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\subsection{Applying the Verifiable Software Toolchain}
\label{subsec:with-VST}

\begin{sloppypar}
We now turn our focus to the formal specification of \TNaCle{crypto_scalarmult}.
We use our definition of X25519 from the RFC in the Hoare triple and prove
its correctness.
\end{sloppypar}

\subheading{Specifications.}
We show the soundness of TweetNaCl by proving a correspondence between
the C version of TweetNaCl and the same code as a pure Coq function.
% why "pure" ?
% 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)
This defines the equivalence between the Clight representation and our Coq
definition of the ladder (\coqe{RFC}).

\begin{lstlisting}[language=CoqVST]
Definition crypto_scalarmult_spec :=
DECLARE _crypto_scalarmult_curve25519_tweet
WITH
  v_q: val, v_n: val, v_p: val, c121665:val,
  sh : share,
  q : list val, n : list Z, p : list Z
(*------------------------------------------*)
PRE [ _q OF (tptr tuchar),
     _n OF (tptr tuchar),
     _p OF (tptr tuchar) ]
PROP (writable_share sh;
      Forall (fun x => 0 <= x < 2^8) p;
      Forall (fun x => 0 <= x < 2^8) n;
      Zlength q = 32; Zlength n = 32;
      Zlength p = 32)
LOCAL(temp _q v_q; temp _n v_n; temp _p v_p;
      gvar __121665 c121665)
SEP  (sh [{ v_q }] <<(uch32)-- q;
      sh [{ v_n }] <<(uch32)-- mVI n;
      sh [{ v_p }] <<(uch32)-- mVI p;
      Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665)
(*------------------------------------------*)
POST [ tint ]
PROP (Forall (fun x => 0 <= x < 2^8) (RFC n p);
      Zlength (RFC n p) = 32)
LOCAL(temp ret_temp (Vint Int.zero))
SEP  (sh [{ v_q }] <<(uch32)-- mVI (RFC n p);
      sh [{ v_n }] <<(uch32)-- mVI n;
      sh [{ v_p }] <<(uch32)-- mVI p;
      Ews [{ c121665 }] <<(lg16)-- mVI64 c_121665
\end{lstlisting}

In this specification we state preconditions like:
\begin{itemize}
  \item[] \VSTe{PRE}: \VSTe{_p OF (tptr tuchar)}\\
  The function \TNaCle{crypto_scalarmult} takes as input three pointers to
  arrays of unsigned bytes (\VSTe{tptr tuchar}) \VSTe{_p}, \VSTe{_q} and \VSTe{_n}.
  \item[] \VSTe{LOCAL}: \VSTe{temp _p v_p}\\
  Each pointer represent an address \VSTe{v_p},
  \VSTe{v_q} and \VSTe{v_n}.
  \item[] \VSTe{SEP}: \VSTe{sh [{ v_p $\!\!\}\!\!]\!\!\!$ <<(uch32)-- mVI p}\\
  In the memory share \texttt{sh}, the address \VSTe{v_p} points
  to a list of integer values \VSTe{mVI p}.
  \item[] \VSTe{PROP}: \VSTe{Forall (fun x => 0 <= x < 2^8) p}\\
  In order to consider all the possible inputs, we assume each
  element of the list \texttt{p} to be bounded by $0$ included and $2^8$
  excluded.
  \item[] \VSTe{PROP}: \VSTe{Zlength p = 32}\\
  We also assume that the length of the list \texttt{p} is 32. This defines the
  complete representation of \TNaCle{u8[32]}.
\end{itemize}

As postcondition we have conditions like:
\begin{itemize}
  \item[] \VSTe{POST}: \VSTe{tint}\\
  The function \TNaCle{crypto_scalarmult} returns an integer.
  \item[] \VSTe{LOCAL}: \VSTe{temp ret_temp (Vint Int.zero)}\\
  The returned integer has value $0$.
  \item[] \VSTe{SEP}: \VSTe{sh [{ v_q $\!\!\}\!\!]\!\!\!$ <<(uch32)-- mVI (RFC n p)}\\
  In the memory share \texttt{sh}, the address \VSTe{v_q} points
  to a list of integer values \VSTe{mVI (RFC n p)} where \VSTe{RFC n p} is the
  result of the \TNaCle{crypto_scalarmult} of \VSTe{n} and \VSTe{p}.
  \item[] \VSTe{PROP}: \VSTe{Forall (fun x => 0 <= x < 2^8) (RFC n p)}\\
  \VSTe{PROP}: \VSTe{Zlength (RFC n p) = 32}\\
  We show that the computation for \VSTe{RFC} fits in  \TNaCle{u8[32]}.
\end{itemize}

computes the same result as \VSTe{RFC} in Coq provided that inputs are within
their respective bounds: arrays of 32 bytes.

The correctness of this specification is formally proven in Coq with
\coqe{Theorem body_crypto_scalarmult}.

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% \begin{sloppypar}
% This specification (proven with VST) shows that \TNaCle{crypto_scalarmult} in C.
% \end{sloppypar}
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% The Verifiable Software Toolchain uses a strongest postcondition strategy.
% The user must first write a formal specification of the function he wants to verify in Coq.
% This should be as close as possible to the C implementation behavior.
% This will simplify the proof and help with stepping through the Clight version of the software.
% With the range of inputs defined, VST mechanically steps through each instruction
% and ask the user to verify auxiliary goals such as array bound access, or absence of overflows/underflows.
% We call this specification a low level specification. A user will then have an easier
% time to prove that his low level specification matches a simpler higher level one.

% In order to further speed-up the verification process, it has to be know that to
% prove the specification \TNaCle{crypto_scalarmult}, a user only need the specification of e.g. \TNaCle{M}.
% This provide with multiple advantages: the verification by the Coq kernel can be done
% in parallel and multiple users can work on proving different functions at the same time.
% For the sake of completeness we proved all intermediate functions.

\subheading{Memory aliasing.}
The semicolon in the \VSTe{SEP} parts of the Hoare triples represents the \emph{separating conjunction} (often written as a star), which means that
the memory shares of \texttt{q}, \texttt{n} and \texttt{p} do not overlap.
In other words,
we only prove correctness of \TNaCle{crypto_scalarmult} when it is called without aliasing.
But for other TweetNaCl functions, like the multiplication function \texttt{M(o,a,b)}, we cannot ignore aliasing, as it is called in the ladder in an aliased manner.

In the VST, a simple specification of this function will assume that the pointer arguments
point to non-overlapping space in memory.
When called with three memory fragments (\texttt{o, a, b}),
the three of them will be consumed. However assuming this naive specification
when \texttt{M(o,a,a)} is called (squaring), the first two memory areas (\texttt{o, a})
are consumed and the VST will expect a third memory section (\texttt{a}) which does not \emph{exist} anymore.
Examples of such cases are illustrated in \fref{tikz:MemSame}.
\begin{figure}[h]%
  \centering%
  \include{tikz/memory_same_sh}%
  \caption{Aliasing and Separation Logic}%
  \label{tikz:MemSame}%
\end{figure}
As a result, a function must either have multiple specifications or specify which
aliasing case is being used.
The first option would require us to do very similar proofs multiple times for a same function.
We chose the second approach: for functions with 3 arguments, named hereafter \texttt{o, a, b},
we define an additional parameter $k$ with values in $\{0,1,2,3\}$:
\begin{itemize}
  \item if $k=0$ then \texttt{o} and \texttt{a} are aliased.
  \item if $k=1$ then \texttt{o} and \texttt{b} are aliased.
  \item if $k=2$ then \texttt{a} and \texttt{b} are aliased.
  \item else there is no aliasing.
\end{itemize}
In the proof of our specification, we do a case analysis over $k$ when needed.
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This solution does not generate all the possible cases of aliasing over 3 pointers
(\eg \texttt{o} = \texttt{a} = \texttt{b}) but it is enough to cover our needs.
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\subheading{Improving speed.}
To make the verification the smoothest, the Coq formal definition of the function
should be as close as possible to the C implementation behavior.
Optimizations of such definitions are often counter-productive as they increase the
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amount of proofs required for \eg bounds checking, loops invariants.
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In order to further speed-up the verification process, to prove the specification
\TNaCle{crypto_scalarmult}, we only need the specification of the subsequently
called functions (\eg \TNaCle{M}).
This provide with multiple advantages: the verification by the Coq kernel can be
done in parallel and multiple users can work on proving different functions at
the same time. For the sake of completeness we proved all intermediate functions.