Commit cbf1658b by Benoit Viguier

### more text

parent 5c754263
 ... @@ -4,26 +4,29 @@ ... @@ -4,26 +4,29 @@ In this section we present our formalization of RFC~7748~\cite{rfc7748}. In this section we present our formalization of RFC~7748~\cite{rfc7748}. \begin{informaltheorem} \begin{informaltheorem} The specification of X25519 in RFC~7748 is formalized by \Coqe{ZCrypto_Scalarmult}. The specification of X25519 in RFC~7748 is formalized by \Coqe{RFC}. \end{informaltheorem} \end{informaltheorem} More precisely, we formalized X25519 with the following definition. More precisely, we formalized X25519 with the following definition. \begin{coq} \begin{lstlisting}[language=Coq] Definition ZCrypto_Scalarmult n p := Definition RFC (n: list Z) (p: list Z) : list Z := let k := decodeScalar25519 n in let u := decodeUCoordinate p in let t := montgomery_rec let t := montgomery_rec 255 (* iterate 255 times *) 255 (* iterate 255 times *) (Zclamp n) (* clamped n *) k (* clamped n *) 1 (* x_2 *) 1 (* x_2 *) (ZUnpack25519 p) (* x_3 *) u (* x_3 *) 0 (* z_2 *) 0 (* z_2 *) 1 (* z_3 *) 1 (* z_3 *) 0 (* dummy *) 0 (* dummy *) 0 (* dummy *) 0 (* dummy *) (ZUnpack25519 p) (* x_1 *) in u (* x_1 *) in let a := get_a t in let a := get_a t in let c := get_c t in let c := get_c t in ZPack25519 (Z.mul a (ZInv25519 c)). let o := ZPack25519 (Z.mul a (ZInv25519 c)) \end{coq} in encodeUCoordinate o. \end{lstlisting} We first present a generic description of the Montgomery ladder (\ref{subsec:spec-ladder}). We first present a generic description of the Montgomery ladder (\ref{subsec:spec-ladder}). Then we turn our attention to the different steps of the computation (\ref{subsec:spec-unpack-clamp-inv-pack}). Then we turn our attention to the different steps of the computation (\ref{subsec:spec-unpack-clamp-inv-pack}). ... @@ -64,6 +67,18 @@ We later prove our ladder correct in that respect (\sref{sec:maths}). ... @@ -64,6 +67,18 @@ We later prove our ladder correct in that respect (\sref{sec:maths}). \subsection{Unpacking, clamping, Inversion and Packing} \subsection{Unpacking, clamping, Inversion and Packing} \label{subsec:spec-unpack-clamp-inv-pack} \label{subsec:spec-unpack-clamp-inv-pack} \begin{lstlisting}[language=Coq] Definition decodeScalar25519 (l: list Z) : Z := ZofList 8 (clamp l). Definition decodeUCoordinate (l: list Z) : Z := ZofList 16 (Unpack25519 l). Definition encodeUCoordinate (x: Z) : list Z := ListofZ32 8 x. \end{lstlisting} Inputs of X25519 Inputs of X25519 % \emph{To implement the X25519(k, u) and X448(k, u) functions (where k is % \emph{To implement the X25519(k, u) and X448(k, u) functions (where k is % the scalar and u is the u-coordinate), first decode k and u and then % the scalar and u is the u-coordinate), first decode k and u and then ... ...
 ... @@ -3,10 +3,11 @@ ... @@ -3,10 +3,11 @@ In this section we prove the following theorem: In this section we prove the following theorem: % In this section we outline the structure of our proofs of the following theorem: % In this section we outline the structure of our proofs of the following theorem: \begin{informaltheorem} \begin{informaltheorem} \label{thm:VST-RFC} \label{thm:VST-RFC} The implementation of X25519 in TweetNaCl (\TNaCle{crypto_scalarmult}) matches The implementation of X25519 in TweetNaCl (\TNaCle{crypto_scalarmult}) matches the specifications of RFC~7748~\cite{rfc7748} (\Coqe{ZCrypto_Scalarmult}) the specifications of RFC~7748~\cite{rfc7748} (\Coqe{RFC}) \end{informaltheorem} \end{informaltheorem} More formally. More formally. ... @@ -27,10 +28,9 @@ used to in some of our more complex proofs (\ref{subsec:inversions-reflections}) ... @@ -27,10 +28,9 @@ used to in some of our more complex proofs (\ref{subsec:inversions-reflections}) \label{subsec:proof-structure} \label{subsec:proof-structure} In order to prove the correctness of X25519 in TweetNaCl code \TNaCle{crypto_scalarmult}, In order to prove the correctness of X25519 in TweetNaCl code \TNaCle{crypto_scalarmult}, we use VST to prove that the code matches our functional Coq specification of \Coqe{Crypto_Scalarmult} we use VST to prove that the code matches our functional Coq specification of \Coqe{RFC}. (to save space we sometimes abbreviate this as \Coqe{CSM}). Then, we prove that Then, we prove that our specification of the scalar multiplication matches the mathematical definition our specification of the scalar multiplication matches the mathematical definition of elliptic curves and Theorem 2.1 by Bernstein~\cite{Ber06} (\tref{thm:Elliptic-RFC}). of elliptic curves and Theorem 2.1 by Bernstein~\cite{Ber06} (\tref{thm:Elliptic-CSM}). \fref{tikz:ProofOverview} shows a graph of dependencies of the proofs. \fref{tikz:ProofOverview} shows a graph of dependencies of the proofs. The mathematical proof of X25519 is presented in \sref{sec:maths}. The mathematical proof of X25519 is presented in \sref{sec:maths}. \begin{figure}[h] \begin{figure}[h] ... @@ -111,7 +111,7 @@ a pure Coq function. ... @@ -111,7 +111,7 @@ a pure Coq function. % A pure function is a function where the return value is only determined by its % A pure function is a function where the return value is only determined by its % input values, without observable side effects (Side effect are e.g. printing) % input values, without observable side effects (Side effect are e.g. printing) This defines the equivalence between the Clight representation and our Coq This defines the equivalence between the Clight representation and our Coq definition of the ladder (\coqe{CSM}). definition of the ladder (\coqe{RFC}). \begin{lstlisting}[language=CoqVST] \begin{lstlisting}[language=CoqVST] Definition crypto_scalarmult_spec := Definition crypto_scalarmult_spec := ... @@ -137,10 +137,10 @@ SEP (sh [{ v_q }] <<(uch32)-- q; ... @@ -137,10 +137,10 @@ SEP (sh [{ v_q }] <<(uch32)-- q; Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665) Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665) (*------------------------------------------*) (*------------------------------------------*) POST [ tint ] POST [ tint ] PROP (Forall (fun x => 0 <= x < 2^8) (CSM n p); PROP (Forall (fun x => 0 <= x < 2^8) (RFC n p); Zlength (CSM n p) = 32) Zlength (RFC n p) = 32) LOCAL(temp ret_temp (Vint Int.zero)) LOCAL(temp ret_temp (Vint Int.zero)) SEP (sh [{ v_q }] <<(uch32)-- mVI (CSM n p); SEP (sh [{ v_q }] <<(uch32)-- mVI (RFC n p); sh [{ v_n }] <<(uch32)-- mVI n; sh [{ v_n }] <<(uch32)-- mVI n; sh [{ v_p }] <<(uch32)-- mVI p; sh [{ v_p }] <<(uch32)-- mVI p; Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665 Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665 ... @@ -172,22 +172,22 @@ As Post-condition we have: ... @@ -172,22 +172,22 @@ As Post-condition we have: The function \TNaCle{crypto_scalarmult} returns an integer. The function \TNaCle{crypto_scalarmult} returns an integer. \item[] \VSTe{LOCAL}: \VSTe{temp ret_temp (Vint Int.zero)}\\ \item[] \VSTe{LOCAL}: \VSTe{temp ret_temp (Vint Int.zero)}\\ The returned integer has value $0$. The returned integer has value $0$. \item[] \VSTe{SEP}: \VSTe{sh [{ v_q $\!\!\}\!\!]\!\!\!$ <<(uch32)-- mVI (CSM n p)}\\ \item[] \VSTe{SEP}: \VSTe{sh [{ v_q $\!\!\}\!\!]\!\!\!$ <<(uch32)-- mVI (RFC n p)}\\ In the memory share \texttt{sh}, the address \VSTe{v_q} points In the memory share \texttt{sh}, the address \VSTe{v_q} points to a list of integer values \VSTe{mVI (CSM n p)} where \VSTe{CSM n p} is the to a list of integer values \VSTe{mVI (RFC n p)} where \VSTe{RFC n p} is the result of the \VSTe{crypto_scalarmult} of \VSTe{n} and \VSTe{p}. result of the \VSTe{crypto_scalarmult} of \VSTe{n} and \VSTe{p}. \item[] \VSTe{PROP}: \VSTe{Forall (fun x => 0 <= x < 2^8) (CSM n p)}\\ \item[] \VSTe{PROP}: \VSTe{Forall (fun x => 0 <= x < 2^8) (RFC n p)}\\ \VSTe{PROP}: \VSTe{Zlength (CSM n p) = 32}\\ \VSTe{PROP}: \VSTe{Zlength (RFC n p) = 32}\\ We show that the computation for \VSTe{CSM} fits in \TNaCle{u8[32]}. We show that the computation for \VSTe{RFC} fits in \TNaCle{u8[32]}. \end{itemize} \end{itemize} This specification shows that \TNaCle{crypto_scalarmult} in C computes the same This specification shows that \TNaCle{crypto_scalarmult} in C computes the same result as \VSTe{CSM} in Coq provided that inputs are within their respective result as \VSTe{RFC} in Coq provided that inputs are within their respective bounds: arrays of 32 bytes. bounds: arrays of 32 bytes. \begin{theorem} \begin{theorem} \label{thm:crypto-vst} \label{thm:crypto-vst} \TNaCle{crypto_scalarmult} in TweetNaCl has the same behavior as \coqe{Crypto_Scalarmult} in Coq. \TNaCle{crypto_scalarmult} in TweetNaCl has the same behavior as \coqe{RFC} in Coq. \end{theorem} \end{theorem} ... @@ -278,7 +278,7 @@ and without returns the same result. ... @@ -278,7 +278,7 @@ and without returns the same result. We formalized this result in a generic way in Appendix~\ref{subsubsec:for}. We formalized this result in a generic way in Appendix~\ref{subsubsec:for}. Using this formalization, we prove that the 255 steps of the Montgomery ladder Using this formalization, we prove that the 255 steps of the Montgomery ladder in C provide the same computations as in \coqe{CSM}. in C provide the same computations as in \coqe{RFC}. ... @@ -645,22 +645,22 @@ By proving that each functions \coqe{Low.M}; \coqe{Low.A}; \coqe{Low.Sq}; \coqe{ ... @@ -645,22 +645,22 @@ By proving that each functions \coqe{Low.M}; \coqe{Low.A}; \coqe{Low.Sq}; \coqe{ \coqe{Unpack25519}; \coqe{clamp}; \coqe{Pack25519}; \coqe{car25519} are behaving over \coqe{list Z} \coqe{Unpack25519}; \coqe{clamp}; \coqe{Pack25519}; \coqe{car25519} are behaving over \coqe{list Z} as their equivalent over \coqe{Z} in \coqe{:GF} (in \Zfield), we prove the correctness of as their equivalent over \coqe{Z} in \coqe{:GF} (in \Zfield), we prove the correctness of \begin{theorem} % \begin{theorem} \label{thm:crypto-rfc} % \label{thm:crypto-rfc} \coqe{Crypto_Scalarmult} matches the specification of RFC~7748. % \coqe{Crypto_Scalarmult} matches the specification of RFC~7748. \end{theorem} % \end{theorem} This is formalized as follows in Coq: % This is formalized as follows in Coq: \begin{lstlisting}[language=Coq] % \begin{lstlisting}[language=Coq] Theorem Crypto_Scalarmult_Eq : % Theorem Crypto_Scalarmult_Eq : forall (n p:list Z), % forall (n p:list Z), Zlength n = 32 -> % Zlength n = 32 -> Zlength p = 32 -> % Zlength p = 32 -> Forall (fun x : Z, 0 <= x /\ x < 2 ^ 8) n -> % Forall (fun x : Z, 0 <= x /\ x < 2 ^ 8) n -> Forall (fun x : Z, 0 <= x /\ x < 2 ^ 8) p -> % Forall (fun x : Z, 0 <= x /\ x < 2 ^ 8) p -> ZofList 8 (Crypto_Scalarmult n p) = % ZofList 8 (Crypto_Scalarmult n p) = ZCrypto_Scalarmult (ZofList 8 n) (ZofList 8 p). % ZCrypto_Scalarmult (ZofList 8 n) (ZofList 8 p). \end{lstlisting} % \end{lstlisting} We prove that \coqe{Crypto_Scalarmult} matches the specification of RFC~7748 (\tref{thm:crypto-rfc}). We prove that \coqe{Crypto_Scalarmult} matches the specification of RFC~7748 (\tref{thm:crypto-rfc}). With the VST we also prove that \coqe{Crypto_Scalarmult} matches the Clight translation of Tweetnacl (\tref{thm:crypto-vst}). With the VST we also prove that \coqe{Crypto_Scalarmult} matches the Clight translation of Tweetnacl (\tref{thm:crypto-vst}). ... ...
 ... @@ -2,12 +2,26 @@ ... @@ -2,12 +2,26 @@ \label{sec:maths} \label{sec:maths} In this section we prove the following theorem: In this section we prove the following theorem: \begin{theorem} \label{thm:Elliptic-CSM} \begin{informaltheorem} \label{thm:Elliptic-RFC} The implementation of X25519 in TweetNaCl (\TNaCle{crypto_scalarmult}) computes the The implementation of X25519 in TweetNaCl (\TNaCle{crypto_scalarmult}) computes the $\F{p}$-restricted $x$-coordinate scalar multiplication on $E(\F{p^2})$ where $p$ is $\p$ $\F{p}$-restricted $x$-coordinate scalar multiplication on $E(\F{p^2})$ where $p$ is $\p$ and $E$ is the elliptic curve $y^2 = x^3 + 486662 x^2 + x$. and $E$ is the elliptic curve $y^2 = x^3 + 486662 x^2 + x$. \end{theorem} \end{informaltheorem} More formally: \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 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}) We extend it to support Montgomery curves (\ref{subsec:ECC-Montgomery}) ... @@ -414,7 +428,8 @@ we can find a $y$ such that $(x,y)$ is either on the curve or on the quadratic t ... @@ -414,7 +428,8 @@ we can find a $y$ such that $(x,y)$ is either on the curve or on the quadratic t over $M_{486662,2}(\F{p})$ such that the $x$-coordinate of $P$ is $x$. over $M_{486662,2}(\F{p})$ such that the $x$-coordinate of $P$ is $x$. \end{lemma} \end{lemma} \begin{lstlisting}[language=Coq] \begin{lstlisting}[language=Coq] Theorem x_is_on_curve_or_twist: forall x : Zmodp.type, Theorem x_is_on_curve_or_twist: forall x : Zmodp.type, (exists (p : mc curve25519_mcuType), p#x0 = x) \/ (exists (p : mc curve25519_mcuType), p#x0 = x) \/ (exists (p' : mc twist25519_mcuType), p'#x0 = x). (exists (p' : mc twist25519_mcuType), p'#x0 = x). \end{lstlisting} \end{lstlisting} ... @@ -430,22 +445,22 @@ Module Zmodp2. ... @@ -430,22 +445,22 @@ Module Zmodp2. Inductive type := Inductive type := Zmodp2 (x: Zmodp.type) (y:Zmodp.type). Zmodp2 (x: Zmodp.type) (y:Zmodp.type). Definition pi (x : Zmodp.type * Zmodp.type) : type := Definition pi (x: Zmodp.type * Zmodp.type) : type := Zmodp2 x.1 x.2. Zmodp2 x.1 x.2. Coercion repr (x : type) : Zmodp.type*Zmodp.type := Coercion repr (x: type) : Zmodp.type*Zmodp.type := let: Zmodp2 u v := x in (u, v). let: Zmodp2 u v := x in (u, v). Definition zero : type := Definition zero : type := pi ( 0%:R, 0%:R ). pi ( 0%:R, 0%:R ). Definition one : type := Definition one : type := pi ( 1, 0%:R ). pi ( 1, 0%:R ). Definition opp (x : type) : type := Definition opp (x: type) : type := pi (- x.1 , - x.2). pi (- x.1 , - x.2). Definition add (x y : type) : type := Definition add (x y: type) : type := pi (x.1 + y.1, x.2 + y.2). pi (x.1 + y.1, x.2 + y.2). Definition sub (x y : type) : type := Definition sub (x y: type) : type := pi (x.1 - y.1, x.2 - y.2). pi (x.1 - y.1, x.2 - y.2). Definition mul (x y : type) : type := Definition mul (x y: type) : type := pi ((x.1 * y.1) + (2%:R * (x.2 * y.2)), pi ((x.1 * y.1) + (2%:R * (x.2 * y.2)), (x.1 * y.2) + (x.2 * y.1)). (x.1 * y.2) + (x.2 * y.1)). \end{lstlisting} \end{lstlisting} ... @@ -556,16 +571,4 @@ Lemma ZCrypto_Scalarmult_curve25519_ladder: ... @@ -556,16 +571,4 @@ Lemma ZCrypto_Scalarmult_curve25519_ladder: \end{lstlisting} \end{lstlisting} From \tref{thm:RFC} and \tref{thm:general-scalarmult}, we prove the correctness From \tref{thm:RFC} and \tref{thm:general-scalarmult}, we prove the correctness of \TNaCle{crypto_scalarmult} (\tref{thm:Elliptic-CSM}). of \TNaCle{crypto_scalarmult} (\tref{thm:Elliptic-RFC}). \begin{lstlisting}[language=Coq] Theorem Crypto_Scalarmult_Correct: forall (n:list Z) (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 (ZUnpack25519 (ZofList 8 p)) = P#x0 -> ZofList 8 (Crypto_Scalarmult n p) = (P *+ (Z.to_nat (Zclamp (ZofList 8 n)))) _x0. \end{lstlisting}
 ... @@ -77,8 +77,8 @@ match t with ... @@ -77,8 +77,8 @@ match t with end. end. \end{lstlisting} \end{lstlisting} \subsubsection{ZCrypto\_Scalarmult} \subsubsection{RFC in Coq} \label{subsubsec:ZCryptoScalarmult} \label{subsubsec:RFC-Coq} ~ ~ Instantiation of the Class \Coqe{Ops} with operations over \Z and modulo \p. Instantiation of the Class \Coqe{Ops} with operations over \Z and modulo \p. \begin{lstlisting}[language=Coq] \begin{lstlisting}[language=Coq] ... @@ -152,21 +152,33 @@ Proof. ... @@ -152,21 +152,33 @@ Proof. apply Mid.getbit. (* instantiate ith bit *) apply Mid.getbit. (* instantiate ith bit *) Defined. Defined. Definition decodeScalar25519 (l: list Z) : Z := ZofList 8 (clamp l). Definition decodeUCoordinate (l: list Z) : Z := ZofList 16 (Unpack25519 l). Definition encodeUCoordinate (x: Z) : list Z := ListofZ32 8 x. (* instantiate montgomery_rec with Z_Ops *) (* instantiate montgomery_rec with Z_Ops *) Definition ZCrypto_Scalarmult n p := Definition RFC (n: list Z) (p: list Z) : list Z := let k := decodeScalar25519 n in let u := decodeUCoordinate p in let t := montgomery_rec let t := montgomery_rec 255 (* iterate 255 times *) 255 (* iterate 255 times *) (Zclamp n) (* clamped n *) k (* clamped n *) 1 (* x_2 *) 1 (* x_2 *) (ZUnpack25519 p) (* x_3 *) u (* x_3 *) 0 (* z_2 *) 0 (* z_2 *) 1 (* z_3 *) 1 (* z_3 *) 0 (* dummy *) 0 (* dummy *) 0 (* dummy *) 0 (* dummy *) (ZUnpack25519 p) (* x_1 *) in u (* x_1 *) in let a := get_a t in let a := get_a t in let c := get_c t in let c := get_c t in ZPack25519 (Z.mul a (ZInv25519 c)). let o := ZPack25519 (Z.mul a (ZInv25519 c)) in encodeUCoordinate o. \end{lstlisting} \end{lstlisting} \subsubsection{CSM} \subsubsection{CSM} ... ...
 ... @@ -8,8 +8,27 @@ ... @@ -8,8 +8,27 @@ preaction = {decorate}, preaction = {decorate}, postaction = {draw,line width=1.4pt, white,shorten >= 4.5pt}] postaction = {draw,line width=1.4pt, white,shorten >= 4.5pt}] \path [thick, dashed] (2.5,1) edge +(0,-6.75); \draw (2.5,1) node[anchor=north east] {\sref{sec:Coq-RFC}}; \draw (2.5,1) node[anchor=north west] {\sref{sec:C-Coq}}; \path [thick, dashed] (0,-5.75) edge +(8.5,0); \draw (8.5,-5.75) node[anchor=north east] {\sref{sec:maths}}; % SECTION III % Definition of RFC \begin{scope}[yshift=-3 cm,xshift=0 cm] \draw (0,0) -- (0.4, 0.4) -- (1.75,0.4) -- (1.75,0) -- cycle; \draw (0,0) -- (1.75,0) -- (1.75,-1) -- (0, -1) -- cycle; \draw (0.3,-0.05) node[textstyle] {Definition}; \draw (0.875,-0.5) node[textstyle, anchor=center] {\texttt{RFC}}; \draw (1.75,-1) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{.V}}; \end{scope} % SECTION IV % C code % C code \begin{scope}[yshift=0 cm,xshift=0 cm] \begin{scope}[yshift=-0.25 cm,xshift=3 cm] \draw (0,0) -- (0.4, 0.4) -- (1.25,0.4) -- (1.25,0) -- cycle; \draw (0,0) -- (0.4, 0.4) -- (1.25,0.4) -- (1.25,0) -- cycle; \draw (0,0) -- (1.25,0) -- (1.25,-1.25) -- (0, -1.25) -- cycle; \draw (0,0) -- (1.25,0) -- (1.25,-1.25) -- (0, -1.25) -- cycle; \draw (0.3,-0.05) node[textstyle] {code}; \draw (0.3,-0.05) node[textstyle] {code}; ... @@ -17,9 +36,8 @@ ... @@ -17,9 +36,8 @@ \draw (1.25,-1.25) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{.C}}; \draw (1.25,-1.25) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{.C}}; \end{scope} \end{scope} % V code % V code \begin{scope}[yshift=0 cm,xshift=2.5 cm] \begin{scope}[yshift=-0.25 cm,xshift=5 cm] \draw (0,0) -- (0.4, 0.4) -- (1.25,0.4) -- (1.25,0) -- cycle; \draw (0,0) -- (0.4, 0.4) -- (1.25,0.4) -- (1.25,0) -- cycle; \draw (0,0) -- (1.25,0) -- (1.25,-1.25) -- (0, -1.25) -- cycle; \draw (0,0) -- (1.25,0) -- (1.25,-1.25) -- (0, -1.25) -- cycle; \draw (0.3,-0.05) node[textstyle] {code}; \draw (0.3,-0.05) node[textstyle] {code}; ... @@ -28,61 +46,63 @@ ... @@ -28,61 +46,63 @@ % \draw (1.25,0) node[anchor=south east, inner sep=0pt] {\includegraphics[width=.0125\textwidth]{img/coq_logo.png}}; % \draw (1.25,0) node[anchor=south east, inner sep=0pt] {\includegraphics[width=.0125\textwidth]{img/coq_logo.png}}; \end{scope} \end{scope} % VST Theorem \path [thick, double, ->] (4.25,-0.75) edge (5, -0.75); \begin{scope}[yshift=0 cm,xshift=6 cm] \draw (0,0) -- (0.4, 0.4) -- (2.5,0.4) -- (2.5,0) -- cycle; \draw (0,0) -- (2.5,0) -- (2.5,-1.25) -- (0, -1.25) -- cycle; \draw (0.3,-0.05) node[textstyle] {Proof}; \draw (1.25,-0.5) node[textstyle, anchor=center] {\{{\color{doc@lstnumbers}\textbf{Pre}}\} \texttt{Prog} \{{\color{doc@lstdirective}\textbf{Post}}\}}; \draw (2.5,-1.25) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{\checkmark}}; \end{scope} % VST Spec % VST Spec \begin{scope}[yshift=-2.5 cm,xshift=3 cm] \begin{scope}[yshift=-3 cm,xshift=3 cm] \draw (0,0) -- (0.4, 0.4) -- (2,0.4) -- (2,0) -- cycle; \draw (0,0) -- (0.4, 0.4) -- (2,0.4) -- (2,0) -- cycle; \draw (0,0) -- (2,0) -- (2,-2) -- (0, -2) -- cycle; \draw (0,0) -- (2,0) -- (2,-2) -- (0, -2) -- cycle; \draw (0.3,-0.05) node[textstyle] {Specification}; \draw (0.3,-0.05) node[textstyle] {Specification}; \draw (1,-1) node[textstyle, anchor=center, align=left] { \draw (1,-1) node[textstyle, anchor=center, align=left] { {\color{doc@lstnumbers}\textbf{Pre}}:\\ {\color{doc@lstnumbers}\textbf{Pre}}:\\ ~~$n \in \N$,\\ ~~$n \in \N$,\\ % ~~$n \in$ \TNaCles{u8[32]},\\ ~~$P \in E(\F{p^2})$\\ ~~$P \in E(\F{p^2})$\\ % ~~$P \in$ \TNaCles{u8[32]}\\ {\color{doc@lstdirective}\textbf{Post}}:\\ {\color{doc@lstdirective}\textbf{Post}}:\\ ~~\texttt{CSM}$(n,P)$}; ~~\texttt{RFC}$(n,P)$}; \end{scope} \end{scope} % Definition of CSM % VST Theorem \begin{scope}[yshift=-4.5 cm,xshift=0 cm] \begin{scope}[yshift=-3 cm,xshift=6 cm] \draw (0,0) -- (0.4, 0.4) -- (2,0.4) -- (2,0) -- cycle; \draw (0,0) -- (0.4, 0.4) -- (2.5,0.4) -- (2.5,0) -- cycle; \draw (0,0) -- (2,0) -- (2,-1) -- (0, -1) -- cycle; \draw (0,0) -- (2.5,0) -- (2.5,-1.25) -- (0, -1.25) -- cycle; \draw (0.3,-0.05) node[textstyle] {Definition}; \draw (0.3,-0.05) node[textstyle] {Proof}; \draw (1,-0.5) node[textstyle, anchor=center] {\texttt{CSM}}; \draw (1.25,-0.5) node[textstyle, anchor=center] {\{{\color{doc@lstnumbers}\textbf{Pre}}\} \texttt{Prog} \{{\color{doc@lstdirective}\textbf{Post}}\}}; \draw (2.5,-1.25) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{\checkmark}}; \end{scope} \end{scope} \path [thick, double] (5.625,-1.5) edge [out=-90, in=90] (5.625, -2.5); \path [thick, double, ->] (5.625, -2.5) edge [out=-90, in=180] (6, -3.5); \path [thick, double, ->] (5,-3.75) edge [out=0, in=180] (6, -3.75); % SECTION V % Spec of Curve nP % Spec of Curve nP \begin{scope}[yshift=-7.5 cm,xshift=0 cm] \begin{scope}[yshift=-7.25 cm,xshift=0 cm] \draw (0,0) -- (0.4, 0.4) -- (2,0.4) -- (2,0) -- cycle; \draw (0,0) -- (0.4, 0.4) -- (1.75,0.4) -- (1.75,0) -- cycle; \draw (0,0) -- (2,0) -- (2,-1) -- (0, -1) -- cycle; \draw (0,0) -- (1.75,0) -- (1.75,-1) -- (0, -1) -- cycle; \draw (0.3,-0.05) node[textstyle] {Definition}; \draw (0.3,-0.05) node[textstyle] {Definition}; \draw (1,-0.5) node[textstyle, anchor=center] {$n \cdot P$}; \draw (0.875,-0.5) node[textstyle, anchor=center] {$n \cdot P$}; \end{scope} \end{scope} % Correctness Theorem % Correctness Theorem \begin{scope}[yshift=-7 cm,xshift=6 cm] \begin{scope}[yshift=-6.75 cm,xshift=6 cm] \draw (0,0) -- (0.4, 0.4) -- (2.5,0.4) -- (2.5,0) -- cycle; \draw (0,0) -- (0.4, 0.4) -- (2.5,0.4) -- (2.5,0) -- cycle; \draw (0,0) -- (2.5,0) -- (2.5,-1.5) -- (0, -1.5) -- cycle; \draw (0,0) -- (2.5,0) -- (2.5,-1.5) -- (0, -1.5) -- cycle; \draw (0.3,-0.05) node[textstyle] {Proof}; \draw (0.3,-0.05) node[textstyle] {Proof}; \draw (2.5,-1.5) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{\checkmark}}; \draw (2.5,-1.5) node[textstyle, anchor = south east] {\color{doc@lstfunctions}{\checkmark}}; \draw (1.25,-0.75) node[textstyle, anchor=center] {{\color{doc@lstnumbers}\textbf{Pre}} $\implies$\\$\text{\texttt{CSM}}(n,P) = n \cdot P$}; \draw (1.25,-0.75) node[textstyle, anchor=center] {{\color{doc@lstnumbers}\textbf{Pre}} $\implies$\\$\text{\texttt{RFC}}(n,P) = n \cdot P$}; \end{scope} \end{scope} \path [thick, double, ->] (1.25,-0.5) edge (2.5, -0.5); \path [thick, double, ->] (1.75,-3.5) edge [out=0, in=-180] (3, -3.5); \path [thick, double, ->] (3.75,-0.5) edge (6, -0.5); \path [thick, double] (1.75,-3.5) edge [out=0, in=90] (2.25, -4); \path [thick, double, ->] (5,-3.5) edge [out=0, in=-90] (6.5, -1.25); \path [thick, double] (2.25, -4) edge [out=-90, in=90] (2.25, -6.75); \path [thick, double, ->] (2,-5) edge [out=0, in=-180] (3, -3.5); \path [thick, double] (2.25, -6.75) edge [out=-90, in=-180] (3, -7.5); \path [thick, double, ->] (2,-5) edge [out=0, in=-180] (6, -7.5); \path [thick, double, ->] (3, -7.5) edge [out=0, in=-180] (6, -7.5); \path [thick, double, ->] (2,-8) edge [out=0, in=-180] (6, -8); \path [thick, double, ->] (1.75,-7.75) edge [out=0, in=-180] (6, -7.75); \path [thick, dashed] (0,-5.75) edge +(8.5,0); \draw (8.5,-5.75) node[anchor=south east] {\sref{sec:C-Coq-RFC}}; \draw (8.5,-5.75) node[anchor=north east] {\sref{sec:maths}}; \end{tikzpicture} \end{tikzpicture}
 ... @@ -67,8 +67,6 @@ Require Import Tweetnacl_verif.verif_crypto_scalarmult_lemmas. ... @@ -67,8 +67,6 @@ Require Import Tweetnacl_verif.verif_crypto_scalarmult_lemmas. Require Import Tweetnacl.Low.Get_abcdef. Require Import Tweetnacl.Low.Get_abcdef. Require Import Tweetnacl.Low.ScalarMult_rev. Require Import Tweetnacl.Low.ScalarMult_rev. Require Import Tweetnacl.Low.Constant. Require Import Tweetnacl.Low.Constant. (* Require Import Tweetnacl.Low.Crypto_Scalarmult. *) (* Require Import Tweetnacl.Low.Crypto_Scalarmult_. *) Require Import Tweetnacl.Mid.Instances. Require Import Tweetnacl.Mid.Instances. Require Import Tweetnacl.rfc.rfc. Require Import Tweetnacl.rfc.rfc. Open Scope Z. Open Scope Z. ... ...
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