From 8b7e53afd367a0c112278bfc8813328807c087fa Mon Sep 17 00:00:00 2001 From: Jean-Louis Huynen Date: Fri, 6 Dec 2019 10:57:17 +0100 Subject: [PATCH] add: [workshop] more content for SOC --- .../hands-on-support/CREDIT.md | 8 + .../workshop/5-snake-oil-crypto/rsabasics.tex | 176 ++++++++++++++++++ docs/workshop/5-snake-oil-crypto/soc.pdf | Bin 904692 -> 904561 bytes 3 files changed, 184 insertions(+) create mode 100644 docs/workshop/5-snake-oil-crypto/rsabasics.tex diff --git a/docs/workshop/5-snake-oil-crypto/hands-on-support/CREDIT.md b/docs/workshop/5-snake-oil-crypto/hands-on-support/CREDIT.md index d6ae89d..3729378 100644 --- a/docs/workshop/5-snake-oil-crypto/hands-on-support/CREDIT.md +++ b/docs/workshop/5-snake-oil-crypto/hands-on-support/CREDIT.md @@ -1,4 +1,12 @@ * Parts of this material (the keys) has been borrowed to Sjoerd Langkemper https://www.sjoerdlangkemper.nl/2019/06/19/attacking-rsa/ + * Some Algorithms were borrowed from Daniel J. Bersntein's website: http://facthacks.cr.yp.to/fermat.html + +* The "shared prime" key material comes from the challenge hosted here: +http://www.loyalty.org/~schoen/rsa/ + +* Pictures +https://www.freecodecamp.org/news/how-does-pretty-good-privacy-work-3f5f75ecea97/ +https://upload.wikimedia.org/wikipedia/commons/4/4e/Web_of_Trust-en.svg diff --git a/docs/workshop/5-snake-oil-crypto/rsabasics.tex b/docs/workshop/5-snake-oil-crypto/rsabasics.tex new file mode 100644 index 0000000..c905269 --- /dev/null +++ b/docs/workshop/5-snake-oil-crypto/rsabasics.tex @@ -0,0 +1,176 @@ +\begin{frame} + \begin{center} + {\bf Understanding RSA} + \end{center} +\end{frame} + +\begin{frame} + \frametitle{RSA Basics} + Ron {\bf R}ivest, Adi {\bf S}hamir, and Leonard {\bf A}dleman in 1977: + \begin{itemize} + \item asymmetric crypto system, + \item can encrypt and sign, + \item messages are big numbers, + \item encryption is basically multiplication of big numbers, + \item creates a \textit{trapdoor permutation}: turning x in y is easy, but + finding x from y is hard. + \end{itemize} + +\end{frame} + +\begin{frame}[fragile] + \frametitle{RSA - Use with openssl} + \begin{itemize} + \item {\bf Hands-on}: + +\begin{lstlisting} +~/hands-on/UsingRSA +\end{lstlisting} + + \item Decrypt message.bin + \item generate a new private key, + \item generate the corresponding public key, + \item use this new key to encrypt a message, + \item use this new key to decrypt a message. + + \end{itemize} +\end{frame} + + +\begin{frame}[fragile] + \frametitle{RSA ``by hand''} +run: sage rsa.sage at the folder's root: +\begin{lstlisting}[basicstyle=\tiny] +PlainText is: 1234567890 +p = random_prime(2^32) = 2312340619 +q = random_prime(2^32) = 2031410981 +n = p*q = 4697314125248937239 +phi = (p-1)*(q-1) = 4697314120905185640 +e = random_prime(phi) = 2588085603940229747 +d = xgcd(e,phi)[1] = -2102894211931680277 +Does d*e == 1? + mod(d*e, phi) = 1 +CipherText y = power_mod(x, e, n) = 1454606910711062745 +Decrypted CT is: 1234567890 +\end{lstlisting} + +\end{frame} + +\begin{frame} + \frametitle{With only one key} + Several potential weaknesses: + \begin{itemize} + \item Key size too small: keys up to 1024 bits are breakable given the + right means, + \item close p and q, + \item unsafe primes, smooth primes, + \item broken primes (FactorDB, Debian OpenSSL bug). + \item signing with RSA-CRT (instead of RSA-PSS) + \end{itemize} + +\end{frame} + +\begin{frame} + \frametitle{With a set of keys} + Several potential weaknesses: + \begin{itemize} + \item share moduli: if n1 = n2 then the keys share p and q, + \item share p or q, + \end{itemize} + \vspace{10mm} + {\bf In both case, it is trivial to recover the private keys.} +\end{frame} + + +\begin{frame}[fragile] + \frametitle{Breaking small keys\footnote{https://www.sjoerdlangkemper.nl/2019/06/19/attacking-rsa/}} + \begin{itemize} +\item {\bf Hands-on}: + +\begin{lstlisting} +~/hands-on/SmallKey +\end{lstlisting} + + \item what is the key size of smallkey? + \item what is n? + \item what is the public exponent? + \item what is n in base10? + \item what are p and q? + + \end{itemize} + + \vspace{8mm} + {\bf Let's generate the private key: }using p, then using q. + +\end{frame} + +\begin{frame}[fragile] + \frametitle{Close Prime Factors} + \begin{itemize} +\item {\bf Hands-on}: + +\begin{lstlisting} +~/hands-on/ClosePQ +\end{lstlisting} + + \item use Fermat Algorithm\footnote{\url{http://facthacks.cr.yp.to/fermat.html}} to find {\bf both p and q:} + +\begin{lstlisting}[basicstyle=\tiny] +def fermatfactor(N): + if N <= 0: return [N] + if is_even(N): return [2,N/2] + a = ceil(sqrt(N)) + while not is_square(a^2-N): + a = a + 1 + b = sqrt(a^2-N) + return [a - b,a + b] +\end{lstlisting} + + \end{itemize} + +\end{frame} + +\begin{frame}[fragile] + \frametitle{Shared prime factors} + Researchers have shown that several devices generated their keypairs + at boot time without enough entropy\footnote{Bernstein, Heninger, and Lange: \url{http://facthacks.cr.yp.to/}}: + +\begin{lstlisting}[language=python, basicstyle=\tiny] +prng.seed(seed) +p = prng.generate_random_prime() +// prng.add_entropy() +q = prng.generate_random_prime() +n = p*q +\end{lstlisting} + +Given n=pq and n' = pq' it is trivial to recover the shared p by computing their +{\bf Greatest Common Divisor (GCD)}, and therefore {\bf both private + keys}\footnote{\url{http://www.loyalty.org/~schoen/rsa/}}.\\ +\vspace{5mm} +``They cracked cracked about 13000 of them'' +\end{frame} + +\begin{frame}[fragile] + \frametitle{Shared prime factors} + \begin{itemize} +\item {\bf Hands-on}: + +\begin{lstlisting} +~/hands-on/SharedPrimeFactor +\end{lstlisting} + +\item Read README.txt, you have a challenge to solve : + + \begin{itemize} + \item the \emph{answers} folder should be left alone for now, + \item \emph{scripts} contains scripts that may be useful + to solve the challenge, + \item \emph{attempts} may hold your attempt are + generating private keys. + \item \emph{bgcd-bd.sage} contains Daniel J. 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