X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/f3efb657ffed3b779ad71dcf80a7149d586f1a38..28690929433ca34390a326790df02387bbae7c6e:/prng_gpu.tex?ds=inline diff --git a/prng_gpu.tex b/prng_gpu.tex index d7e4664..7629e10 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1326,10 +1326,72 @@ been used. -\subsection{A Secure Asymetric Cryptosystem} +\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem} +We finish this research work by giving some thoughts about the use of +the proposed PRNG in an asymmetric cryptosystem. +This first approach will be further investigated in a future work. +\subsubsection{Recalls of the Blum-Goldwasser Probabilistic Cryptosystem} +The Blum-Goldwasser cryptosystem is a cryptographically secure asymmetric key encryption algorithm +proposed in 1984~\cite{Blum:1985:EPP:19478.19501}. The encryption algorithm +implements a XOR-based stream cipher using the BBS PRNG, in order to generate +the keystream. Decryption is done by obtaining the initial seed thanks to +the final state of the BBS generator and the secret key, thus leading to the + reconstruction of the keystream. + +The key generation consists in generating two prime numbers $(p,q)$, +randomly and independently of each other, that are + congruent to 3 mod 4, and to compute the modulus $N=pq$. +The public key is $N$, whereas the secret key is the factorization $(p,q)$. + + +Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Alice: +\begin{enumerate} +\item Bob picks an integer $r$ randomly in the interval $\llbracket 1,N\rrbracket$ and computes $x_0 = r^2~mod~N$. +\item He uses the BBS to generate the keystream of $L$ pseudorandom bits $(b_0, \dots, b_{L-1})$, as follows. For $i=0$ to $L-1$, +\begin{itemize} +\item $i=0$. +\item While $i \leqslant L-1$: +\begin{itemize} +\item Set $b_i$ equal to the least-significant\footnote{BBS can securely output up to $\mathsf{N} = \lfloor log(log(N)) \rfloor$ of the least-significant bits of $x_i$ during each round.} bit of $x_i$, +\item $i=i+1$, +\item $x_i = (x_{i-1})^2~mod~N.$ +\end{itemize} +\end{itemize} +\item The ciphertext is computed by XORing the plaintext bits $m$ with the keystream: $ c = (c_0, \dots, c_{L-1}) = m \oplus b$. This ciphertext is $[c, y]$, where $y=x_{0}^{2^{L}}~mod~N.$ +\end{enumerate} + + +When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ as follows: +\begin{enumerate} +\item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$. +\item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$. +\item She recomputes the bit-vector $b$ by using BBS and $x_0$. +\item Alice computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. +\end{enumerate} + + +\subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser} + +We propose to adapt the Blum-Goldwasser protocol as follows. +Let $\mathsf{N} = \lfloor log(log(N)) \rfloor$ be the number of bits that can +be obtained securely with the BBS generator using the public key $N$ of Alice. +Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too, and +her new public key will be $(S^0, N)$. + +To encrypt his message, Bob will compute +\begin{equation} +c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right) +\end{equation} +instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. + +The same decryption stage as in Blum-Goldwasser leads to the sequence +$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$. +Thus, with a simple use of $S^0$, Alice can obtained the plaintext. +By doing so, the proposed generator is used in place of BBS, leading to +the inheritance of all the properties presented in this paper. \section{Conclusion}