6-conclusion.tex 4.98 KB
Newer Older
1
\section{Conclusion}
Benoit Viguier's avatar
Benoit Viguier committed
2
\label{sec:Conclusion}
3

Benoit Viguier's avatar
Benoit Viguier committed
4
5
Any formal system relies on a trusted base. In this section we describe our
chain of trust.
6

7
8
9
10
\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.
Benoit Viguier's avatar
Benoit Viguier committed
11
12
13

In our case we rely on:
\begin{itemize}
14
  \item \textbf{Calculus of Inductive Constructions}. The intuitionistic logic
Benoit Viguier's avatar
Benoit Viguier committed
15
  used by Coq must be consistent in order to trust the proofs. As an axiom,
16
  we assume that the functional extensionality is also consistent with that logic.
Benoit Viguier's avatar
Benoit Viguier committed
17
  $$\forall x, f(x) = g(x) \implies f = g$$
Benoit Viguier's avatar
Benoit Viguier committed
18
19
20
21
22
23
\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}

Benoit Viguier's avatar
Benoit Viguier committed
24
  \item \textbf{Verifiable Software Toolchain}. This framework developed at
Benoit Viguier's avatar
Benoit Viguier committed
25
  Princeton allows a user to prove that a Clight code matches pure Coq
26
  specification.
Benoit Viguier's avatar
Benoit Viguier committed
27

Freek Wiedijk's avatar
Freek Wiedijk committed
28
  \item \textbf{CompCert}.
29
30
31
32
  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.
Benoit Viguier's avatar
Benoit Viguier committed
33

Freek Wiedijk's avatar
Freek Wiedijk committed
34
35
  \item \textbf{\texttt{clightgen}}. The tool making the translation from {C} to
  {Clight}. It is the first step of the compilation.
Benoit Viguier's avatar
Benoit Viguier committed
36
37
38
39
40
41
42
  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}
43
  The trust of the proof relies on the trust of a correct translation from the
44
45
  initial version of \emph{TweetNaCl} to \emph{TweetNaclVerifiableC}.
  \texttt{clightgen} comes with \texttt{-normalize} flag which
46
  factors out function calls and assignments from inside subexpressions.
47
  The changes required for C code to make it verifiable are now minimal.
Benoit Viguier's avatar
Benoit Viguier committed
48

49
  \item Finally, we must trust the \textbf{Coq kernel} and its
Benoit Viguier's avatar
Benoit Viguier committed
50
51
52
53
  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}
54

Benoit Viguier's avatar
Benoit Viguier committed
55
56
\todo{NEW}

57
\subheading{Corrections in TweetNaCl.}
Benoit Viguier's avatar
Benoit Viguier committed
58
As a result of this verification, we removed superfluous code.
59
60
61
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.
62
63

Peter Wu and Jason A. Donenfeld brought to our attention that the original
Benoit Viguier's avatar
WIP    
Benoit Viguier committed
64
\TNaCle{car25519} function carried a risk of undefined behavior if \texttt{c}
65
66
67
68
69
is a negative number.
\begin{lstlisting}[language=Ctweetnacl]
c=o[i]>>16;
o[i]-=c<<16; // c < 0 = UB !
\end{lstlisting}
70
71
We replaced this statement with a logical \texttt{and}, proved correctness,
and thus solved this problem.
72
73
74
\begin{lstlisting}[language=Ctweetnacl]
o[i]&=0xffff;
\end{lstlisting}
Benoit Viguier's avatar
Benoit Viguier committed
75

76
\todo{NEW}
77

78
79
80
81
82
83
Aside from the modications above mentionned, all subsequent alteration
---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
84
85
86
87
88
89
90
91
92
mentionned will soon be integrated in a new version of the library.

\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
assuming the same forumlas are used.
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.

93
94
95
96
97
98
99
100
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
multiplication, carry propagations and reductions).
101
102
103
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.
Benoit Viguier's avatar
Benoit Viguier committed
104

105
\subheading{A complete proof.}
Benoit Viguier's avatar
Benoit Viguier committed
106
107
108
109
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.
Benoit Viguier's avatar
typos    
Benoit Viguier committed
110
In a second step we extended the Coq library for elliptic curves \cite{BartziaS14}
111
by Bartzia and Strub to support Montgomery curves. Using this extension we
112
proved that the X25519 from the RFC and therefore its implementation in TweetNaCl matches
Benoit Viguier's avatar
Benoit Viguier committed
113
the mathematical definitions as given in~\cite[Sec.~2]{Ber06}.