--- /dev/null
+@misc {Boneh-DeMillo-Lipton-attack,
+ author = "Boneh, Dan and DeMillo, Richard A. and Lipton, Richard J.",
+ title = "On the Importance of Eliminating Errors in Cryptographic Computations",
+ year = "1999",
+ howpublished = "\url{http://crypto.stanford.edu/~dabo/abstracts/faults.html}"
+}
+
+@misc {Barenghi-Breveglieri-Koren-Naccache-fault-injection,
+ author = "Barenghi, Alessandro and Breveglieri, Luca and Koren, Israel and Naccache, David",
+ title = "Fault Injection Attacks on Cryptographic Devices: Theory, Practice and Countermeasures",
+ year = "2012",
+ howpublished = "\url{http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=6178001}"
+}
+
+@misc {wiki-rsa,
+ author = "Wikipedia",
+ title = "RSA (cryptosystem) --- {W}ikipedia{,} The Free Encyclopedia",
+ year = "2014",
+ howpublished = "\url{http://en.wikipedia.org/wiki/RSA_%28cryptosystem%29}"
+ }
+
\ No newline at end of file
\usepackage[T1]{fontenc}
\usepackage{lmodern}
+\usepackage{amssymb,amsmath,amsthm}
+
\usepackage{graphicx}
\usepackage{listings}
\usepackage{url}
\subsubsection*{Question 1.3 : Quels sont les valeurs que l'on peut pré-calculer est stocker hormis $n$ et $d$ afin d'améliorer la vitesse de calcul d'une signature avec \emph{RSA-CRT} ?}
-[TODO]
+Les valeurs de $p$, $q$, $d_p$, $d_q$ et $q_{inv}$ sont mémorisées en tant que clef privée. Celles ci sont calculées comme suit.
+
+\begin{flalign*}
+ e &= 65537 \\
+ \mathbf{p, q} &&\text{deux nombres premiers de 512 bits choisis de manière aléatoire} \\
+ n &= p * q \\
+ \varphi(n) &= (p - 1) * (q - 1) \\
+ d &= e^{-1} ~(mod ~\varphi(n)) \\
+ \mathbf{d_p} &= d ~(mod ~p - 1) \\
+ \mathbf{d_q} &= d ~(mod ~q - 1) \\
+ \mathbf{q_{inv}} &= q^{-1} ~(mod p)
+\end{flalign*}
+
+
+La signature $sig$ du message $m$ peut être ensuite calculée comme suit.
+
+\begin{flalign*}
+ s_p &= m^{d_p} ~(mod ~p) &\\
+ s_q &= m^{d_q} ~(mod ~q) \\
+ \mathbf{sig} &= s_q + ((q_{inv} \cdot (s_p - s_q)) ~mod ~p) \cdot q
+\end{flalign*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fonctionnement}
-http://crypto.stanford.edu/~dabo/abstracts/faults.html
+D'après le document \cite{Boneh-DeMillo-Lipton-attack} :
-(maths)
+\begin{flalign*}
+ q &= gcd(m - sign'^e, n) &
+\end{flalign*}
+
+Où :
+
+\begin{itemize}
+ \item $m$ : le message signé avec $sign'$
+ \item $sign'$ : la signature calculé avec un $p$ altéré.
+\end{itemize}
+
+
+Nous pouvons alors facilement retrouver $p$:
+
+\begin{flalign*}
+ p &= n / q &
+\end{flalign*}
+
+Il est alors trivial de reconstituer la clef privée à partir de $p$ et $q$.
\subsubsection*{Question 2.1 : En pratique, comment est-il possible d'introduire des fautes dans l'implémentation d'un algorithme cryptographique ?}
+Voici une liste de techniques issues du document \cite{Barenghi-Breveglieri-Koren-Naccache-fault-injection} :
+\begin{itemize}
+ \item Variation du niveau de voltage de l'alimentation électrique ;
+ \item Injection d’irrégularités dans le \emph{clock} de l'horloge ;
+ \item Champs magnétique ;
+ \item Émission de radiations ;
+ \item Surchauffe de l'appareil ;
+ \item Exposition à une lumière intense.
+\end{itemize}
+
+
+\subsubsection*{Est-ce que cette attaque fonctionne dans le cas d'un bourrage non déterministe ?}
-\subsubsection*{Est-ce que cette attaque fonctionne dans le cas d'un bourrage non détérministe ?}
+
+
+
+\subsection{Implémentation}
+
+Cette attaque est illustrée dans la fonction \texttt{Tests::doAttack()}. Pour tester cette attaque :
+
+\begin{verbatim}
+qbs run -- attack
+\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
-% http://en.wikipedia.org/wiki/RSA_%28cryptosystem%29
+%
+% Fault Injection Attacks on Cryptographic Devices: Theory, Practice and Countermeasures
+
-%\bibliographystyle{plain}
-%\bibliography{main}
+\bibliographystyle{plain}
+\bibliography{main}
\end{document}
return sq + ((kPriv.qInv * (sp - sq)) % kPriv.p) * kPriv.q;
}
+
+mpz_class RsaCrt::signWithFaultySp(const mpz_class& m, const KeyPriv& kPriv)
+{
+ mpz_class sp, sq;
+
+ mpz_powm_sec(sp.get_mpz_t(), m.get_mpz_t(), kPriv.dp.get_mpz_t(), kPriv.p.get_mpz_t());
+ mpz_powm_sec(sq.get_mpz_t(), m.get_mpz_t(), kPriv.dq.get_mpz_t(), kPriv.q.get_mpz_t());
+
+ mpz_combit(sp.get_mpz_t(), 42); // Flip the fourty second bit.
+
+ return sq + ((kPriv.qInv * (sp - sq)) % kPriv.p) * kPriv.q;
+}
+
+
mpz_class qInv;
};
+ /**
+ * Generate a pair of keys (public, private).
+ */
static std::pair<Rsa::KeyPub, KeyPriv> generateRSAKeys(uint exponent, uint keySizeBits);
/**
* m must not be greater or equal than kPriv.n.
*/
static mpz_class sign(const mpz_class& m, const KeyPriv& kPriv);
+
+ static mpz_class signWithFaultySp(const mpz_class& m, const KeyPriv& kPriv);
};
#endif
cout << "Speedup: " << (double(timeRsaStd) / double(timeRsaCRT)) << endl;
}
+void Tests::doAttack()
+{
+ const auto& keys = RsaCrt::generateRSAKeys(RSA_PUBLIC_EXPONENT, KEY_SIZE_BITS);
+ const auto& kPub = keys.first;
+ const auto& kPriv = keys.second;
+ mpz_class message = Rand::randSize(128);
+ mpz_class faultySignature = RsaCrt::signWithFaultySp(message, kPriv);
+ mpz_class correctSignature = RsaCrt::sign(message, kPriv);
+
+ bool attackOK = true;
+
+ cout << "Original:" << endl;
+ cout << " p = " << kPriv.p << endl;
+ cout << " q = " << kPriv.q << endl;
+
+ // At this point the attacker doesn't know the private key but he has intercepted the message and the faulty signature.
+ {
+ mpz_class faultySignaturePowerE;
+ mpz_pow_ui(faultySignaturePowerE.get_mpz_t(), faultySignature.get_mpz_t(), RSA_PUBLIC_EXPONENT);
+ mpz_class messageMinuxFaultySignaturePowerE = message - faultySignaturePowerE;
+ mpz_class q;
+ mpz_gcd(q.get_mpz_t(), messageMinuxFaultySignaturePowerE.get_mpz_t(), kPub.n.get_mpz_t());
+ mpz_class p = kPub.n / q;
+
+ cout << "Found with a faulty signature:" << endl;
+ cout << " p = " << p << endl;
+ cout << " q = " << q << endl;
+
+ attackOK = attackOK && kPriv.p == p && kPriv.q == q; // With p and q we can recreate the original private key.
+ }
+
+ // Try the attack with a correct signature.
+ {
+ mpz_class correctSignaturePowerE;
+ mpz_pow_ui(correctSignaturePowerE.get_mpz_t(), correctSignature.get_mpz_t(), RSA_PUBLIC_EXPONENT);
+ mpz_class messageMinuxCorrectSignaturePowerE = message - correctSignaturePowerE;
+ mpz_class q;
+ mpz_gcd(q.get_mpz_t(), messageMinuxCorrectSignaturePowerE.get_mpz_t(), kPub.n.get_mpz_t());
+ mpz_class p = kPub.n / q;
+
+ cout << "Found with a correct signature:" << endl;
+ cout << " p = " << p << endl; // Equal to 1.
+ cout << " q = " << q << endl; // Equal to n.
+
+ attackOK = attackOK && kPriv.p != p && kPriv.q != q;
+ }
+
+ if (attackOK)
+ cout << "Attack successful" << endl;
+ else
+ cout << "Attack failed" << endl;
+}
+
bool Tests::rsaStandard()
{
const auto& keys = RsaStd::generateRSAKeys(RSA_PUBLIC_EXPONENT, KEY_SIZE_BITS);
void runTests();
void runTimeMeasures();
+ void doAttack();
private:
bool rsaStandard();
void printUsage(const string& progName)
{
- cout << "Usage: " << progName << " [tests|time-measures]" << endl;
-
-// mpz_class n = 10;
-// mpz_class d = 3;
-// mpz_class q;
-// mpz_fdiv_q(q.get_mpz_t(), n.get_mpz_t(), d.get_mpz_t());
-
-// cout << "q: " << q << endl;
-// cout << "q: " << (n / d) << endl;
+ cout << "Usage: " << progName << " <command>" << endl;
+ cout << " <command> can be one of the following:" << endl;
+ cout << " * tests: Do some tests for RSA and RSA-CRT" << endl;
+ cout << " * time-measures: Compute the ratio between RSA and RSA-CRT" << endl;
+ cout << " * attack: Simulate the Boneh-DeMillo-Lipton attack against RSA-CRT" << endl;
+ cout << " * attack-fixed: [TODO]" << endl;
}
int main(int argc, char** argv)
Tests(KEY_SIZE_BITS, RSA_PUBLIC_EXPONENT).runTests();
else if (args.size() >= 2 && args[1] == "time-measures")
Tests(KEY_SIZE_BITS, RSA_PUBLIC_EXPONENT).runTimeMeasures();
+ else if (args.size() >= 2 && args[1] == "attack")
+ Tests(KEY_SIZE_BITS, RSA_PUBLIC_EXPONENT).doAttack();
else
printUsage(args[0]);
projects / crypto_lab3.git / commitdiff