar multiples are representing the same point, \ie for all $\lambda \neq 0$, $(X:Y:Z)$ are $(\lambda X:\lambda Y:\lambda Z)$ defining for all $\lambda \neq 0$, the triples $(X:Y:Z)$ and $(\lambda X:\lambda Y:\lambda Z)$ represent the same point. For $Z\neq 0$, the projective point $(X:Y:Z)$ corresponds to the point $(X/Z,Y/Z)$ on the Euclidean plane, likewise the point $(X,Y)$ on the point $(X/Z,Y/Z)$ on the Euclidean plane. Likewise the point $(X,Y)$ on the Euclidean plane corresponds to $(X:Y:1)$ on the projective plane. Using fractions as coordinates, the equation for a Montgomery curve $M_{a,b}(\K)$ ... ...
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