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On Melissa O’Neill’s PCG random number generator (lemire.me)
104 points by ozanonay on Aug 15, 2017 | hide | past | web | favorite | 83 comments

A few notes:

The author writes "Meanwhile, at least one influential researcher (whose work I respect) had harsh words publicly for her result", and then quotes some of these words:

   Note that (smartly enough) the PCG author avoids
   carefully to compare with xorshift128+ or xorshift1024*.
However, the author fails to note that said "influential researcher", Sebastiano Vigna, is the author of xorshift128+ and related PRNG.

In the linked test [2] by John D. Cook (who uses PactRand, a test similar to the (obsolete) DIEHARD), xorshift128+ and xoroshir0128+ fail within 3 seconds, while PCG ran 16 hours producing 2 TB of pseudo-random numbers without any suspicious p-value detected.

On the other hand, Vigna claims that the xoroshiro family does "pass" PactRand.

I've submitted an answer to StackOverflow a while ago [1], recommending xoroshiro and PCG, thus I'd be concerned if PCG turns out to be flawed. It's actually quite hard to get academics in the field to give an authoritative recommendation (I've tried) - their response is typically along the line "It's complicated"...

[1] https://stackoverflow.com/questions/4720822/best-pseudo-rand...

[2] https://www.johndcook.com/blog/2017/08/14/testing-rngs-with-...

Edit: remove italics due to asterisk in PRNG name, & add link to John. D Cook's test.

I think Vigna's claim is that if you ignore the PractRand tests that fail, it passes. (Really!)

O'Neill has instructions on how to test with PractRand and with TestU01 on her blog (http://www.pcg-random.org/blog/). I had a go with TestU01 on Vigna's generators, and when you test the low 32 bits reversed (for 64-bit PRNGs, you have to test the high 32, the low 32, both forwards and reversed), I found that all Vigna's generators fail.

Given the PractRand results it makes sense, I guess, but I had read that Vigna's generators were supposed to pass TestU01.

Does anyone else wants to have a go at testing so I can know if I screwed up somehow?

I think Vigna's claim is that if you ignore the PractRand tests that fail, it passes. (Really!)

The code does explain exactly what the issue is, i.e. that the last bit isn't random:

   This generator passes the PractRand test suite
   up to (and included) 16TB, with the exception of binary rank tests,
   which fail due to the lowest bit being an LFSR; all other bits pass all
   tests. We suggest to use a sign test to extract a random Boolean value.
But I'm tempted to agree this isn't a desirable property for a generic RNG.

How many users of JavaScript know about this property? (it's the default RNG for most browser engines) Or does it not matter because they return 53-bit floats?

In the comment section to the V8 JavaScript blog post [1], Vigna writes:

  - Technically, it would be better if you used the upper
  52 bits, rather than the lower 52 bits, to generate a
  double. The lowest bit of a xorshift128+ generator is an
  LSFR, and while people has been happy using LSFR for
  decades, it is slightly inferior in quality to all other
  bits. This is really OCD, as computational errors makes
  the lowest bit almost irrelevant, but now you know.
I don't know whether that's been implemented, but the maintainer replied:

  Thanks for the suggestions! I will definitely revisit the 
  current implementation with your tips in mind.

[1] https://v8project.blogspot.nl/2015/12/theres-mathrandom-and-...

I love O'Neill's work on PCG, and loved the talks by her I watched online.

As a tenured professor I want to say two things about this piece:

1. I think academic publishing will be forced to change. I'm not sure what it's going to look like in the end, but traditional journals are starting to seem really quaint and outdated now.

2. As far as I can tell from what she's written on the PCG page, the submission to TOMS is a poor example, because no one I know expects to be done with one submission. That is, no one I know submits a paper to one journal, even one reputable journal, and is done. They submit and it gets rejected and revise it and resubmit it, maybe three or even four times. After the fourth or fifth time, you might give up, but not necessarily even then.

I have mixed feelings about the PCG paper as an example, because in some ways it's great: an example of how something very influential has superceded traditional academic publishing. In other ways, though, it's horrible, because it's misleading about the typical academic publishing experience. Yes, academic publishing is full of random nonsense, and corruption, but yes, you can also get past it (usually) with just a little persistence. In still other ways, it's a good example of what we might see increasingly, which is a researcher having a lower threshold for the typical bullshit out there.

Here's the site for the random number generator.[1] It's basically a simple linear congruential random number generator (well known, but not very good) fed into a mixer. The mixer is new.

Most of the analysis is about the LCG or the final output. The suggested mixer is just

    output = rotate64(uint64_t(state ^ (state >> 64)), state >> 122);
That's simple, and the insight in this paper is that something that simple helps a lot. I would have thought that you'd want a mixer where changing one bit of the input changes, on average, half the bits of the output. The mixer above won't do that. DES as a mixer would probably be better, but it's slower. The new result here is that something this simple passes many statistical tests.

This isn't crypto-grade; both that mixer and a LCG generator are reversible with enough work.

[1] http://www.pcg-random.org/

>linear congruential random number generator (well known, but not very good)

Relevant quotes from the paper:

"But if you began reading the section with the belief that “linear congruential generators are bad” (a fairly widely-held belief amongst people who know a little about random number generation), you may have been surprised by how well they performed. We’ve seen that they are fast, fairly space efficient, and at larger sizes even make it through statistical tests that take down other purportedly better generators. And that’s without an improving step."


"Despite their flaws, LCGs have endured as one of the most widely used random- number generation schemes, with good reason. They are fast, easy to implement, and fairly space efficient. As we saw in Section 3.3, despite poor performance at small bit sizes, they continue to improve as we add bits to their state, and at larger bit sizes, they pass stringent statistical tests (provided that we discard the low-order bits), actually outperforming many more-complex generators. And in a surprise upset, they can even rival the Mersenne Twister at its principle claims to fame, long period and equidistribution."

"Nevertheless, there is much room for improvement. From the empirical evidence we saw in Section 3.3 (and the much more thorough treatment of L’Ecuyer & Simard [28], who observe that LCGs are only free of birthday-test issues if n < 16p1/3, where n is the number of numbers used and p is the period), we can surmise that we may observe statistical flaws in a 128-bit LCG after reading fewer than 247 numbers (which is more than BigCrush consumes but nevertheless isn’t that many—an algorithm could plausibly use one number per nanosecond and 247 nanoseconds is less than two days)."

> 247 nanoseconds is less than two days

Yes, yes it is.

For those just as confused as I was, replace all instances of 247 with 247

Hmmm... is HN removing ^carets?

Replace 247 with '2^47', Which is ~'1.28x10^14' or ~36 hours in ns.

Hah. I used double asterisks, should have seen that coming in hindsight. Muphry's Law.

I've often been frustrated trying to put more than one asterisk (not italic tag) in an HN comment.

Serious question, what happened to WYSIWYG? The Web has supported it for like 20 years, why have we standardized on clunky in-line markup for rich text entry, pretty much everywhere?

FWIW, you can indent text in a HN comment (by a few spaces), and you get

  a Blockquote like this, 
  and *asterisks* survive: 2**47

WYSIWYG is kinda crap on mobile. A checkbox saying "Don't format" would be nice, though.

It'd be nice if the formatter would realise that two asterisks in a row aren't emphasising any text and therefore just leave them as is.

> replace all instances of 247 with 247

That doesn't appear to help ;-)

Two-to-the-forty-seven nanoseconds is a little over thirty-nine hours.

(Let's see HN mangle that!)

I feel bad for the poor NLP tokenizer that will process this later and decide that two-to-the-forty-seven is a word, strangely unattested in any corpora.

It's a weird paper. It's quite long and takes an eternity to reach this simple point. It's also littered with "security concerns" which are confusing at best and misleading at worst; in reality, none of the generators it discusses are suitable for "sensitive applications", even in the corner-case scenario it discusses of needing permutations of all the b-bit integers, a problem we already have cryptographic tools to solve.

But I'll confess to not really understanding what all the fuss is about insecure generators.

Linear congruential generators (x = k1*x + k2, unsigned with truncation on overflow) have some strange properties, some of which the paper explores. If you repeatedly take three sequential values from one and treat them as 3D coordinates, the points line up in parallel planes. (There's an explanation of why in Knuth.) Something has to be done to destroy that order. The solution here is to XOR the upper and lower halves of the 128-bit state to get 64 bits, then circular shift that by the high 6 bits of the 128-bit input. This passes most of the classic tests for random number generators.

That's the paper, basically.

The paper is really two papers. The first paper is a thoughtful and careful explanation of how we should consider judging pseudorandomness, and how to measure various criteria, along with a survey of many popular ten-lines-or-less PRNGs which PCG competes with. I found it interesting.

> But I'll confess to not really understanding what all the fuss is about insecure generators.

Based on things she's said on her site and in comments on John D. Cook's blog, it's all about algorithmic complexity attacks on randomized algorithms.

In other words, if you're doing quicksort on external input with a random pivot, and someone knows the PRNG state, they can make a pathological input that'll trigger quadratic behavior.

I don't know how likely this is to happen, but I know there were similar attacks on hash tables a few years ago.

And we addressed it with actual cryptography: SipHash. Since cryptographic random number generators are generated with very similar primitives, why wouldn't SipHash be the answer to those problems as well?

> I wonder whether the academic publications are growing ever less relevant to practice.

I think there are two topics here. One is whether academic research and work is becoming less relevant to practice. The other is whether the formalism of academic-style publishing are becoming less relevant to the modern world where more and more venues for publishing, rating, and discovering work.

On the former, I believe that academic work is as relevant as ever. There are some areas (like systems) where I'm doubtful about relevance from the point of view of a practitioner, but other areas (like hardware and ML where work remains extremely relevant). I haven't noticed a trend there over the last decade, except in some areas of systems where the industrial practice tends to happen on cluster sizes that are often not approachable for academia.

On the latter, academic publication does indeed seem to be getting less relevant. There are other (often better) ways to discover work. There are other ways to tell whether a piece of work is relevant, or credible. There are other, definitely better, ways to publish and distribute work. In some sense I think this is a pity: as an academic-turned-practitioner I like academic-style publications. Still, I think they are going to either change substantially or die.

This article raises another very good point: sometimes the formalism of academic publication makes the work harder to understand, less approachable, or less valuable. That's clear harm, and it seems like this professor was right to avoid that.

As an engineer who switched to PCG:

- PCG is not crypto, everybody should understand that. It's for simulation and rendering.

- PCG mainly replaces Mersenne Twister which is in c++11. The Twister has a LOT more state and is a LOT slower for less randomness.

- In rendering and simulation speed really matters, and PCG excels there.

- Xorshift is another algorithm in the same class. I would really like to see an objective comparison. In my cursory engineering look PCG seemed better.

- Fast PRNG is almost a new field again: It's not crypto, but immensely useful. How did the Twister get into C++11 while it is so much worse than PCG or Xorshift? Nobody cared!

- Maybe PCG should have been a paper at SigGraph.

- For the style of the paper, I think one contribution is rethinking PRNG outside crypto. That deserves and requires a lot of exposition.

Yeah, it's not for crypto. But I think it's for more than simulation and rendering. It's meant as a general purpose PRNG. It's just as good for randomized algorithms like picking the pivot in quicksort, or playing games, procedural content generation (PCG!), or whatever you want to use it for.

I think that the whole point of the prediction difficulty stuff is that a library (e.g., C++11's) with general purpose PRNGs can't know how they'll be used. Maybe some idiot write code for a gambling machine in C++ and use whatever PRNG is to hand. There was a story in the news the other week about people going around casinos predicting slot machines, so maybe this has already happened! PCG is trying to make your simulation and rendering code fast while trying to offer at least some defense against egregious misuse.

Basically PCG is trying to be a good all rounder. As you say, it's meant as a replacement for the Mersenne Twister.

I found the story: https://www.wired.com/2017/02/russians-engineer-brilliant-sl...

And yes, PCG is harder to exploit in this way than the Twister, but you still really should not bet money on it!

I like this because she is a professor at Harvey Mudd. They took steps to make CS more inclusive, with great results. I appreciate her attitude on accessibility, which is in keeping with that institution's philosophy.

That she ran into a paper wall doesn't bother her because she's openly publishing is even better.

There's room for accessible and for abstruse literature. Usually what happens with novel work is that initial publications are abstruse but met with excitement by the scholarly community and gradually more accessible works (as the number of collaborators/coauthors grows too) are published.

That said, even if multi-culti math means that top-line researchers are going to be spending time with song-and-dance introductions, she should still have put a grad student onto the task of making the short paper that experts will actually read.

If the whole thing is a matter of style and not of obfuscation, this would have given a grad student an easy, cool first publication.

Well, Harvey Mudd is an undergrad-only STEM school, so I'm not sure having a grad student edit it would have been possible.

Well, her peers in her research program probably have grad students...

I mean, she's working in collaboration with someone, right?

In the history of the paper she mentions some summer research, but that is for undergrads. Harvey Mudd is all undergrad and tends to emphasize teaching over research (at least, that was the philosophy 20 years ago o_O).

What is multi-culti?

It's a common sarcastic term hurled at things like "inclusive CS".

For all I know this iteration of "inclusive" could be the Right Stuff -- proper intellectual discipline even as it advances secondary goals of "inclusiveness". But stiiill... she should have by now had the common courtesy of producing a short document aimed at experts, possibly prepared by her underlings. This would also have helped the career of the underlings.

"Physics for poets".


Regardless of her attitude on academic accessibility, it is inappropriate for a paper introducing a novel primitive with proposed security considerations to spend the time explaining why determinism is a concern in functions dealing with randomness. This paper could have been 10 pages.

If you want to make your research more accessible, there are ways to do that without assuming that your reader is coming in from a dead start on the field.

I am not sure why you are being downvoted. I think this is clearly true.

An easy way to make research accessible is to write a monograph!

I watched her EE380 talk [1] and thought she was a fine professor for a difficult (to me) subject. That she didn't get published for at best stylistic reasons means nothing.

[1] https://ee.stanford.edu/event/seminar/ee380-computer-systems...

It sounds like an interesting result, I look forward to reading the paper more carefully. That said, it's clearly not written for an academic journal. Section 2.4.3 is entitled "The Importance of Code Size", and explains why shorter code is better. I think you can argue that some academic papers are excessively concise, but this is a 58-page paper about an RNG. It is clearly not a journal paper and has a ton of extraneous content. I have to sympathize with the commenter that the author has made a trade-off and written a paper that's less rigorous than it should be (for peer review). I wonder why she didn't write 2 versions.

> I wonder why she didn't write [two] versions.

Because the reviewers took over 10 months to respond with a rejection mainly citing the length of the paper. And more importantly, "By that point, everyone who might have wanted to read it had almost certainly found it here and done so, so I saw little merit in drastically shortening the paper."[1]

She has updated the blog post which discusses all the nuanced details of the whole affair last month (2017-07-25)[2].

[1] http://www.pcg-random.org/paper.html [2] http://www.pcg-random.org/posts/history-of-the-pcg-paper.htm...

There is no excuse for the journal to take so long to provide a response. At the same time, it seems to me that their response was entirely predictable. Or does the journal usually post such long articles directed at a general audience?

Physicist here.


> And it is not even entirely clear what “really random” would mean. It is not clear that we live in a randomized universe…

At the quantum level it really is clear that we live in a really random universe. What's the meaning of really random? The outcome of a quantum process.

On-topic. Yeah, you have to know your audience. As OP mentions, just because the paper wasn't published doesn't prevent anyone from thinking about it and even building on it. On the other hand these scientific publications have styles and target audiences, and maybe she got rejected not due to lack of relevance or rigor, but because the paper didn't match the publication's non-scientific criteria for publication.

A quantum process is a random process, but isn't it still an open philosophical question as to whether the "random process" we observe is truly random, or is instead governed by deterministic hidden state?

No, all theories of (local) hidden variables have been experimentally ruled out.


> Bell's theorem states that any physical theory that incorporates local realism cannot reproduce all the predictions of quantum mechanical theory. Because numerous experiments agree with the predictions of quantum mechanical theory, and show differences between correlations that could not be explained by local hidden variables, the experimental results have been taken by many as refuting the concept of local realism as an explanation of the physical phenomena under test. For a hidden variable theory, if Bell's conditions are correct, the results that agree with quantum mechanical theory appear to indicate superluminal effects, in contradiction to the principle of locality.

Interesting. But it sounds like non-local hidden variables are still possible. My assumption is that non-local hidden variables is non-falsifiable, which is why I said this was a philosophical question.

Non-local hidden variables have not been ruled out, as evinced by the de Broglie–Bohm theory.

Have quantum outcomes been tested for the kinds of statistical not-quite-randomness that PRNGs have? Aren't the measurements far too imprecise?

This is an interesting article, but it's more about the changing landscape of academic publishing than it is about random number generators.

[EDIT] The actual paper is here: http://www.pcg-random.org/pdf/hmc-cs-2014-0905.pdf

Only if it turns out the work is actually good. If not this is exactly what should happen in peer review.

That is exactly what peer review should do, to determine whether the idea has merit. But that isn't what has happened. Peer review which is a cornerstone to the academic/scientific community stumbled on style and didn't get to merit.

I disagree, I think the peer review functioned exactly as it was supposed to.

Her paper doesn't pass the sniff test for me whatsoever when it comes to security analysis. She spent close to no time analyzing the primitives she introduced (and with no proofs or rigor!), meanwhile the thing is 58 pages because she takes the time to explain what "determinism" and "seeds" are to her audience.

"Exposition" is, in my opinion, a fully valid reason to reject a paper. I'm not going to sit and read your 60 page paper that could have been compressed to 10 pages if you just got to the point and assumed your audience understood the field well enough to assess your results. Rewrite it and send it back without the assumption that your audience needs to be reminded of everything they'd need to learn just to properly assess your result. It's not as though they rejected the paper on empirical grounds without a meritocratic review; they rejected it because they have a finite amount of time and (speaking as someone in the field) it's sort of annoying to read after page 10.

I think academia frequently gets lost in the ivory tower and loses touch with what an accessible paper looks like; this is not an answer to that, it's a swing in the other direction, where papers with truly novel results will suddenly be hundreds of pages and tens of pages of setup.

> papers with truly novel results will suddenly be hundreds of pages and tens of pages of setup.

We're already there with Inter-universal Teichmüller Theory (https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCll...), the entire mathematical field singlehandedly created by Mochizuki to prove the ABC Conjecture. Mochizuki worked in isolation and astounded the world by revealing all of this at once.

Now, the world's top mathematicians have been so impressed with whatever they've managed to understand from Mochizuki's papers that big efforts are being made to unravel it, getting Mochizuki to teach lectures, etc. As best as I know it, the entire thing hasn't been independently verified -- it's such a tall stack of novel mathematics.

Now: we can chastise Mochizuki for not playing within the ordinary rules of math research (publish ongoing research, etc.), or acclaim him as a genius having produced fundamental, discontinuous advances in his field... or we can do both.

I wouldn't blame people for holding off on "using" Mochizuki's results (they're too abstract for that, but anyway) because the whole thing is so obscure still. What's more: I wouldn't hold skepticism on Mochizuki as an example of academia being too self-referential. The academic rat race is supposed to keep these things from happening by imposing some structure on the production of knowledge.

If you don't like it then don't read it or don't publish it. But sniff test and exposition are style. Ding it for not analyzing her primitives but don't then call that security analysis. Do the security analysis.

In patents there's the concept of enablement: did the inventor enable/teach the idea? Yes she did. Did she flesh out every idea? No she didn't.

This was rejected on style grounds. Ignoring it then on 'security analysis' is not improving matters; it certainly isn't helping security. Do the security analysis.

> Sniff test and exposition are style.

Peer review journals and conferences have (typically explicit) style guides. The argument that peer review arbitrarily rejects papers unsubstantively is a strawman; peer review councils do not maintain that meritocratic results-driven analysis are the only barriers to entry.

>Ding it for not analyzing her primitives but don't then call that security analysis. Do the security analysis.

Authors introducing novel results with cryptographic considerations typically perform their own analysis and publish that in the paper with the result. I was not referring to my observation of her own insufficient analysis as if it is a formal analysis on my part, I was observing (correctly) that she didn't do enough of her own formal analysis. There is a modicum of author-provided proof-based assurance that 1) is considered in the peer review process and 2) forms the foundation for formal cryptanalysis by peers in the community. In other words, there isn't yet enough for cryptanalysts to attack (or more precisely, the author has put the onus of analysis on other researchers, instead of providing specific, rigorous claims which they can empirically refute).

It comes down to respecting time. In attempting to be accessible, the author is not being respectful of other researchers' time. We don't need to have the birthday paradox explained to us, we've understood that since Intro to Statistics. The paper is 58 pages because, "think of the unwashed masses who can't understand our work!", but in attempting to appeal to those beyond the ivory tower, it's just become circuitous and over-indulgent. She didn't have to write in this style to make it more accessible (and the relevant mathematics has a lower bound on how accessible it can be, anyway).

It is not respectful of qualified cryptanalysts to not even provide a precise set of proofs for your claims. She uses phrases like, "PCG is a middle ground between security and performance." We cannot analyze that, and it's not for cryptanalysts to review every single claim that comes across their desks. This is why the author of xorshift+ didn't even provide a cryptanalysis in his scathing review.

If you're not going to sufficiently specify your claims, don't be surprised when the academic community ignores them (even if they're valid!). You are arguing your point from a first principles approach to meritocracy and the fairness of style-based assessment; I am arguing that as a practical matter in the research community, there is a reason this is not how novel cryptographic primitives are introduced.

It's not a security paper. Or at least, I think it isn't? It's not clear to me what journal it was submitted to.

ACM Transactions on Mathematical Software (TOMS)

You don't think the peer review process ever produces false negatives?

The peer review process absolutely produces false negatives, but that doesn't really change the fact that this paper doesn't need to be nearly 60 pages, doesn't sufficiently analyze one of its central premises (6.2.2 Security Considerations) and in general focuses on treatise and levity rather than rigor.

Peer review is basically just proof-reading, and taking 60 pages to present 10 pages of findings is exactly the kind of thing a proof-reader should catch.

You didn't ask me, but I'll confirm that it definitely does.

As a general comment, I dislike deliberately obtuse writing in papers. In my current work, I came across a very in-depth survey of our industry (sex work). Excellent study, very helpful. But some of the sentences seemed to over-complicate the math. Example: "Consider the set P {p1, p2, ... pN} representing providers and the set C {c1, c2, ... cN} representing customers". I am pretty sure this kind of stuff is filler or pretends to make things look more rigorous than they are.

On the other hand, maybe spending more than a line explaining what the birthday paradox is should be cut out and put in a backgrounder paper or appendix so that the paper can focus on the actual novel ideas.

> On the other hand, maybe spending more than a line explaining what the birthday paradox is should be cut out and put in a backgrounder paper or appendix so that the paper can focus on the actual novel ideas.

That was my annoyance with the paper as well. Add to that explanations that amount to, "What even is determinism?" or "What's a seed?" and I'm unsurprised it's nearly 60 pages.

are you joking? That is your example of abstruse mathematical notation? Some variables with names?!

No. I am saying that giving the definition of a set each time is just extra verbosity. The whole paper had that extra verbiage, everywhere. Kind of why-use-one-word-when-ten-will-do feeling.

Could you share the paper? I mean if they were making a mathematical argument...it is kind of hard to do that without defining things..

"Consider the set P {p1, p2, ... pN} representing providers and the set c {c1, c2, ... cN} representing customers"


"Consider the set P of providers and the set C of customers"

The former is about 2x the length of the latter. This is the verbosity I read into the original comment.

The first is required when you want to later refer to an individual element from the set.

"Consider the set P of providers" means when you eventually refer to p_2, you'll have to note that you mean an element (the second, in some sense) of P. That moved the verbosity around rather than eliminating it.

To me it seems like a much bigger error that "Consider the set P {p_1, p_2, ... p_N} representing providers and the set c {c_1, c_2, ... c_N} representing customers" states outright that the sets are equal in size. I would expect C to be much larger than P.

If this is a commonly used notation in the paper, then it would make sense to state once up front "here's how we refer to sets and their members". DRY and all that.

mmm I see.

Melissa O'Neill gave a talk describing the PCG random number generator at Stanford in EE380. https://www.youtube.com/watch?v=45Oet5qjlms

I once tried to develop my own fast random number generate using nothing but bitwise operations. On the theory they were the fastest/simplest. I had a program generate thousands of random combinations of bitwise functions. And then used statistical tests to see which ones produced the most "random" seeming behavior.

It worked as far as I can tell. But I don't trust the statistical tests. Who is to say there isn't a very obvious pattern in the numbers that I didn't test for or notice? How do you prove a random number generator is good?

> How do you prove a random number generator is good?

You can't; that's the nature of randomness. You can prove they're bad, though.

A theoretical approach based on computability theory is https://en.m.wikipedia.org/wiki/Algorithmically_random_seque....

> How do you prove a random number generator is good?

Compress its output.

I have a few comments:

1. The paper itself[1] is extremely 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 very long for its result (58 pages!), and it could easily do without passages like this one:

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.

That'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'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 "meat." 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't know these details very well, how can I properly assess the author's results?

2. The author's tone in her writing is something I take issue with. For example, passages like this one...

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 32/32, 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.*

...go a bit beyond levity. If you're trying to establish rigorous definitions and use cases to distinguish between generators, functions and permutations, this isn't the way to do it. This isn't appropriate because it doesn't go far enough to formalize the point. It makes it intuitive, sure, and that's a great educational tool! But it'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's motivated by novel results that exhibit superior qualities over existing primitives.

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:

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/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.

That's not a rigorous analysis of a primitive's security. It is an informal explanation of why the primitive may be secure, but it so high level that there is no proof based on a significant hardness assumption. Compare this with Dan Boneh's recent paper, "Constrained Keys for Invertible Pseudorandom Functions"[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.

Novel research exploring functions with (pseudo)random properties is inherently mathematical; it'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.


1. http://www.pcg-random.org/pdf/hmc-cs-2014-0905.pdf

2. https://eprint.iacr.org/2017/477.pdf

Just to be clear: it's not a cryptography paper, is it? Did you figure out what journal it was submitted to?

She submitted it to ACM Transactions on Mathematical Software. I would personally consider it a cryptography paper, for three reasons:

1. She purports to introduce a novel result that bridges "medium-grade" performance characteristics and security characteristics in one primitive. In fact, if you look at the PCG Random website (pcg-random.org), she very clearly compares and emphasizes both performance and security characteristics with functions like xorshift and ChaCha.

2. We see cryptography papers submitted to all manner of theoretical CS conferences and journals, for example Symposium on the Theory of Computing, which are not uniformly crypto-focused.

3. She acknowledges herself that she found it hard to categorize her paper (it could be relevant for simulstion, it could be relevant for stream ciphers, etc) in a blog post about how she chose the venue: http://www.pcg-random.org/posts/history-of-the-pcg-paper.htm...

As a meta point I read the whole thing, and I actually think it would be a nice publishable result if it were, say 10 - 20 pages. But 60 is wild! It took me longer to get through this "accessible" paper than it did for me to get through any of Boneh's papers on constrained and puncturable pseudorandom functions!

It's definitely interesting, and sure, why not explore "medium-grade security" that makes explicit tradeoffs with performance and security. But the presentation seems like it was written by someone writing for a non-academic audience, and the content of 6.2.2 "Security Considerations" is really light on provable security.

TOMS has a long and storied history, mostly on numerical codes. I was looking through it in grad school in the late 80s/early 90s for better saner methods in numerical software.

Better PRNGs for simulation is definitely in its bailiwick. Crypto ... maybe less so.

If you stripped out all the (weird, random) cryptographic stuff from this paper, it would read pretty much the same and make pretty much the same points, which tells me: it's not a cryptographic paper.

That's a fair point; it was mostly the presence of the random crypto that annoyed me :)

O'Neill recently mentioned the crypto aspects of PCG in an comment on another post by John D. Cook. I'll just quote it below. But it looks to me like she thought that any analysis she did on the prediction difficulty of PCG wouldn't be well regarded.

Also, I notice that lots of people seem to think her paper was too long, but they also claim that it doesn't say enough about their favorite topic. That seems to be happening with you.

Direct quote from O'Neill's comment here https://www.johndcook.com/blog/2017/07/07/testing-the-pcg-ra...

John, in your post, you said that PCG has “excellent statistical and cryptographic properties”. I’ve learn it best never to say “cryptographic security” or “cryptographic properties” when trying to place something on a spectrum of prediction difficulty. Too many misunderstandings. Saying “prediction difficulty” still causes some crossed wires, but it’s about as good as we can go.

Dan & Dmitriy, just to be 100% clear, I HAVE NEVER RECOMMENDED PCG FOR CRYPTOGRAPHY. I do however, care about prediction difficulty, and I don’t like trivially predictable generators. Because my viewpoint is often misunderstood, I have some more blog posts in the pipeline about these issues, but the key thing is that general-purpose PRNGs get used for almost anything, from using the low-order bit to toss a coin, to supporting randomized algorithms. If someone can predict your generator, they can mount an algorithmic complexity attack on your randomized algorithm, tanking its performance. If predicting the generator is more costly than the algorithmic complexity attack itself, people will try their attacks on easier targets. But if the generator spends to much time being trying hard to be unpredictable, we also tank our performance, thus we have to try to strike a balance in a different place than we do for traditional cryptographic applications. We already live in a world where hash table implementations in scripting languages need to be hardened because of algorithmic complexity attacks; this is the next logical step.

All that said, there are members of the pcg family (not pcg32 and especially not pcg32_fast!) that I personally think would be really challenging to predict. I also find it frustrating that people don’t compare like with like. If you want to compare the prediction difficulty of PCG against a cryptographically secure PRNG, you need to compare a PCG variant at least broadly similar in size. Say, for example, we wanted to contrast PCG against the ChaCha PRNG from Orson Peters (perhaps with four rounds rather than the full 20), that PRNG is 104 bytes in size, so it’s fairest to compare it to pcg64_c8, which is 80 bytes, or pcg64_c16, which is 144 bytes.

Regarding prediction difficulty, I’m very well aware of Bruce Schneier’s law, “Anyone, from the most clueless amateur to the best cryptographer, can create an algorithm that he himself can’t break. It’s not even hard. What is hard is creating an algorithm that no one else can break, even after years of analysis. And the only way to prove that is to subject the algorithm to years of analysis by the best cryptographers around.”, and his subsequent elaboration “Anyone can invent a security system that he himself cannot break. I’ve said this so often that Cory Doctorow has named it “Schneier’s Law”: When someone hands you a security system and says, “I believe this is secure,” the first thing you have to ask is, “Who the hell are you?” Show me what you’ve broken to demonstrate that your assertion of the system’s security means something.” Thus my personal thoughts on how difficult it is mean NOTHING in a cryptography context. I also can’t expect cryptographers to spend their time on my education, but if someone out does know a simple and efficient algorithm that can reliably break pcg64_c8, I really would love to see how. (Also, hey Bruce, gender-neutral language, it’s a thing.)

I can do at least one thing that adds a tiny tiny tiny bit of credibility in the eyes of folks like Bruce Schneier , I can show other PRNGs I have broken that might have seemed hard to predict to a casual observer. Mostly that won’t actually help though, because a cryptographer would say “Ha! That’s toddler level stuff!” and a mathematician might say “I can’t understand why you care about prediction at all”. Sometimes I feel there should be more people trying to occupy the middle ground. Meh.


I think we've surrendered some of the "clarity" argument with the sentences in that paper describing which exact instantiation of the non-cryptographic RNG to select for "sensitive" applications.

Even here, in this comment, it's really hard to follow what you're saying. For instance, you've taken the time to compare the "prediction difficulty" of the PCG PRNG to that of a ChaCha20-based DRBG. But ChaCha20 is a stream cipher, a cryptographic primitive. To be competitive with it at producing uncorrelated bits is to be yourself a cryptographic primitive; that is, to argue that an LCG and a trivial mixer is all we ever needed to encrypt data. That would be... newsworthy?

Also: if you're making an appeal to the cryptographic literature, there are better people to cite than Bruce Schneier.

ACM Transactions on Mathematical Software.

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