conclusion.tex 5.85 KB
 Peter Schwabe committed Jun 20, 2019 1 \section{Conclusion}  Benoit Viguier committed Aug 12, 2019 2 \label{sec:Conclusion}  Peter Schwabe committed Jun 20, 2019 3   Benoit Viguier committed Jul 10, 2019 4 5 Any formal system relies on a trusted base. In this section we describe our chain of trust.  Peter Schwabe committed Jun 20, 2019 6   Benoit Viguier committed Sep 02, 2019 7 8 \subheading{Trusted Code Base of the proof.} Our proof relies on a trusted base, i.e. a foundation of definitions that must be  Benoit Viguier committed Feb 15, 2020 9 correct. One should not be able to prove a false statement in that system, \eg by  Benoit Viguier committed Sep 02, 2019 10 proving an inconsistency.  Benoit Viguier committed Jul 10, 2019 11 12 13  In our case we rely on: \begin{itemize}  Benoit Viguier committed Sep 27, 2019 14  \item \textbf{Calculus of Inductive Constructions}. The intuitionistic logic  Benoit Viguier committed Jul 10, 2019 15  used by Coq must be consistent in order to trust the proofs. As an axiom,  Peter Schwabe committed Sep 29, 2019 16  we assume that the functional extensionality is also consistent with that logic.  Benoit Viguier committed Aug 16, 2019 17  $$\forall x, f(x) = g(x) \implies f = g$$  Benoit Viguier committed Feb 06, 2020 18 \begin{lstlisting}[language=Coq,belowskip=-0.25 \baselineskip]  Benoit Viguier committed Jul 10, 2019 19 20 21 22 23 Lemma f_ext: forall (A B:Type), forall (f g: A -> B), (forall x, f(x) = g(x)) -> f = g. \end{lstlisting}  Benoit Viguier committed Aug 01, 2019 24  \item \textbf{Verifiable Software Toolchain}. This framework developed at  Benoit Viguier committed Aug 21, 2019 25  Princeton allows a user to prove that a Clight code matches pure Coq  Benoit Viguier committed Sep 02, 2019 26  specification.  Benoit Viguier committed Jul 10, 2019 27   Benoit Viguier committed Feb 06, 2020 28  \item \textbf{CompCert}. When compiling with CompCert we only need to trust  Benoit Viguier committed Feb 15, 2020 29  CompCert's {assembly} semantics, as the compilation chain has been formally proven correct.  Benoit Viguier committed Jan 16, 2020 30 31  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 committed Jul 10, 2019 32   Freek Wiedijk committed Oct 01, 2019 33  \item \textbf{\texttt{clightgen}}. The tool making the translation from {C} to  Benoit Viguier committed Feb 06, 2020 34  {Clight}, the first step of the CompCert compilation.  Benoit Viguier committed Feb 15, 2020 35  VST does not support the direct verification of \texttt{o[i] = a[i] + b[i]}.  Benoit Viguier committed Feb 06, 2020 36 37 38  This needs to be rewritten into: \begin{lstlisting}[language=Ctweetnacl,stepnumber=0,belowskip=-0.5 \baselineskip] aux1 = a[i]; aux2 = b[i];  Benoit Viguier committed Jul 10, 2019 39 40 o[i] = aux1 + aux2; \end{lstlisting}  Benoit Viguier committed Feb 06, 2020 41  The \texttt{-normalize} flag is taking care of this  Benoit Viguier committed Feb 12, 2020 42  rewriting and factors out assignments from inside subexpressions.  Benoit Viguier committed Feb 06, 2020 43 44 45  % The trust of the proof relies on a correct translation from the % initial version of \emph{TweetNaCl} to \emph{TweetNaClVerifiableC}. % The changes required for C code to make it verifiable are now minimal.  Benoit Viguier committed Jul 10, 2019 46   Benoit Viguier committed Jan 16, 2020 47  \item Finally, we must trust the \textbf{Coq kernel} and its  Benoit Viguier committed Jul 10, 2019 48 49 50 51  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}  Peter Schwabe committed Jun 20, 2019 52   Benoit Viguier committed Sep 02, 2019 53 \subheading{Corrections in TweetNaCl.}  Benoit Viguier committed Aug 17, 2019 54 As a result of this verification, we removed superfluous code.  Benoit Viguier committed Sep 02, 2019 55 56 57 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.  Benoit Viguier committed Aug 17, 2019 58 59  Peter Wu and Jason A. Donenfeld brought to our attention that the original  Benoit Viguier committed Oct 01, 2019 60 \TNaCle{car25519} function carried a risk of undefined behavior if \texttt{c}  Benoit Viguier committed Aug 17, 2019 61 is a negative number.  Benoit Viguier committed Feb 06, 2020 62 \begin{lstlisting}[language=Ctweetnacl,stepnumber=0]  Benoit Viguier committed Aug 17, 2019 63 64 65 c=o[i]>>16; o[i]-=c<<16; // c < 0 = UB ! \end{lstlisting}  Peter Schwabe committed Sep 29, 2019 66 67 We replaced this statement with a logical \texttt{and}, proved correctness, and thus solved this problem.  Benoit Viguier committed Feb 06, 2020 68 \begin{lstlisting}[language=Ctweetnacl,stepnumber=0]  Benoit Viguier committed Aug 17, 2019 69 70 o[i]&=0xffff; \end{lstlisting}  Benoit Viguier committed Aug 12, 2019 71   Peter Schwabe committed Feb 15, 2020 72 73 Aside from this modifications, all subsequent alterations to the TweetNaCl code% ---such as the type change of loop indexes (\TNaCle{int} instead of \TNaCle{i64})---%  Benoit Viguier committed Feb 15, 2020 74 were required for VST to step through the code properly. We believe that those  Benoit Viguier committed Jan 17, 2020 75 76 adjustments do not impact the trust of our proof.  Peter Schwabe committed Feb 15, 2020 77 78 We contacted the authors of TweetNaCl and expect that the changes described above will soon be integrated in a new version of the library.  Benoit Viguier committed Jan 16, 2020 79   Benoit Viguier committed Feb 15, 2020 80 81 82 83 84 85  % Do we want to say that ? % \subheading{Verification Effort.} % In addition to the time required to get familiar with % research software, we faced a few bugs which we reported  Benoit Viguier committed Feb 15, 2020 86 % to the developers of VST to get them fixed.  Benoit Viguier committed Feb 15, 2020 87 88 89 90 91 92 93 % It is very hard to work with a tool without being involved % in the development loop. Additionally newer versions often % broke some of our proofs and it was often needed to adapt % to the changes. % As a result we do not believe the metric person-month to be % a good representation of the verification effort.  Peter Schwabe committed Oct 01, 2020 94 95 96 \subheading{Lessons learned.} \todo{Write something about VST etc.}  Benoit Viguier committed Jan 16, 2020 97 98 99 \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  Benoit Viguier committed Jan 17, 2020 100 assuming the same formulas are used.  Benoit Viguier committed Jan 16, 2020 101 102 103 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.  Benoit Viguier committed Jan 17, 2020 104 With respect to the C code verification (\sref{sec:C-Coq}), the extension of  Peter Schwabe committed Feb 15, 2020 105 the verification effort to Ed25519 would make directly use of the low-level  Benoit Viguier committed Jan 17, 2020 106 107 108 arithmetic. The ladder steps formula being different this would require a high level verification similar to \tref{thm:montgomery-ladder-correct}.  Benoit Viguier committed Feb 15, 2020 109 The verification of \eg X448~\cite{cryptoeprint:2015:625,rfc7748} in C would  Peter Schwabe committed Feb 15, 2020 110 require the adaptation of most of the low-level arithmetic (mainly the  Benoit Viguier committed Jan 17, 2020 111 multiplication, carry propagation and reduction).  Benoit Viguier committed Jan 16, 2020 112 Once the correctness and bounds of the basic operations are established,  Benoit Viguier committed Feb 15, 2020 113 reproving the full ladder would make use of our generic definition.  Benoit Viguier committed Aug 12, 2019 114   Benoit Viguier committed Sep 02, 2019 115 \subheading{A complete proof.}  Benoit Viguier committed Sep 21, 2019 116 We provide a mechanized formal proof of the correctness of the X25519  Benoit Viguier committed Feb 06, 2020 117 118 implementation in TweetNaCl from C up the mathematical definitions with a single tool. We first formalized X25519 from RFC~7748~\cite{rfc7748} in Coq.  Peter Schwabe committed Feb 15, 2020 119 We then proved that TweetNaCl's implementation of X25519 matches our formalization.  Benoit Viguier committed Sep 02, 2019 120 In a second step we extended the Coq library for elliptic curves \cite{BartziaS14}  Benoit Viguier committed Feb 06, 2020 121 122 123 124 125 126 by Bartzia and Strub to support Montgomery curves. Using this extension we proved that the X25519 from the RFC matches the mathematical definitions as given in~\cite[Sec.~2]{Ber06}. Therefore in addition to proving the mathematical correctness of TweetNaCl, we also increases the trust of other works such as \cite{zinzindohoue2017hacl,Erbsen2016SystematicSO} which rely on RFC~7748.