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8 \textit{As the reviewers point out, the paper is well written, is interesting, but there are some major concerns about both the practical aspects of the paper, as well as more theoretical aspects. While the paper has only been reviewed by two reviewers, their concerns are enough to recommend that the author consider them carefully and then resubmit this paper as a new paper.}
11 \textit{Most of the issues raised are related to cryptography, and not to the acceleration work on a GPU. The issue may be that during their preparation of this paper the authors were too focused on the acceleration work, and did not spend enough time being precise about the cryptography discussion. The two reviewers are experts on cryptography, as well as acceleration techniques, and the review indicate that the analysis needs to be strengthened.}
19 \textit{The authors should include a summary of test measurements showing their method passes the test sets mentioned (NIST, Diehard, TestU01) instead of the one sentence saying it passed that is in section 1.}
21 \begin{color}{red} In section 1, we have added a small summary of test measurements performed with BigCrush of TestU01.
22 As other tests (NIST, Diehard, SmallCrush and Crush of TestU01 ) are deemed less selective, in this paper we did not use them.
28 The authors say they replace the xor-like PRNG with a cryptographically secure one, BBS, but then proceed to use extremely small values, as far as a cryptographer is concerned (modulus of $2^{16}$), in the computation due to the need to use 32 bit integers in the GPU and combine bits from multiple BBS generated values, but they never prove (or even discuss) how this can be considered cryptographically secure due to the small individual values. At the end of 9.1, the authors say $S^n$ is secure because it is formed from bits from the BBS generator, but do not consider if the use of such small values will lead to exhaust searches to determine individual bits. The authors either need to remove all of section 9 and or prove the resulting PRNG is cryptographically secure.}
30 A new section (namely, the Section 9.2) has been added to measure practically the security of the generator.
33 \textit{In the conclusion:
34 Reword last sentence of 1st paragraph
35 In the 2nd paragraph, change "these researches" to "this research" in "we plan to extend ..."}
44 \textit{The paper is, overall, well written and clear, with appropriate references to the relevant concepts and prior work. The motivation of the work, however, is not quite clear: the authors present (provable) chaotic properties of a PRNG as a security improvement, but provide no convincing argument beyond opinion (or hope).}
48 \textit{There seems to have been no effort in showing how the new PRNG improves on a single (say) xorshift generator, considering the slowdown of calling 3 of them per iteration (cf. Listing 1). This could be done, if not with the mathematical rigor of chaos theory, then with simpler bit diffusion metrics, often used in cryptography to evaluate building blocks of ciphers.}
50 A large section (Section 5) has been added, using and extending some previous works, explains with more detail why topological chaos
51 is useful to pass statistical tests. This new section contains both qualitative explanations and quantitative (experimental) evaluations.
52 Using several examples, this section illustrates that defective PRNGs are always improved, according
53 to the NIST, DieHARD, and TestU01 batteries.
56 \textit{The generator of Listing 1, despite being proved chaotic, has several problems. First, it doesn't seem to be new; using xor to mix the states of several independent generators is standard procedure (e.g., [1]).}
58 The novelty of the approach is not in the discovery of a new kind of operator, but on the way to combine existing PRNGs. We propose
59 to realize a post-treatment based on chaotic iterations on these generators, in order to add topological properties that improve
60 their statistics while preserving their cryptographical security. In this document, generators that use XOR or BBS are only
61 illustrative examples using the vectorial negation as iterative function in the chaotic iterations. Theorems 1 and 2 explain how to
62 replace this negation function, that leads to well known forms of generators, by more exotic ones. However, the choice of the vectorial
63 negation for illustrations has been motivated for speed.
65 Indeed, to the best of our knowledge, all the generators proposed in the literature mix only a few operations on previously obtained states:
66 arithmetic operations, exponentiation, shift, exclusive or. It is impossible to define a fast PRNG or to prove its security when
67 using more complicated operations, and the number of such operations that are mixed is necessary very low. Thus almost all
68 up-to-date fast or secure generators are very simple, like the BBS or all the XORshift-like ones. In a certain extend, they are all similar,
69 due to the very reduced number of efficient elementary operations offered to define them.
73 \textit{Secondly, the periods of the 3 xorshift generators are not coprime --- this reduces the useful period of combining the sequences.}
76 Raph, c'est pour toi ça : soit tu changes tes xorshits, soit tu justifies ton choix ;)
80 \textit{Thirdly, by combining 3 linear generators with xor, another linear operation, you still get a linear generator, potentially vulnerable to stringent high-dimensional spectral tests.}
82 This first generator has not been designed for security reasons, but for speed: the
83 idea was to provide a very efficient version of our former generator that can pass
85 operations are a necessity when speed with pseudorandomness are desired. If the desire is to use a fast and statistically perfect PRNG, then simulations
86 proposed in this document show that this first PRNG is suitable. However, we have neither
87 claimed nor proved that this generator is secure. Indeed, we have only shown that some
88 chaotic iteration based post-treatment, like the one that use the vectorial negation,
89 can preserve the cryptographically secure property (while adding chaos), if this property has been established
90 for the inputted generator. As the inputted generator is not
91 cryptographically secure in the example disputed by the reviewer, we cannot apply this
92 result. Indeed the first part of the document does not deal with security,
93 but it investigates the speed, chaos, and statistical quality of PRNGs.
94 A sentence has been added to clarify this point at the end of Section 5.4.
98 \textit{The BBS-based generator of section 9 is anything but cryptographically secure. A 16-bit modulus (trivially factorable) gives out a period of at most $2^{16}$, which is neither useful nor secure. Its speed is irrelevant, as this generator as no practical applications whatsoever (a larger modulus, at least 1024-bit long, might be useful in some situations, but it will be a terrible GPU performer, of course).}
101 This claim is surprising, as this result is mathematically proven in the article:
102 either there is something wrong in the proof, or the generator is cryptographically
103 secure. Indeed, there is probably a misunderstanding of this notion, which does
104 not deal with the practical aspects of security. For instance, BBS is
105 cryptographically secure, but whatever the size of the keys, a brute force attack always
106 achieve to break it. It is only a question of time: with sufficiently large primes,
107 the time required to break it is astronomically large, making this attack completely
108 impracticable in practice. To sum up, being cryptographically secure is not a
109 question of key size.
112 Most of theoretical cryptographic definitions are somehow an extension of the
113 notion of one-way function. Intuitively a one way function is a function
114 easy to compute but which is practically impossible to
115 inverse (i.e. from $f(x)$ it is not possible to compute $x$).
116 Since the size of $x$ is known, it is always possible to use a brute force
117 attack, that is computing $f(y)$ for all $y$'s of the good size until
118 $f(y)\neq f(x)$. Informally, if a function is one-way, it means that every
119 algorithm that can compute $x$ from $f(x)$ with a good probability requires
120 a similar amount of time than the brute force attack. It is important to
121 note that if the size of $x$ is small, then the brute force attack works in
122 practice. The theoretical security properties don't guaranty that the system
123 cannot be broken, it guaranty that if the keys are large enough, then the
124 system still works (computing $f(x)$ can be done, even if $x$ is large), and
125 cannot be broken in a reasonable time. The theoretical definition of a
126 secure PRNG is more technical than the one on one-way function but the
127 ideas are the same: a cryptographically secured PRNG can be broken
128 by a brute force prediction, but not in a reasonable time if the
129 keys/seeds are large enough.
133 \textit{To sum it up, while the theoretical part of the paper is interesting, the practical results leave much to be desired, and do not back the thesis that chaos improves some quality metric of the generators.}
136 We hope now that, with the new sections added to the document, we have convinced the reviewers that to add chaotic properties in
137 existing generators can be of interest.
140 \textit{On the theoretical side, you may be interested in Vladimir Anashin's work on ergodic theory on p-adic (specifically, 2-adic) numbers to prove uniform distribution and maximal period of generators. The $d_s(S, \check{S})$ distance loosely resembles the p-adic norm.}
142 Thank you for this information. However, we have already established the uniform distribution in \cite{bcgr11:ip} (recalled in Theorem 2).
145 \textit{Typos and other nitpicks:\\
146 - Blub Blum Shub is misspelled in a few places as "Blum Blum Shum";}
148 These misspells have been corrected (sorry for that).
151 \textit{ - Page 12, right column, line 54: In "$t<<=4$", the $<<$ operation is using the `` character instead.}
154 \textit{ [1] Howes, L., and Thomas, D. "Efficient random number generation and application using CUDA." In GPU Gems 3, H. Nguyen, Ed. NVIDIA, 2007, Ch. 37. }
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