\section{Proving that X25519 in Coq matches the mathematical model} \label{sec:maths} In this section we prove the following informal theorem: \begin{informaltheorem} The implementation of X25519 in TweetNaCl computes the $\F{p}$-restricted \xcoord scalar multiplication on $E(\F{p^2})$ where $p$ is $\p$ and $E$ is the elliptic curve $y^2 = x^3 + 486662 x^2 + x$. \end{informaltheorem} More precisely, we prove that our formalization of the RFC matches the definitions of Curve25519 by Bernstein: \begin{lstlisting}[language=Coq] Theorem RFC_Correct: forall (n p : list Z) (P:mc curve25519_Fp2_mcuType), Zlength n = 32 -> Zlength p = 32 -> Forall (fun x => 0 <= x /\ x < 2 ^ 8) n -> Forall (fun x => 0 <= x /\ x < 2 ^ 8) p -> Fp2_x (decodeUCoordinate p) = P#x0 -> RFC n p = encodeUCoordinate ((P *+ (Z.to_nat (decodeScalar25519 n))) _x0). \end{lstlisting} 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 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}). \subsection{Formalization of elliptic curves} \label{subsec:ECC} \fref{tikz:ProofHighLevel1} presents the structure of the proof of the ladder's correctness. The white tiles are definitions, the orange ones are hypothesis and the green tiles represent major lemmas and theorems. % The plan is as follows. % (This is part of the description of the picture). 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 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}). \begin{figure}[h] \centering \include{tikz/highlevel1} \caption{Overview of the proof of Montgomery ladder's correctness} \label{tikz:ProofHighLevel1} \end{figure} % this is for the flow of the text otherwise someone will again complain of a definition poping out of nowhere. We now turn our attention to the details of the proof of the ladder's correctness. \begin{dfn} 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 defined over \K, written $E(\K)$, is formed by the solutions $(x,y)$ of $E$ together with a distinguished point $\Oinf$ called point at infinity: $$E(\K) = \{ (x,y) \in \K \times \K ~|~E(x,y)\} \cup \{\Oinf\}$$ \end{dfn} \subsubsection{Short Weierstra{\ss} curves} \label{subsec:ECC-Weierstrass} For the remainder of this text, we assume that the characteristic of $\K$ is neither 2 nor 3. Then, this equation $E(x,y)$ can be reduced into its short Weierstra{\ss} form. \begin{dfn} Let $a \in \K$ and $b \in \K$ such that $$\Delta(a,b) = -16(4a^3 + 27b^2) \neq 0.$$ The \textit{elliptic curve} $E_{a,b}$ is defined by the equation $$y^2 = x^3 + ax + b.$$ $E_{a,b}(\K)$ is the set of all points $(x,y) \in \K^2$ satisfying the $E_{a,b}$ 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 represents the points on a specific curve. It is parameterized by a \texttt{K : ecuFieldType} ---the type of fields whose characteristic is neither 2 nor 3--- and \texttt{E : ecuType} ---a record that packs the curve parameters $a$ and $b$--- along with the proof that $\Delta(a,b) \neq 0$. \begin{lstlisting}[language=Coq] Inductive point := EC_Inf | EC_In of K * K. Notation "(| x, y |)" := (EC_In x y). Notation "\infty" := (EC_Inf). Record ecuType := { A : K; B : K; _ : 4 * A^3 + 27 * B^2 != 0}. Definition oncurve (p : point) := if p is (| x, y |) then y^2 == x^3 + A * x + B else true. 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.% \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 $P$ 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 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. Definition add (p1 p2 : point) := match p1, p2 with | \infty , _ => p2 | _ , \infty => p1 | (| x1, y1 |), (| x2, y2 |) => if x1 == x2 then ... else let s := (y2 - y1) / (x2 - x1) in let xs := s^2 - x1 - x2 in (| xs, - s * (xs - x1 ) - y1 |) end. \end{lstlisting} 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). Definition addec (p1 p2 : ec) : ec := EC p1 p2 (addO p1 p2) \end{lstlisting} \subsubsection{Montgomery curves} \label{subsec:ECC-Montgomery} 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} Let $a \in \K \backslash \{-2, 2\}$, and $b \in \K \backslash \{ 0\}$. The \textit{elliptic curve} $M_{a,b}$ is defined by the equation: $$by^2 = x^3 + ax^2 + x,$$ $M_{a,b}(\K)$ is the set of all points $(x,y) \in \K^2$ satisfying the $M_{a,b}$ along with an additional formal point $\Oinf$, at infinity''. \end{dfn} Similar to the definition of \texttt{ec}, we define 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 whose characteristic is neither 2 nor 3--- and \texttt{M : mcuType} ---a record that packs the curve parameters $a$ and $b$ along with the proofs that $b \neq 0$ and $a^2 \neq 4$. \begin{lstlisting}[language=Coq,belowskip=-0.1 \baselineskip] Record mcuType := { cA : K; cB : K; _ : cB != 0; _ : cA^2 != 4}. Definition oncurve (p : point K) := if p is (| x, y |) then cB * y^+2 == x^+3 + cA * x^+2 + x else true. 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 in a similar way as for the Weierstra{\ss} form. \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] Definition add (p1 p2 : point K) := match p1, p2 with | \infty, _ => p2 | _, \infty => p1 | (|x1, y1|), (|x2, y2|) => if x1 == x2 then if (y1 == y2) && (y1 != 0) then ... else \infty else let s := (y2 - y1) / (x2 - x1) in let xs := s^+2 * cB - cA - x1 - x2 in (| xs, - s * (xs - x1) - y1 |) end. \end{lstlisting} And again we prove the result is on the curve: \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 define a bijection between a Montgomery curve and its short Weierstra{\ss} form (\lref{lemma:bij-ecc}) and prove that it respects the addition as defined on the respective curves. In this way we get all the group laws for Montgomery curves from the Weierstra{\ss} ones. After having verified the group properties, it follows that the bijection is a group isomorphism. \begin{lemma} \label{lemma:bij-ecc} Let $M_{a,b}$ be a Montgomery curve, define %\vspace{-0.3em} $$a' = \frac{3-a^2}{3b^2} \text{\ \ \ \ and\ \ \ \ } b' = \frac{2a^3 - 9a}{27b^3}.$$ then $E_{a',b'}$ is a Weierstra{\ss} curve, and the mapping $\varphi : M_{a,b} \mapsto E_{a',b'}$ defined as: %\vspace{-0.5em} \begin{align*} \varphi(\Oinf_M) & = \Oinf_E \\ \varphi( (x , y) ) & = \left( \frac{x}{b} + \frac{a}{3b} , \frac{y}{b} \right) \end{align*} is a group isomorphism between elliptic curves. \end{lemma} % \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] % Definition ec_of_mc_point p := % match p with % | \infty => \infty % | (|x, y|) => (|x/(M#b) + (M#a)/(3%:R * (M#b)), y/(M#b)|) % end. % Lemma ec_of_mc_point_ok p : % oncurve M p -> % ec.oncurve E (ec_of_mc_point p). % Definition ec_of_mc p := % EC (ec_of_mc_point_ok (oncurve_mc p)). % Lemma ec_of_mc_bij : bijective ec_of_mc. % \end{lstlisting} \subsubsection{Projective coordinates} \label{subsec:ECC-projective} In a projective plane, points are represented by the triples $(X:Y:Z)$ excluding $(0:0:0)$. Scalar multiples of triples are identified with each other, \ie for all $\lambda \neq 0$, the triples $(X:Y:Z)$ and $(\lambda X:\lambda Y:\lambda Z)$ represent the same point in the projective plane. For $Z\neq 0$, the point $(X:Y:Z)$ corresponds to the point $(X/Z,Y/Z)$ in the affine plane. Likewise, the point $(X,Y)$ in the affine plane corresponds to $(X:Y:1)$ in the projective plane. % The points $(X : Y : 0)$ can be considered as points at infinity. Using fractions as coordinates, the equation for a Montgomery curve $M_{a,b}$ becomes $$b \bigg(\frac{Y}{Z}\bigg)^2 = \bigg(\frac{X}{Z}\bigg)^3 + a \bigg(\frac{X}{Z}\bigg)^2 + \bigg(\frac{X}{Z}\bigg).$$ Multiplying both sides by $Z^3$ yields $$b Y^2Z = X^3 + a X^2Z + XZ^2.$$ Setting $Z = 0$ in this equation, we derive $X = 0$. Hence, $(0 : 1 : 0)$ is the unique point on the curve at infinity. By restricting the parameter $a$ of $M_{a,b}(\K)$ such that $a^2-4$ is not a square in \K (Hypothesis \ref{hyp:a_minus_4_not_square}), we ensure that $(0,0)$ is the only point with a $y$-coordinate of $0$. \begin{hypothesis} \label{hyp:a_minus_4_not_square} The number $a^2-4$ is not a square in \K. \end{hypothesis} \begin{lstlisting}[language=Coq] Hypothesis mcu_no_square : forall x : K, x^+2 != (M#a)^+2 - 4%:R. \end{lstlisting} We define $\chi$ and $\chi_0$ to return the \xcoord of points on a curve. \begin{dfn} Let $\chi : M_{a,b}(\K) \mapsto \K \cup \{\infty\}$ and $\chi_0 : M_{a,b}(\K) \mapsto \K$ such that %\vspace{-0.5em} \begin{align*} \chi((x,y)) & = x, & \chi(\Oinf) & = \infty, & & \text{and} \\[-0.5ex] \chi_0((x,y)) & = x, & \chi_0(\Oinf) & = 0. \end{align*} \end{dfn} 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 let $X_1, Z_1, X_2, Z_2, X_4, Z_4 \in \K$, such that $(X_1,Z_1) \neq (0,0)$, $(X_2,Z_2) \neq (0,0)$, $X_4 \neq 0$ and $Z_4 \neq 0$. Define %\vspace{-0.5em} \begin{align*} X_3 & = Z_4((X_1 - Z_1)(X_2+Z_2) + (X_1+Z_1)(X_2-Z_2))^2 \\[-0.5ex] Z_3 & = X_4((X_1 - Z_1)(X_2+Z_2) - (X_1+Z_1)(X_2-Z_2))^2 \end{align*} then for any point $P_1$ and $P_2$ in $M_{a,b}(\K)$ such that $X_1/Z_1 = \chi(P_1), X_2/Z_2 = \chi(P_2)$, and $X_4/Z_4 = \chi(P_1 - P_2)$, we have $X_3/Z_3 = \chi(P_1+P_2)$.\\ \textbf{Remark:} These definitions should be understood in $\K \cup \{\infty\}$. If $x\ne 0$ then we define $x/0 = \infty$. \end{lemma} Similarly, we also prove the formula for point doubling.% (\lref{lemma:xDBL}). \begin{lemma} \label{lemma:xDBL} Let $M_{a,b}$ be a Montgomery curve such that $a^2-4$ is not a square in \K, and let $X_1, Z_1 \in \K$, such that $(X_1,Z_1) \neq (0,0)$. Define \begin{align*} c & = (X_1 + Z_1)^2 - (X_1 - Z_1)^2 \\ X_3 & = (X_1 + Z_1)^2(X_1-Z_1)^2 \\ Z_3 & = c\Big((X_1 + Z_1)^2+\frac{a-2}{4}\times c\Big), \end{align*} then for any point $P_1$ in $M_{a,b}(\K)$ such that $X_1/Z_1 = \chi(P_1)$, we have $X_3/Z_3 = \chi(2P_1)$. \end{lemma} With \lref{lemma:xADD} and \lref{lemma:xDBL}, we are able to compute efficiently differential additions and point doublings using projective coordinates. \subsubsection{Scalar multiplication algorithms} \label{subsec:ECC-ladder} By taking \aref{alg:montgomery-ladder} and replacing \texttt{xDBL\&ADD} by a combination of the formulae from \lref{lemma:xADD} and \lref{lemma:xDBL}, we define a ladder \coqe{opt_montgomery} (in which $\K$ has not been fixed yet). % similar to the one used in TweetNaCl (with \coqe{montgomery_rec}). % shown above. % We prove its correctness for any point whose \xcoord is not 0. % % \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] % Lemma opt_montgomery_x : % forall (n m : nat) (x : K), % n < 2^m -> x != 0 -> % forall (p : mc M), p#x0 = x -> % opt_montgomery n m x = (p *+ n)#x0. % \end{lstlisting} % We can remark that for an input $x = 0$, the ladder returns $0$. % \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] % Lemma opt_montgomery_0: % forall (n m : nat), opt_montgomery n m 0 = 0. % \end{lstlisting} % Also \Oinf\ is the neutral element of $M_{a,b}(\K)$. % \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] % Lemma p_x0_0_eq_0 : forall (n : nat) (p : mc M), % p #x0 = 0%:R -> (p *+ n) #x0 = 0%R. % \end{lstlisting} % This gives us the theorem of the correctness of the Montgomery ladder. This gives us the theorem of the correctness of the Montgomery ladder. \begin{theorem} \label{thm:montgomery-ladder-correct} For all $n, m \in \N$, $x \in \K$, $P \in M_{a,b}(\K)$, if $\chi_0(P) = x$ then \coqe{opt_montgomery} returns $\chi_0(n \cdot P)$ \end{theorem} \begin{lstlisting}[language=Coq] Theorem opt_montgomery_ok (n m: nat) (x : K) : n < 2^m -> forall (p : mc M), p#x0 = x -> opt_montgomery n m x = (p *+ n)#x0. \end{lstlisting} The definition of \coqe{opt_montgomery} is similar to \coqe{montgomery_rec_swap} that was used in \coqe{RFC}. We proved their equivalence, and used it in our final proof of \coqe{Theorem RFC_Correct}. \subsection{Curves, twists and extension fields} \label{subsec:curve_twist_fields} \fref{tikz:ProofHighLevel2} gives a high-level view of the proofs presented here. The white tiles are definitions while green tiles are important lemmas and theorems. \begin{figure}[h] \centering \include{tikz/highlevel2} \caption{Proof dependencies for the correctness of X25519.} \label{tikz:ProofHighLevel2} \end{figure} A brief overview of the complete proof is described below. We first pose $a = 486662$, $b = 1$, $b' = 2$, $p = 2^{255}-19$, with the equations $C = M_{a,b}$, and $T = M_{a,b'}$. We prove the primality of $p$ and define the field $\F{p}$. Subsequently, we show that neither $2$ nor $a^2 - 2$ are square in $\F{p}$. We consider $\F{p^2}$ and define $C(\F{p})$, $T(\F{p})$, and $C(\F{p^2})$. We prove that for all $x \in \F{p}$ there exists a point of \xcoord $x$ either on $C(\F{p})$ or on the quadratic twist $T(\F{p})$. We prove that all points in $M(\F{p})$ and $T(\F{p})$ can be projected in $M(\F{p^2})$ and derive that computations done in $M(\F{p})$ and $T(\F{p})$ yield to the same results if projected to $M(\F{p^2})$. Using \tref{thm:montgomery-ladder-correct} we prove that the ladder is correct for $M(\F{p})$ and $T(\F{p})$; with the previous results, this results in the correctness of the ladder for $M(\F{p^2})$, in other words the correctness of X25519. Now that we have an red line for the proof, we turn our attention to the details. Indeed \tref{thm:montgomery-ladder-correct} cannot be applied directly to prove that X25519 is doing the computations over $M(\F{p^2})$. This would infer that $\K = \F{p^2}$ and we would need to satisfy hypothesis~\ref{hyp:a_minus_4_not_square}:% % $a^2-4$ is not a square in \K: $$\forall x \in \K,\ x^2 \neq a^2-4.$$ which is not possible as there always exists $x \in \F{p^2}$ such that $x^2 = a^2-4$. Consequently, we first study Curve25519 and one of its quadratic twists Twist25519, both defined over \F{p}. \subsubsection{Curves and twists} \label{subsec:Zmodp} We define $\F{p}$ as the numbers between $0$ and $p = \p$. We create a \coqe{Zmodp} module to encapsulate those definitions. \begin{lstlisting}[language=Coq] Module Zmodp. Definition betweenb x y z := (x <=? z) && (z 0. Inductive type := Zmodp x of betweenb 0 p x. \end{lstlisting} We define the basic operations ($+, -, \times$) with their respective neutral elements ($0, 1$) and prove \lref{lemma:Zmodp_field}. \begin{lemma} \label{lemma:Zmodp_field} $\F{p}$ is a field. \end{lemma} For $a = 486662$, by using the Legendre symbol we prove that $a^2 - 4$ and $2$ are not squares in $\F{p}$. \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] Fact a_not_square : forall x: Zmodp.type, x^+2 != (Zmodp.pi 486662)^+2 - 4%:R. \end{lstlisting} \begin{lstlisting}[language=Coq,label=two_not_square,belowskip=-0.1 \baselineskip] Fact two_not_square : forall x: Zmodp.type, x^+2 != 2%:R. \end{lstlisting} We now consider $M_{486662,1}(\F{p})$ and $M_{486662,2}(\F{p})$, one of its quadratic twists. % \begin{dfn}Let the following instantiations of \aref{alg:montgomery-double-add}:\\ \begin{dfn} %Let the following instantiations of \aref{alg:montgomery-ladder}:\\ We instantiate \coqe{opt_montgomery} in two specific ways: \begin{itemize} \item[--] $Curve25519\_Fp(n,x)$ for $M_{486662,1}(\F{p})$. \item[--] $Twist25519\_Fp(n,x)$ for $M_{486662,2}(\F{p})$. \end{itemize} \end{dfn} With \tref{thm:montgomery-ladder-correct} we derive the following two lemmas: \begin{lemma} For all $x \in \F{p},\ n \in \N,\ P \in \F{p} \times \F{p}$,\\ such that $P \in M_{486662,1}(\F{p})$ and $\chi_0(P) = x$. $$Curve25519\_Fp(n,x) = \chi_0(n \cdot P)$$ \end{lemma} \begin{lemma} For all $x \in \F{p},\ n \in \N,\ P \in \F{p} \times \F{p}$\\ such that $P \in M_{486662,2}(\F{p})$ and $\chi_0(P) = x$. $$Twist25519\_Fp(n,x) = \chi_0(n \cdot P)$$ \end{lemma} As the Montgomery ladder does not depend on $b$, it is trivial to see that the computations done for points in $M_{486662,1}(\F{p})$ and in $M_{486662,2}(\F{p})$ are the same. \begin{lstlisting}[language=Coq] Theorem curve_twist_eq: forall n x, curve25519_Fp_ladder n x = twist25519_Fp_ladder n x. \end{lstlisting} Because $2$ is not a square in $\F{p}$, it allows us split $\F{p}$ into two sets. \begin{lemma} \label{lemma:square-or-2square} For all $x$ in $\F{p}$, there exists $y$ in $\F{p}$ such that $$y^2 = x\ \ \ \lor\ \ 2y^2 = x$$ \end{lemma} For all $x \in \F{p}$, we can compute $x^3 + ax^2 + x$. Using \lref{lemma:square-or-2square} we can find a $y$ such that $(x,y)$ is either on the curve or on the quadratic twist: \begin{lemma} \label{lemma:curve-or-twist} For all $x \in \F{p}$, there exists a point $P$ in $M_{486662,1}(\F{p})$ or in $M_{486662,2}(\F{p})$ such that the \xcoord of $P$ is $x$. \end{lemma} \begin{lstlisting}[language=Coq,belowskip=-0.5 \baselineskip] Theorem x_is_on_curve_or_twist: forall x : Zmodp.type, (exists (p : mc curve25519_mcuType), p#x0 = x) \/ (exists (p' : mc twist25519_mcuType), p'#x0 = x). \end{lstlisting} \subsubsection{Curve25519 over \F{p^2}} \label{subsec:curvep2} The quadratic extension $\F{p^2}$ is defined as $\F{p}[\sqrt{2}]$ by~\cite{Ber06}. The theory of finite fields already has been formalized in the Mathematical Components library, %ref? but this formalization is rather abstract, and we need concrete representations of field elements here. For this reason we decided to formalize a definition of $\F{p^2}$ ourselves. We can represent $\F{p^2}$ as the set $\F{p} \times \F{p}$, % in other words, representing polynomials with coefficients in $\F{p}$ modulo $X^2 - 2$. In a similar way as for $\F{p}$ we use a module in Coq. \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] Module Zmodp2. Inductive type := Zmodp2 (x: Zmodp.type) (y:Zmodp.type). Definition pi (x: Zmodp.type * Zmodp.type) : type := Zmodp2 x.1 x.2. Definition mul (x y: type) : type := pi ((x.1 * y.1) + (2%:R * (x.2 * y.2)), (x.1 * y.2) + (x.2 * y.1)). \end{lstlisting} % Definition zero : type := % pi ( 0%:R, 0%:R ). % Definition one : type := % pi ( 1, 0%:R ). % Definition opp (x: type) : type := % pi (- x.1 , - x.2). % Definition add (x y: type) : type := % pi (x.1 + y.1, x.2 + y.2). % Definition sub (x y: type) : type := % pi (x.1 - y.1, x.2 - y.2). We define the basic operations ($+, -, \times$) with their respective neutral elements $0$ and $1$. Additionally we verify that for each element of in $\F{p^2}\backslash\{0\}$, there exists a multiplicative inverse. \begin{lemma} \label{lemma:Zmodp2_inv} For all $x \in \F{p^2}\backslash\{0\}$ and $a,b \in \F{p}$ such that $x = (a,b)$, $$x^{-1} = \Big(\frac{a}{a^2-2b^2}\ , \frac{-b}{a^2-2b^2}\Big)$$ \end{lemma} As in $\F{p}$, we define $0^{-1} = 0$ and prove \lref{lemma:Zmodp2_field}. \begin{lemma} \label{lemma:Zmodp2_field} $\F{p^2}$ is a commutative field. \end{lemma} %% TOO LONG %% If need remove this paragraph We then specialize the basic operations in order to speed up the verification of formulas by using rewrite rules: \begin{equation*} \begin{split} (a,0) + (b,0) &= (a+b, 0)\\ (a, 0)^{-1} &= (a^{-1}, 0) \end{split} \qquad \begin{split} (a,0) \cdot (b,0) &= (a \cdot b, 0)\\ (0,a)^{-1} &= (0,(2\cdot a)^{-1}) \end{split} \end{equation*} The injection $a \mapsto (a,0)$ from $\F{p}$ to $\F{p^2}$ preserves $0, 1, +, -, \times$. Thus $(a,0)$ can be abbreviated as $a$ without confusion. We define $M_{486662,1}(\F{p^2})$. With the rewrite rule above, it is straightforward to prove that any point on the curve $M_{486662,1}(\F{p})$ is also on the curve $M_{486662,1}(\F{p^2})$. Similarly, any point on the quadratic twist $M_{486662,2}(\F{p})$ also corresponds to a point on the curve $M_{486662,1}(\F{p^2})$. As direct consequence, using \lref{lemma:curve-or-twist}, we prove that for all $x \in \F{p}$, there exists a point $P \in \F{p^2}\times\F{p^2}$ on $M_{486662,1}(\F{p^2})$ such that $\chi_0(P) = (x,0) = x$. \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip] Lemma x_is_on_curve_or_twist_implies_x_in_Fp2: forall (x:Zmodp.type), exists (p: mc curve25519_Fp2_mcuType), p#x0 = Zmodp2.Zmodp2 x 0. \end{lstlisting} We now study the case of the scalar multiplication and show similar proofs. \begin{dfn} Define the functions $\varphi_c$, $\varphi_t$ and $\psi$ \begin{itemize} \item[--] $\varphi_c: M_{486662,1}(\F{p}) \mapsto M_{486662,1}(\F{p^2})$\\ such that $\varphi((x,y)) = ((x,0), (y,0))$. \item[--] $\varphi_t: M_{486662,2}(\F{p}) \mapsto M_{486662,1}(\F{p^2})$\\ such that $\varphi((x,y)) = ((x,0), (0,y))$. \item[--] $\psi: \F{p^2} \mapsto \F{p}$ such that $\psi(x,y) = x$. \end{itemize} \end{dfn} \begin{lemma} \label{lemma:proj} For all $n \in \N$, for all point $P\in\F{p}\times\F{p}$ on the curve $M_{486662,1}(\F{p})$ (respectively on the quadratic twist $M_{486662,2}(\F{p})$), we have %\vspace{-0.3em} \begin{align*} P & \in M_{486662,1}(\F{p}) & \implies \varphi_c(n \cdot P) & = n \cdot \varphi_c(P), & & \text{and} \\ P & \in M_{486662,2}(\F{p}) & \implies \varphi_t(n \cdot P) & = n \cdot \varphi_t(P). \end{align*} \end{lemma} Notice that %\vspace{-0.5em} \begin{align*} \forall P \in M_{486662,1}(\F{p}), & & \psi(\chi_0(\varphi_c(P))) & = \chi_0(P), & & \text{and} \\ \forall P \in M_{486662,2}(\F{p}), & & \psi(\chi_0(\varphi_t(P))) & = \chi_0(P). \end{align*} In summary, for all $n \in \N$, $n < 2^{255}$, for any point $P\in\F{p}\times\F{p}$ in $M_{486662,1}(\F{p})$ or $M_{486662,2}(\F{p})$, $Curve25519\_Fp$ computes $\chi_0(n \cdot P)$. We have proved that for all $P \in \F{p^2}\times\F{p^2}$ such that $\chi_0(P) \in \F{p}$, there exists a corresponding point on the curve or the twist over $\F{p}$. Moreover, we have proved that for any point on the curve or the twist, we can compute the scalar multiplication by $n$ and obtain the same result as if we did the computation in $\F{p^2}$. % As a result we have proved theorem 2.1 of \cite{Ber06}: % No: missing uniqueness ! \begin{theorem} \label{thm:general-scalarmult} For all $n \in \N$, such that $n < 2^{255}$, for all $x \in \F{p}$ and $P \in M_{486662,1}(\F{p^2})$ such that $\chi_0(P) = x$, $Curve25519\_Fp(n,x)$ computes $\chi_0(n \cdot P)$. \end{theorem} % which is formalized in Coq as: % \begin{lstlisting}[language=Coq,belowskip=-0.1 \baselineskip] % Theorem curve25519_Fp2_ladder_ok: % forall (n : nat) (x:Zmodp.type), % (n < 2^255)%nat -> % forall (p : mc curve25519_Fp2_mcuType), % p #x0 = Zmodp2.Zmodp2 x 0 -> % curve25519_Fp_ladder n x = (p *+ n)#x0 /p. % \end{lstlisting} We then prove the equivalence of operations between $\Ffield$ and $\Zfield$, in other words between \coqe{Zmodp} and \coqe{:GF}. This allows us to show that given a clamped value $n$ and normalized \xcoord of $P$, \coqe{RFC} gives the same results as $Curve25519\_Fp$. All put together, this finishes the proof of the mathematical correctness of X25519: the fact that the code in X25519, both in the RFC~7748 and in TweetNaCl versions, correctly computes multiplication in the elliptic curve.