Relecture du prng cryptosur

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 7629e10fa7f50075e8cfed1877352ce68cc46f2f..ff2d42a110bfe8a8b3b45363cbc0922d4b7f0eb4 100644 (file)
--- 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()\;
   o2 = threadIdx-offset+tab2[offset]\;
   \For{i=1 to n} {
     t=xor-like()\;

-    t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\;
+    t=t $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;

     shared\_mem[threadId]=t\;
     shared\_mem[threadId]=t\;

-    x = x $\hat{ }$ t\;
+    x = x $\wedge$ t\;

 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
 
     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
 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,
 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 $D$. 
 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:
 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
 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
 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
 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.
 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
 \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
 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
 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}
 
 variable for BBS number 8 is stored in place 1.
 \end{itemize}
 

-

 \begin{algorithm}
 
 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 \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|=BBS1(bbs1)\&7\;
      t<<=BBS7(bbs7)\&3\;
     t|=BBS2(bbs2)\&7\;

-    t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\;
+    t=t $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;

     shared\_mem[threadId]=t\;
     shared\_mem[threadId]=t\;

-    x = x $\hat{ }$ t\;
+    x = x $\wedge$ t\;

 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }


@@ -1318,11 +1317,31 @@ tab: 2D Arrays containing 16 combinations (in first dimension)  of size combinat
 \label{algo:bbs_gpu}
 \end{algorithm}
 
 \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