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\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
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First of all, we would thank the reviewers for the detailled comments and suggestions for our paper.


\section*{Reviewer \#1:} 
\begin{enumerate}
\item ``In my view, this is an outstanding paper that reports on outstanding work. The detail provided is extensive, well 
thought out, and supports the authors' research thoroughly. In particular, I found the equations and algorithms to be understandable 
(albeit on the upper edge of my understanding... I'm a compiler optimization guy by training) and in keeping with the highly technical
 nature of PRNGs and the topic of RNGs in general. I'd suggest that the direction of future work be carefully planned based on 
technological trends. GPUs are becoming more and more the standard technology for parallel processing as they are widely available,
 cost effective and can get the job done. So, certainly including more work on larger scale GPU clusters would be well advised in addition
 to the other parallel frameworks mentioned.

Any specific comments I might make would only be minor quibbles with wording, and in this case it would simply be matters of writing
 style. It wouldn't hurt the paper to have it gone over once by an editor who hasn't seen it previously to perhaps catch any slight
 wording changes that might improve the overall style... but this paper is so strong anyway that if that didn't happen it wouldn't hurt either.

One minor piece that I didn't find addressed in the paper is the issue of how the seed is acquired. Isn't it always important for the
 seed to be sufficiently random and unpredictable, and if so, perhaps a mention about how important the seed is to the strength of a 
 RNG and any cryptographic technique should be added?
 
Excellent excellent work!''

\medskip
{\it In practice the seeds are generated using ISAAC which is a cryptographic PRNG. So the security of our seeds lies on the security of ISAAC. We think that this is enough to consider that our seeds are sufficiently secure.}

\end{enumerate}


\section*{Reviewer \#2:} 
\begin{enumerate}
\item ``The paper "Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU" presents a new
pseudo number generator comparing favorably with the CURAND library provided by NVIDIA.  The need for high quality
number generation is ubiquitous in  computer simulations based on stochastic methods and in cryptographic security.
It justifies the importance of the paper for readers of the journal.
The nice feature of the article is combined topological considerations on the notion of chaos with practical 
implementation of a working algorithm and its tests. Large parts of the paper are recapitulation of an earlier
work and may seem redundant for a specialist in the field, however an average reader may need it as a review 
on what has been done. The paper is quite clearly written in an understandable English so I recommend to publish 
it in principle as is. However there is one point that I would optionally recommend the authors to adrees.

In the paper I do not find information on the period of the resultant chaotic iteration pseudonumber
 generator compared to the periods of input  generators. It would be nice if it turned out that, among other 
features their procedure to combine a few weaker but fast generators into a much stronger and also fast one, yielded a much
longer period of the resultant generator compared to any of the input generators.''

\medskip
{\it We have added the following text in Section 4.2:\\
Obviously, when $S$ is periodic of period $p$, then $x$ is periodic too of
period either $p$ or $2p$, depending on the fact that, after $p$ iterations,
the state of the system may or not be the same than before these iterations.}
\end{enumerate}




\section*{Reviewer \#4:} 
\begin{enumerate}
\item ``Comments to Authors

This paper describes the mathematical basis for pseudorandom number generators PRNG and implements a GPU accelerated version 
for execution and measurements on two Nvidia based GPGPU systems.
The paper is very well written and has only a few minor spelling mistakes, although some grammar formulations could be 
reworked by a native speaker.''

\medskip
{\it The paper has been reread by a native speaker.}

\item ``The mathematical introductions are short and concise, enabling a reader to gather all relevant knowledge directly from this publication.
 This always raises the question of novelty of the paper, as new aspects are arising only at a later stage in the paper itself 
(i.e. the parallelisation part). Nevertheless, I find the introduction necessary to this extent.
While working through the paper a few minor questions arised:
-	page 7, Algorithm 3: is the increment of i in the while loop implicit, or where is it incremented?''

{\it In fact this is implicit with the ''while'' instruction.}

\item ``In Algorithm 5 and 6 threadIdx is written sometimes as threadId''

{\it We have uniformized that.}

 \item ``Page 19, Algorithm 6: there are 8 random numbers and their corresponding 8 states. You state that only a few of them are used, which is reflected in the algorithm itself. Are always the same ones used, or do you also rotate there?''

{\it   In practice, 8 BBS random numbers are used but for each of them only the four last bits are used. So we used $8 \times 4=32$ bits. If we chose a bigger number of BBS, the computation would slower. That is why we have used 8 BBS which seems to be a good trade-off.}
   
 \item ``In the references starting on page 21, some references need a bit more work. Probably BibTeX did some magic there and needs 
   to be tweaked manually. Example Ref [7] nist needs to be NIST.''

{\it   We have corrected that.}

   \item ``Another nice aspect to consider in future might be different accelerator-based systems such as Intel Xeon Phi cards and speed measurements
 using such cards. As supercomputers tend to get more and more heterogeneous (Tianhe-2, Stampede) using other accelerators than GPGPUs,
 a Xeon Phi solution might be very beneficial for the community.''

{\it Thank you for this remark, we have added that in the future work.}


  \item ``With the strong mathematical focus, the authors could also think about submitting this paper to a different, more 
mathematically oriented journal, as the focus on supercomputers and supercomputing capabilities is not as strong in this paper. 
Nevertheless, I recommend the paper for publication.''

{\it In fact there are different parts in this paper, we have chosen the Journal of Supercomputing because, for previous submissions to this journal, we had interesting reviews and because this journal has a wide audience and is of really high standards.}
\end{enumerate}




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