On Melissa O’Neill’s PCG random number generator
I have a few comments:<p>1. The paper itself[1] is <i>extremely</i> readable by the standards of most cryptography research. On one hand, this is great because I was able to follow the whole thing in essentially one pass. On the other hand, the paper is <i>very</i> long for its result (58 pages!), and it could easily do without passages like this one:<p><i>Yet because the algorithms that we are concerned with are deterministic, their
behavior is governed by their inputs, thus they will produce the same stream of “random”
numbers from the same initial conditions—we might therefore say that they are
only random to an observer unaware of those initial conditions or unaware of how
the algorithm has iterated its state since that point. This deterministic behavior is
valuable in a number of fields, as it makes experiments reproducible.  As a result, the
parameters that set the initial state of the generator are usually known as the seed. If
we want reproducible results we should pick an arbitrary seed and remember it to
reproduce the same random sequence later, whereas if we want results that cannot
be easily reproduced, we should select the seed in some inscrutable (and, ideally, nondeterministic) way, and keep it secret. Knowing the seed, we can predict the output, but for many generators even without
the seed it is possible to infer the current state of the generator from its output. This
property is trivially true for any generator where its output is its entire internal state—a
strategy used by a number of simple random number generators. For some other
generators, such as the Mersenne Twister [35], we have to go to a little more trouble and
invert its tempering function (which is a bijection; see Section 5), but nevertheless
after only 624 outputs, we will have captured its entire internal state.</i><p>That&#x27;s a lot of setup for what is frankly a very basic idea. A cryptographer being verbose in their writing might briefly remind the reader of these properties with the first sentence, but they&#x27;d still likely do that with much more brevity than this. I understand wanting to make your research accessible, but for people who understand the field this detracts from getting to the &quot;meat.&quot; It might make it harder to get through, but a 10-30 page result is preferable to a nearly 60-page one that assumes I know nearly nothing about the field. If I don&#x27;t know these details very well, how can I properly assess the author&#x27;s results?<p>2. The author&#x27;s <i>tone</i> in her writing is something I take issue with. For example, passages like this one...<p><i>Suppose that, excited by the idea of permutation functions, you decide to always
improve the random number generators you use with a multiplicative step. You turn
to L’Ecuyer’s excellent paper [25], and without reading it closely (who has time to
read papers these days!), you grab the last 32-bit constant he lists, 204209821. You
are then surprised to discover that your “improvement” makes things worse! The
problem is that you were using XorShift</i> 32&#x2F;32, a generator that already includes
multiplication by 747796405 as an improving step. Unfortunately, 204209821 is the
multiplicative inverse of 747796405 (mod 2
32), so you have just turned it back into
the far-worse–performing XorShift generator! Oops.*<p>...go a bit beyond levity. If you&#x27;re trying to establish rigorous definitions and use cases to distinguish between generators, functions and permutations, this isn&#x27;t the way to do it. This isn&#x27;t appropriate because it doesn&#x27;t go far enough to <i>formalize</i> the point. It makes it intuitive, sure, and that&#x27;s a great educational tool! But it&#x27;s a poor scenario to use as the basis for a problem statement - research is not motivated by the failure of an engineer to properly read and understand existing primitives, it&#x27;s motivated by novel results that exhibit superior qualities over existing primitives.<p>3. The biggest grievance I have with this paper is the way in which it analyzes its primitives for cryptographic security. For example, this passage under 6.2.2 Security Considerations:<p><i>In addition, most of the PCG variations presented in the next section have an
output function that returns only half as many bits as there are in the generator state.
But the mere use of a 2
b&#x2F;2-to -1 function does not guarantee that an adversary cannot
reconstruct generator state from the output. For example, Frieze et al. [12] showed
that if we simply drop the low-order bits, it is possible for an adversary to discover
what they are. Our output functions are much more complex than mere bit dropping,
however, with each adding at least some element of additional challenge. In addition,
one of the generators, PCG-XSL-RR (described in Section 6.3.3), is explicitly designed
to make any attempt at state reconstruction especially difficult, using xor folding to
minimize the amount of information about internal state that leaks out.17 It should be
used when a fast general-purpose generator is needed but enhanced security would
also be desirable. It is also the default generator for 64-bit output.</i><p>That&#x27;s not a rigorous analysis of a primitive&#x27;s security. It <i>is</i> an informal explanation of why the primitive <i>may</i> be secure, but it so high level that there is no proof based on a significant hardness assumption. Compare this with Dan Boneh&#x27;s recent paper, &quot;Constrained Keys for Invertible Pseudorandom Functions&quot;[2]. Appendices A and B after the list of references occupy nearly 20 pages of theorems used to analyze and prove the security of primitives explored in the paper under various assumptions.<p>Novel research exploring functions with (pseudo)random properties is inherently mathematical; it&#x27;s absolutely insufficient to use a bunch of statistical tests, then informally assess the security of a primitive based on the abbreviated references to one or two papers.<p>_________<p>1. <a href="http:&#x2F;&#x2F;www.pcg-random.org&#x2F;pdf&#x2F;hmc-cs-2014-0905.pdf" rel="nofollow">http:&#x2F;&#x2F;www.pcg-random.org&#x2F;pdf&#x2F;hmc-cs-2014-0905.pdf</a><p>2. <a href="https:&#x2F;&#x2F;eprint.iacr.org&#x2F;2017&#x2F;477.pdf" rel="nofollow">https:&#x2F;&#x2F;eprint.iacr.org&#x2F;2017&#x2F;477.pdf</a>