X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/0cafd2bf1c7bf4758d8f1aa902d08d880419b67d..568ccd6776e38446e0338d420f423c1d53aa4475:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 74b78a9..ff2d42a 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -996,9 +996,9 @@ tab1, tab2: Arrays containing combinations of size combination\_size\;}
   o2 = threadIdx-offset+tab2[offset]\;
   \For{i=1 to n} {
     t=xor-like()\;
-    t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\;
+    t=t $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;
     shared\_mem[threadId]=t\;
-    x = x $\oplus$ t\;
+    x = x $\wedge$ t\;
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
@@ -1118,15 +1118,15 @@ In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any
-seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
-$\ell_G(N)$ with $\ell_G(N)>N$.
+seed $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size
+$\ell_G(k)$ with $\ell_G(k)>k$.
 The notion of {\it secure} PRNGs can now be defined as follows. 
 
 \begin{definition}
 A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
 algorithm $D$, for any positive polynomial $p$, and for all sufficiently
 large $k$'s,
-$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(N)},$$
+$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$
 where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the
 probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the
 internal coin tosses of $D$. 
@@ -1230,56 +1230,55 @@ algorithm (Algorithm~\ref{algo:gpu_kernel2}).   Due to Proposition~\ref{cryptopr
 it simply consists  in replacing
 the  {\it  xor-like} PRNG  by  a  cryptographically  secure one.  
 We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form:
-$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers. These
-prime numbers  need to be congruent  to 3 modulus  4. BBS is
+$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers (these
+prime numbers  need to be congruent  to 3 modulus  4). BBS is known to be
 very slow and only usable for cryptographic applications. 
 
   
 The modulus operation is the most time consuming operation for current
 GPU cards.  So in order to obtain quite reasonable performances, it is
 required to use only modulus  on 32 bits integer numbers. Consequently
-$x_n^2$ need  to be less than $2^{32}$  and the number $M$  need to be
-less than $2^{16}$.  So in practice we can choose prime numbers around
+$x_n^2$ need  to be lesser than $2^{32}$,  and thus the number $M$ must be
+lesser than $2^{16}$.  So in practice we can choose prime numbers around
 256 that are congruent to 3 modulus 4.  With 32 bits numbers, only the
 4 least significant bits of $x_n$ can be chosen (the maximum number of
 indistinguishable    bits    is    lesser    than   or    equals    to
-$log_2(log_2(x_n))$). So to generate a  32 bits number, we need to use
-8 times  the BBS  algorithm with different  combinations of  $M$. This
-approach is  not sufficient to pass  all the tests  of TestU01 because
-the fact  of having chosen  small values of  $M$ for the BBS  leads to
-have a  small period. So, in  order to add randomness  we proceed with
+$log_2(log_2(M))$). In other words, to generate a  32 bits number, we need to use
+8 times  the BBS  algorithm with possibly different  combinations of  $M$. This
+approach is  not sufficient to be able to pass  all the TestU01,
+as small values of  $M$ for the BBS  lead to
+  small periods. So, in  order to add randomness  we proceed with
 the followings  modifications. 
 \begin{itemize}
 \item
-First we  define 16 arrangement arrays  instead of 2  (as described in
-algorithm \ref{algo:gpu_kernel2}) but only 2  are used at each call of
-the  PRNG kernels. In  practice, the  selection of  which combinations
-arrays will be used is different for all the threads and is determined
+Firstly, we  define 16 arrangement arrays  instead of 2  (as described in
+Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of
+the  PRNG kernels. In  practice, the  selection of   combinations
+arrays to be used is different for all the threads. It is determined
 by using  the three last bits  of two internal variables  used by BBS.
-This approach  adds more randomness.   In algorithm~\ref{algo:bbs_gpu},
-character  \& performs the  AND bitwise.  So using  \&7 with  a number
-gives the last 3 bits, so it provides a number between 0 and 7.
+%This approach  adds more randomness.   
+In Algorithm~\ref{algo:bbs_gpu},
+character  \& is for the  bitwise AND. Thus using  \&7 with  a number
+gives the last 3 bits, providing so a number between 0 and 7.
 \item
-Second, after the  generation of the 8 BBS numbers  for each thread we
-have a 32 bits number for which the period is possibly quite small. So
-to add randomness,  we generate 4 more BBS numbers  which allows us to
-shift  the 32 bits  numbers and  add upto  6 new  bits.  This  part is
-described  in algorithm~\ref{algo:bbs_gpu}.  In  practice, if  we call
-{\it strategy}, the number representing  the strategy, the last 2 bits
-of the first new BBS number are  used to make a left shift of at least
+Secondly, after the  generation of the 8 BBS numbers  for each thread, we
+have a 32 bits number whose period is possibly quite small. So
+to add randomness,  we generate 4 more BBS numbers   to
+shift  the 32 bits  numbers, and  add up to  6 new  bits.  This  improvement is
+described  in Algorithm~\ref{algo:bbs_gpu}.  In  practice, the last 2 bits
+of the first new BBS number are  used to make a left shift of at most
 3 bits. The  last 3 bits of the  second new BBS number are  add to the
 strategy whatever the value of the first left shift. The third and the
 fourth new BBS  numbers are used similarly to apply  a new left shift
 and add 3 new bits.
 \item
-Finally, as  we use 8 BBS numbers  for each thread, the  store of these
+Finally, as  we use 8 BBS numbers  for each thread, the  storage of these
 numbers at the end of the  kernel is performed using a rotation. So,
 internal  variable for  BBS number  1 is  stored in  place  2, internal
-variable  for BBS  number 2  is  store ind  place 3,  ... and  internal
+variable  for BBS  number 2  is  stored in  place 3,  ..., and finally, internal
 variable for BBS number 8 is stored in place 1.
 \end{itemize}
 
-
 \begin{algorithm}
 
 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
@@ -1305,9 +1304,9 @@ tab: 2D Arrays containing 16 combinations (in first dimension)  of size combinat
     t|=BBS1(bbs1)\&7\;
      t<<=BBS7(bbs7)\&3\;
     t|=BBS2(bbs2)\&7\;
-    t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\;
+    t=t $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;
     shared\_mem[threadId]=t\;
-    x = x $\oplus$ t\;
+    x = x $\wedge$ t\;
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
@@ -1318,18 +1317,100 @@ tab: 2D Arrays containing 16 combinations (in first dimension)  of size combinat
 \label{algo:bbs_gpu}
 \end{algorithm}
 
-In algorithm~\ref{algo:bbs_gpu}, t<<=4 performs a left shift of 4 bits
-on the variable  t and stores the result  in t. BBS1(bbs1)\&15 selects
-the last  four bits of the result  of BBS1. It should  be noticed that
-for the two new shifts, we use arbitrarily 4 BBSs that have previously
-been used.
+In Algorithm~\ref{algo:bbs_gpu}, $n$ is for the quantity
+of random numbers that a thread has to generate.
+The operation t<<=4 performs a left shift of 4 bits
+on the variable  $t$ and stores the result  in $t$, and 
+$BBS1(bbs1)\&15$ selects
+the last  four bits of the result  of $BBS1$. 
+Thus an operation of the form $t<<=4; t|=BBS1(bbs1)\&15\;$
+realizes in $t$ a left shift of 4 bits, and then puts
+the 4 last bits of $BBS1(bbs1)$ in the four last
+positions of $t$.
+Let us remark that to initialize $t$ is not a necessity as we
+fill it 4 bits by 4 bits, until having obtained 32 bits.
+The two last new shifts are realized in order to enlarge
+the small periods of the BBS used here, to introduce a variability.
+In these operations, we make twice a left shift of $t$ of \emph{at most}
+3 bits and we put \emph{exactly} the 3 last bits from a BBS into 
+the 3 last bits of $t$, leading possibly to a loss of a few 
+bits of $t$. 
+
+It should  be noticed that this generator has another time the form $x^{n+1} = x^n \oplus S^n$,
+where $S^n$ is referred in this algorithm as $t$: each iteration of this
+PRNG ends with $x = x \wedge t;$. This $S^n$ is only constituted
+by secure bits produced by the BBS generator, and thus, due to
+Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically
+secure
+
+
+
+\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}
 
 
-\subsection{A Secure Asymetric Cryptosystem}
+\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}
 
