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\section{Conclusion}
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\label{sec:Conclusion}
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Any formal system relies on a trusted base. In this section we describe our
chain of trust.
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\subheading{Trusted Code Base of the proof.}
Our proof relies on a trusted base, i.e. a foundation of definitions that must be
correct. One should not be able to prove a false statement in that system, \eg, by
proving an inconsistency.
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In our case we rely on:
\begin{itemize}
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  \item \textbf{Calculus of Inductive Constructions}. The intuitionistic logic
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  used by Coq must be consistent in order to trust the proofs. As an axiom,
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  we assume that the functional extensionality is also consistent with that logic.
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  $$\forall x, f(x) = g(x) \implies f = g$$
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\begin{lstlisting}[language=Coq]
Lemma f_ext: forall (A B:Type),
  forall (f g: A -> B),
  (forall x, f(x) = g(x)) -> f = g.
\end{lstlisting}

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  \item \textbf{Verifiable Software Toolchain}. This framework developed at
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  Princeton allows a user to prove that a Clight code matches pure Coq
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  specification.
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  \item \textbf{CompCert}.
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  When compiling with CompCert we only need to trust CompCert's {assembly}
  semantics, because it has been formally proven correct.
  However, when compiling with other C compilers like Clang or GCC, we need to
  trust that the CompCert's Clight semantics matches the C17 standard.
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  \item \textbf{\texttt{clightgen}}. The tool making the translation from {C} to
  {Clight}. It is the first step of the compilation.
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  VST does not support the direct verification of \texttt{o[i] = a[i] + b[i]}.
  This required us to rewrite the lines into:
\begin{lstlisting}[language=C]
aux1 = a[i];
aux2 = b[i];
o[i] = aux1 + aux2;
\end{lstlisting}
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  The trust of the proof relies on the trust of a correct translation from the
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  initial version of \emph{TweetNaCl} to \emph{TweetNaclVerifiableC}.
  \texttt{clightgen} comes with \texttt{-normalize} flag which
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  factors out function calls and assignments from inside subexpressions.
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  The changes required for C code to make it verifiable are now minimal.
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  \item Finally, we must trust the \textbf{Coq kernel} and its
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  associated libraries; the \textbf{Ocaml compiler} on which we compiled Coq;
  the \textbf{Ocaml Runtime} and the \textbf{CPU}. Those are common to all proofs
  done with this architecture \cite{2015-Appel,coq-faq}.
\end{itemize}
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\subheading{Corrections in TweetNaCl.}
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As a result of this verification, we removed superfluous code.
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Indeed indexes 17 to 79 of the \TNaCle{i64 x[80]} intermediate variable of
\TNaCle{crypto_scalarmult} were adding unnecessary complexity to the code,
we removed them.
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Peter Wu and Jason A. Donenfeld brought to our attention that the original
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WIP    
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\TNaCle{car25519} function carried a risk of undefined behavior if \texttt{c}
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is a negative number.
\begin{lstlisting}[language=Ctweetnacl]
c=o[i]>>16;
o[i]-=c<<16; // c < 0 = UB !
\end{lstlisting}
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We replaced this statement with a logical \texttt{and}, proved correctness,
and thus solved this problem.
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\begin{lstlisting}[language=Ctweetnacl]
o[i]&=0xffff;
\end{lstlisting}
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Aside from the modifications above mentioned, all subsequent alteration
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---such as the type change of loop indexes (\TNaCle{int} instead of \TNaCle{i64})---
were required for VST to parse properly the code. We believe those
adjustments do not impact the trust of our proof.

We contacted the authors of TweetNaCl and expect that the changes above
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mentioned will soon be integrated in a new version of the library.
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\subheading{Extending our work.}
The high-level definition (\sref{sec:maths}) can easily be ported to any
other Montgomery curves and with it the proof of the ladder's correctness
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assuming the same formulas are used.
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In addition to the curve equation, the field \F{p} would need to be redefined
as $p=2^{255}-19$ is hard-coded in order to speed up some proofs.

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With respect to the C code verification (\sref{sec:C-Coq}), the extension of
the verification effort to Ed25519 would makes directly use of the low level
arithmetic. The ladder steps formula being different this would require a high
level verification similar to \tref{thm:montgomery-ladder-correct}.

The verification \eg X448~\cite{cryptoeprint:2015:625,rfc7748} in C would
require the adaptation of most of the low level arithmetic (mainly the
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multiplication, carry propagation and reductions).
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Once the correctness and bounds of the basic operations are established,
reproving the full ladder would make use of our generic definition and lower
the workload.
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\subheading{A complete proof.}
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We provide a mechanized formal proof of the correctness of the X25519
implementation in TweetNaCl.
We first formalized X25519 from RFC~7748~\cite{rfc7748} in Coq. Then we proved
that TweetNaCl's implementation of X25519 matches our formalization.
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typos    
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In a second step we extended the Coq library for elliptic curves \cite{BartziaS14}
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by Bartzia and Strub to support Montgomery curves. Using this extension we
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proved that the X25519 from the RFC and therefore its implementation in TweetNaCl matches
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the mathematical definitions as given in~\cite[Sec.~2]{Ber06}.