### 9.75 pages

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 ... ... @@ -296,7 +296,7 @@ in Coq (for the sake of simplicity we do not display the conversion in the theor For all $n \in \N, n < 2^{255}$ and where the bits 1, 2, 5 248, 249, 250 are cleared and bit 6 is set, for all $P \in E(\F{p^2})$, for all $p \in \F{p}$ such that $P.x = p$, there exists $Q \in E(\F{p^2})$ such that $Q = nP$, $Q.x = q$ and $q$ = \VSTe{CSM} $n$ $p$. there exists $Q \in E(\F{p^2})$ such that $Q = nP$ where $Q.x = q$ and $q$ = \VSTe{CSM} $n$ $p$. \end{theorem} A more complete description in Coq of Theorem \ref{CSM-correct} with the associated conversions is as follow: ... ...
 \section{Linking C and Coq} \section{Linking C and Reflections} In this section we describe techniques used to prove the equivalence between the Clight description of TweetNaCl and Coq functions producing similar behaviors. ... ... @@ -8,27 +8,38 @@ Clight description of TweetNaCl and Coq functions producing similar behaviors. As described in Section \ref{sec:impl}, numbers in \TNaCle{gf} are represented in base $2^{16}$ and we can use a direct mapping to represent that array as a list integers in Coq. However in order to show the correctness of the basic operations, we need to convert this number as a full integer. For this reason we define \Coqe{ZofList : Z -> list Z -> Z}: \begin{Coq} we need to convert this number as a full integer. \begin{definition} Let \Coqe{ZofList} : $\Z \rightarrow \texttt{list} \Z \rightarrow \Z$, a parametrized map by $n$ betwen a list $l$ and its it's little endian representation with a base $2^n$. \end{definition} We define it in Coq as: \begin{lstlisting}[language=Coq] Fixpoint ZofList {n:Z} (a:list Z) : Z := match a with | [] => 0 | h :: q => h + (pow 2 n) * ZofList q end. \end{lstlisting} We define a notation where $n$ is $16$. \begin{lstlisting}[language=Coq] Notation "Z16.lst A" := (ZofList 16 A). \end{Coq} It converts any list of integer from a base $2^n$ to an single integer. In our case we define a notation where $n$ is $16$. The following Coq notation is defined to convert any integer to its representant in \Zfield. \begin{Coq} \end{lstlisting} We also define a notation to do the modulo, projecting any numbers in $\Zfield$. \begin{lstlisting}[language=Coq] Notation "A :GF" := (A mod (2^255-19)). \end{Coq} \end{lstlisting} Remark that this representation is different from \Coqe{Zmodp}. However the equivalence between operations over $\Zfield$ and $\F{p}$ is easily proven. Using these two definitions, we proved lemmas such as the correctness of the multiplication \Coqe{M}: Using these two definitions, we proved intermediates lemmas such as the correctness of the multiplication \Coqe{M} where \Coqe{M} replicate the computations and steps done in C. \begin{lemma} For all list of integers \texttt{a} and \texttt{b} of length 16 representing $A$ and $B$ in $\Zfield$, the number represented in $\Zfield$ by the list \Coqe{(M a b)} is equal to $A \times B \bmod \p$. \end{lemma} And seen in Coq as follows: \begin{Coq} Lemma mult_GF_Zlength : forall (a:list Z) (b:list Z), ... ... @@ -38,14 +49,10 @@ Lemma mult_GF_Zlength : (Z16.lst a * Z16.lst b) :GF. \end{Coq} For all list of integers \texttt{a} and \texttt{b} of length 16 representing $A$ and $B$ in \Zfield, the number represented in \Zfield by the list \Coqe{(M a b)} is equal to $A \times B \bmod 2^{255}-19$. \subsection{Inversions in \Zfield} In a similar fashion we can define a Coq version of the inversion mimicking the behavior of \TNaCle{inv25519} over \Coqe{list Z}. \Coqe{step_pow} contains the body of the for loop. the behavior of \TNaCle{inv25519} over \Coqe{list Z}. \begin{lstlisting}[language=Ctweetnacl] sv inv25519(gf o,const gf a) { ... ... @@ -59,18 +66,15 @@ sv inv25519(gf o,const gf a) FOR(i,16) o[i]=c[i]; } \end{lstlisting} \begin{Coq} We specify this with 2 functions: a recursive \Coqe{pow_fn_rev} to to simulate the for loop and a simple \Coqe{step_pow} containing the body. Note the off by one for the loop. \begin{lstlisting}[language=Coq] Definition step_pow (a:Z) (c g:list Z) : list Z := let c := Sq c in if a <>? 1 && a <>? 3 then M c g else c. \end{Coq} \Coqe{pow_fn_rev} is responsible of the iteration of the loop by making recursive calls. \begin{Coq} Function pow_fn_rev (a:Z) (b:Z) (c g: list Z) {measure Z.to_nat a} : (list Z) := if a <=? 0 ... ... @@ -78,23 +82,46 @@ Function pow_fn_rev (a:Z) (b:Z) (c g: list Z) else let prev := pow_fn_rev (a - 1) b c g in step_pow (b - 1 - a) prev g. \end{Coq} % This \Coqe{Function} requires a proof of termination. It is done by proving the % Well-foundness of the decreasing argument: \Coqe{measure Z.to_nat a}. \end{lstlisting} Calling \Coqe{pow_fn_rev} 254 times allows us to reproduce the same behavior as the Clight definition. \begin{Coq} This \Coqe{Function} requires a proof of termination. It is done by proving the Well-foundness of the decreasing argument: \Coqe{measure Z.to_nat a}. Calling \Coqe{pow_fn_rev} 254 times allows us to reproduce the same behavior as the \texttt{Clight} definition. \begin{lstlisting}[language=Coq] Definition Inv25519 (x:list Z) : list Z := pow_fn_rev 254 254 x x. \end{Coq} \end{lstlisting} Similarily we define the same function over $\Z$. \begin{lstlisting}[language=Coq] Definition step_pow_Z (a:Z) (c:Z) (g:Z) : Z := let c := c * c in if a <>? 1 && a <>? 3 then c * g else c. \subsection{Inversions and Reflections} Function pow_fn_rev_Z (a:Z) (b:Z) (c:Z) (g: Z) {measure Z.to_nat a} : Z := if (a <=? 0) then c else let prev := pow_fn_rev_Z (a - 1) b c g in step_pow_Z (b - 1 - a) prev g. In TweetNaCl, \TNaCle{inv25519} computes an inverse in \Zfield. It uses the Fermat's little theorem by the exponentiation to $2^{255}-21$. To prove the correctness of the result we can use multiple strategy such as: Definition Inv25519_Z (x:Z) : Z := pow_fn_rev_Z 254 254 x x. \end{lstlisting} And prove their equivalence in $\Zfield$. \begin{lstlisting}[language=Coq] Lemma Inv25519_Z_GF : forall (g:list Z), length g = 16 -> (Z16.lst (Inv25519 g)) :GF = (Inv25519_Z (Z16.lst g)) :GF. \end{lstlisting} In TweetNaCl, \TNaCle{inv25519} computes an inverse in $\Zfield$. It uses the Fermat's little theorem by doing an exponentiation to $2^{255}-21$. This is done by applying a square-and-multiply algorithm. The binary representation of $p-2$ implies to always do a multiplications aside for bit 2 and 4, thus the if case. To prove the correctness of the result we can use multiple strategies such as: \begin{itemize} \item Proving it is special case of square-and-multiply algorithm applied to a specific number and then show that this number is indeed $2^{255}-21$. ... ... @@ -102,94 +129,44 @@ correctness of the result we can use multiple strategy such as: $x^a \times x^b = x^{(a+b)}$ and $(x^a)^2 = x^{(2 \times a)}$. We can prove that the resulting exponent is $2^{255}-21$. \end{itemize} We choose the second method. The only drawback is that it requires to apply the unrolling and exponentiation lemmas 255 times. This can be automated in Coq with tacticals such as \Coqe{repeat}, but it generates a big proof object which We use the second method for the benefits of simplicity. However it requires to apply the unrolling and exponentiation formulas 255 times. This can be automated in Coq with tacticals such as \Coqe{repeat}, but it generates a proof object which will take a long time to verify. \subsubsection{Speeding up with Reflections} \subsection{Speeding up with Reflections} In order to speed up the verification, we use a technique called reflection. It provides us with flexibility such as we don't need to know the number of times the lemmas needs to be applied. It provides us with flexibility such as we don't need to know the number of times nor the order in which the lemmas needs to be applied (chapter 15 in \cite{CpdtJFR}). The idea is to \textit{reflect} the goal into a decidable environment. In our case, we show that for a property $P$, we can define a decidable boolean property $P_{bool}$ such that if $P_{bool}$ is \Coqe{true} then $P$ holds. $$P_{bool} = true \implies P$$ With VST we proved that \TNaCle{inv25519} in \texttt{Clight} is equivalent to its Coq functional definition \Coqe{Inv25519}. We also proved that \Coqe{Inv25519} over \Coqe{list Z} is equivalent to applying \Coqe{Inv25519_Z} over \Coqe{Z} % With VST we proved that \TNaCle{inv25519} in \texttt{Clight} is equivalent to its Coq % functional definition \Coqe{Inv25519}. We also proved that \Coqe{pow_fn_rev} % over \Coqe{list Z} is equivalent to applying % \Coqe{pow_fn_rev_Z} over \Coqe{Z} \begin{Coq} Lemma Inv25519_Z_GF : forall (g:list Z), length g = 16 -> (Z16.lst (Inv25519 g)) :GF = (Inv25519_Z (Z16.lst g)) :GF. \end{Coq} % \begin{Coq} % Lemma pow_fn_rev_Z_GF : % forall (a:Z) (b:Z) (c:list Z) (g:list Z), % Zlength c = 16 -> % Zlength g = 16 -> % (Z16.lst (pow_fn_rev a b c g)) :GF = % (pow_fn_rev_Z a b (Z16.lst c) (Z16.lst g)) :GF . % \end{Coq} where \Coqe{Inv25519_Z} reproduces the same pattern as \Coqe{Inv25519}. \begin{Coq} Definition Inv25519_Z (x:Z) : Z := pow_fn_rev_Z 254 254 x x. \end{Coq} \Coqe{pow_fn_rev_Z} mimicks \Coqe{pow_fn_rev} over \Coqe{Z} instead of \Coqe{list Z}. The structure is the same: the application \Coqe{step_pow_Z} and the loop itself \Coqe{pow_fn_rev_Z} \begin{Coq} Definition step_pow_Z (a:Z) (c:Z) (g:Z) : Z := let c := c * c in if a <>? 1 && a <>? 3 then c * g else c. Function pow_fn_rev_Z (a:Z) (b:Z) (c:Z) (g: Z) {measure Z.to_nat a} : Z := if (a <=? 0) then c else let prev := pow_fn_rev_Z (a - 1) b c g in step_pow_Z (b - 1 - a) prev g. \end{Coq} % % In line with the definition of \Coqe{Inv25519}, we define the inversion modulo % $2^{255}-19$ as an instance of \Coqe{pow_fn_rev_Z} where \Coqe{a} and \Coqe{b} are 254. \subsubsection{A Simple Domain Specific Language} We show that for a property $P$, we can define a decidable boolean property $P_{bool}$ such that if $P_{bool}$ is \Coqe{true} then $P$ holds. $$reify\_P : P_{bool} = true \implies P$$ By applying $reify\_P$ on $P$ our goal become $P_{bool} = true$. We can then compute the result of $P_{bool}$. If the decision goes well we are left with the tautology $true = true$. To prove that the \Coqe{Inv25519_Z} is computing $x^{2^{255}-21}$, we define a Domain Specific Language: \begin{Coq} we define a Domain Specific Language. \begin{definition} Let \Coqe{expr_inv} denote an expression which can be either a term; a multiplication of expressions; a squaring of an expression or a power of an expression. And Let \Coqe{formula_inv} denote an equality between two expressions. \end{definition} \begin{lstlisting}[language=Coq] Inductive expr_inv := | R_inv : expr_inv | M_inv : expr_inv -> expr_inv -> expr_inv | S_inv : expr_inv -> expr_inv | P_inv : expr_inv -> positive -> expr_inv. \end{Coq} An expression can be either a term, a multiplication, a squaring or a power. This is denoted as follow: \begin{Coq} Inductive formula_inv := | Eq_inv : expr_inv -> expr_inv -> formula_inv. \end{lstlisting} The denote functions are defined as follows: \begin{lstlisting}[language=Coq] Fixpoint e_inv_denote (m:expr_inv) : Z := match m with | R_inv => ... ... @@ -201,37 +178,13 @@ Fixpoint e_inv_denote (m:expr_inv) : Z := | P_inv x p => pow (e_inv_denote x) (Z.pos p) end. \end{Coq} We defined \Coqe{step_inv} and \Coqe{pow_inv} to mirror the behavior of \Coqe{step_pow_Z} and respectively \Coqe{pow_fn_rev_Z} over our new domain. \begin{Coq} Lemma step_inv_step_pow_eq : forall (a:Z) (c:expr_inv) (g:expr_inv), e_inv_denote (step_inv a c g) = step_pow_Z a (e_inv_denote c) (e_inv_denote g). \end{Coq} \begin{Coq} Lemma pow_inv_pow_fn_rev_eq : forall (a:Z) (b:Z) (c:expr_inv) (g:expr_inv), e_inv_denote (pow_inv a b c g) = pow_fn_rev_Z a b (e_inv_denote c) (e_inv_denote g). \end{Coq} We also define what is a formula in our language: a simple equality. \begin{Coq} Inductive formula_inv := | Eq_inv : expr_inv -> expr_inv -> formula_inv. \end{Coq} This is denoted as follow: \begin{Coq} Definition f_inv_denote (t : formula_inv) : Prop := match t with | Eq_inv x y => e_inv_denote x = e_inv_denote y end. \end{Coq} With such Domain Specific Language we have the equivalence between: \end{lstlisting} With such Domain Specific Language we have the equality between: \begin{lstlisting}[backgroundcolor=\color{white}] f_inv_denote (Eq_inv (M_inv R_inv (S_inv R_inv)) ... ... @@ -239,22 +192,48 @@ f_inv_denote = (x * x^2 = x^3) \end{lstlisting} \subsubsection{Deciding formulas} We define \Coqe{step_inv} and \Coqe{pow_inv} to mirror the behavior of \Coqe{step_pow_Z} and respectively \Coqe{pow_fn_rev_Z} over our DSL and we prove their equality. \begin{lstlisting}[language=Coq] Lemma step_inv_step_pow_eq : forall (a:Z) (c:expr_inv) (g:expr_inv), e_inv_denote (step_inv a c g) = step_pow_Z a (e_inv_denote c) (e_inv_denote g). Lemma pow_inv_pow_fn_rev_eq : forall (a:Z) (b:Z) (c:expr_inv) (g:expr_inv), e_inv_denote (pow_inv a b c g) = pow_fn_rev_Z a b (e_inv_denote c) (e_inv_denote g). \end{lstlisting} We can then derive the following lemma. \begin{lemma} \label{reify} With an appropriate choice of variables, \Coqe{pow_inv} denotes \Coqe{Inv25519_Z}. \end{lemma} In order to prove formulas in \Coqe{formula_inv}, we define a decidable procedure. We first compute the power of each side of the formula and then check their equality. we have the following a decidable procedure. We define \Coqe{pow_expr_inv}, a function which returns the power of an expression. We can then compare the two values and decide over their equality. \begin{Coq} Fixpoint pow_expr_inv (t:expr_inv) : Z := match t with | R_inv => 1 (* power of a term is 1. *) | M_inv x y => (pow_expr_inv x) + (pow_expr_inv y) (* power of a multiplication is the sum of the exponents. *) | S_inv x => 2 * (pow_expr_inv x) (* power of a squaring is the double of the exponent. *) | P_inv x p => (Z.pos p) * (pow_expr_inv x) (* power of a power is the multiplication of the exponents. *) end. Definition decide_e_inv (l1 l2:expr_inv) : bool := ... ... @@ -265,20 +244,24 @@ Definition decide_f_inv (f:formula_inv) : bool := | Eq_inv x y => decide_e_inv x y end. \end{Coq} We proved that our procedure is correct: for all formulas in \Coqe{formula_inv}, if \Coqe{decide_f_inv} returns \Coqe{true}, then the denoted equality is correct. We prove our decision procedure correct. \begin{lemma} \label{decide} For all formulas $f$, if the decision over $f$ returns \Coqe{true}, then the denoted equality by $f$ is true. \end{lemma} Which can be formalized as: \begin{Coq} Lemma decide_formula_inv_impl : forall (f:formula_inv), decide_f_inv f = true -> f_inv_denote f. \end{Coq} By reification to our Domain Specific Language and then by applying \texttt{decide\_formula\_inv\_impl} and computing \texttt{decide\_f\_inv}, we proved that \Coqe{Inv25519} is indeed computing an inverse in modulo $2^{255}-19$. By reification to over DSL (lemma \ref{reify}) and by applying our decision (lemma \ref{decide}). we proved the following theorem. \begin{theorem} \Coqe{Inv25519_Z} computes an inverse in \Zfield. \end{theorem} \begin{Coq} Theorem Inv25519_Z_correct : forall (x:Z), ... ... @@ -287,7 +270,9 @@ Theorem Inv25519_Z_correct : From \Coqe{Inv25519_Z_correct} and \Coqe{Inv25519_Z_GF}, we conclude the functionnal correctness of the inversion over \Zfield. \begin{corollary} \Coqe{Inv25519} computes an inverse in \Zfield. \end{corollary} \begin{Coq} Corollary Inv25519_Zpow_GF : forall (g:list Z), ... ... @@ -296,18 +281,19 @@ Corollary Inv25519_Zpow_GF : (pow (Z16.lst g) (2^255-21)) :GF. \end{Coq} \subsubsection{Packing and other applications} \subsection{Packing and other applications of reflection} Reflection can also be used where proofs requires computing and a small and We prove the functional correctness of \Coqe{Inv25519} with reflections. This technique can also be used where proofs requires some computing or a small and finite domain of variable to test e.g. for all $i$ such that $0 \le i < 16$. Using reflections we proved that we can split the for loop in \TNaCle{pack25519} into two parts. Using reflection we prove that we can split the for loop in \TNaCle{pack25519} into two parts. \begin{lstlisting}[language=Ctweetnacl] for(i=1;i<15;i++) { m[i]=t[i]-0xffff-((m[i-1]>>16)&1); m[i-1]&=0xffff; } \end{lstlisting} The first is computing the substraction while the second is applying the carrying. The first loop is computing the substraction while the second is applying the carrying. \begin{lstlisting}[language=Ctweetnacl] for(i=1;i<15;i++) { m[i]=t[i]-0xffff ... ... @@ -318,8 +304,8 @@ for(i=1;i<15;i++) { } \end{lstlisting} This loop separation allows simpler proofs. The first loop is seen as the substraction of a number in \Zfield. We then proved than with the iteration of the second loop, the number represented in \Zfield stays the same. This lead to the proof that \TNaCle{pack25519} is effectively reducing mod $2^{255}-19$ and returning a number in base $2^8$. We then prove that with the iteration of the second loop, the number represented in \Zfield stays the same. This leads to the proof that \TNaCle{pack25519} is effectively reducing mod $\p$ and returning a number in base $2^8$. \begin{Coq} Lemma Pack25519_mod_25519 : ... ...
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