Romu – Fine random number generators
romu-random.org
romu-random.org
One prime example: the algo for ROMU is of the form new_state64 = ROTATE_LEFT(old_state64,32) * constant, which decomposes the space {1,2,..,2^64-1} into a permutation. From a given start seed, your cycle may be short, and you won't know this until it bites you.
The author incorrectly addresses this using the theory of random permutations, but his is most definitely NOT a random permutation chosen uniformly from S_n. His analysis most does not follow. The proofs he cites are not applicable to his since he has a fixed permutation.
Among all cycles, among all permutations, the results hold. They do not hold for a fixed permutation. For example his results fail for the n-cycle and they fail for the identity permutation. They similarly fail for many, many other permutations, and he has no proof that his is not a badly chosen one. Based on the structure of it (rotate and mult), I suspect from past experience that his permutation has lots of structure that leads to all sorts of pathologies for this use.This is a basic math error, and the paper is full of such basic errors.
For why to listen (somewhat) to me, check out my name - I've done a bit of writing (and a lot of teaching) on PRNGs for decades. I've followed the field for quite some time, and am well aware of much of the research in it.
There is no need with all the good choices out there to choose a PRNG that cannot provide a solid proof of cycle length (or at least correct bounds) for any seed value. This one fails this basic test.
If I get time (and remember) later, I've written some SAT solver code to explicitly find holes in PRNGs, and I'll try to dig it out and analyze the cycles in this fixed permutation. If anyone else wants to try it the idea is you can assign variables to bit positions in the state, represent the permutation as boolean expressions in these bits, and find cycles using SAT solvers without having to run through steps incrementally. I suspect in this case it will break this PRNG pretty quickly.
This is something I'm concerned about (hence https://news.ycombinator.com/item?id=22451307), but I lack the know-how to demonstrate the difference.
> I get time (and remember) later, I've written some SAT solver code to explicitly find holes in PRNGs, and I'll try to dig it out and analyze the cycles in this fixed permutation.
Thanks, I would love to see this. Can I expect to see the results on your blog or here or somewhere else?
Presumably you've done this for other PRNGs in the past. If you've got links to share, I'd like to take a look, and hopefully learn a thing or two. :-)
Unfortunately, I rarely write stuff up. Too busy doing :) Writing up well takes considerable time away from making things.
Important request: Please post every specific error in reasoning or math you saw in the paper. You said there are several. I expect that you are not like many internet posters who reply with vagueness or discouragement when pressed for specifics. I need the specific errors so I can fix all of them before submitting the paper to a peer-reviewed journal.
Also, a SAT solver that finds cycles in these generators would be very helpful!
In fact, I think your method is a weaker subset of PCG generators, which are as fast and have all the good theoretical guarantees.
What do you mean by “true random numbers”?
There are 64! permutations on 64 bits. All Roma permutations are a vanishingly small subset of these with very specific structure. How can you prove that structure doesn’t introduce all sorts of bad cases?
Don’t worry about all the errors. I doubt this one is fixable.
Incorrect. You are misunderstanding what a probability says. Suppose a ball has a 1/3 chance of landing in bin 1 and a 2/3 chance of bin 2. You throw one ball. But you did not see which bin it landed in. What do you know? You know it has a 1/3 and 2/3 chance of being in bin 1 or 2. Those probabilities apply to just one throw. A Romu generator uses one permutation. But you don't know the cycle-lengths. What do you know? You know the probabilities of various lengths. Those probabilities apply to just one permutation, just as they do for one ball-toss.
Bob Jenkins created the JSF generator (http://burtleburtle.net/bob/rand/smallprng.html), which is similar to Romu except it uses additions instead of a multiplication. He wrote an article on designing PRNGs (http://burtleburtle.net/bob/rand/talksmall.html#reversible) in which he explains theory briefly and clearly, so this article might help you.
> [64! possible permutations] All Roma permutations are a vanishingly small subset of these
Yes. In fact, only one permutation is used. But that's irrelevant because the cycles are what matter. And I showed above that the cycles obey the equations for their probable lengths.
> with very specific structure.
Incorrect. Page 7 of the paper contains graphs of all the cycles in several generators with 32 bits of state, with measurements of their randomness. The cycles performed nearly ideally -- the 45-degree line in the graph. So their structure is not "very specific" (nonrandom); rather, they pass tough tests of randomness.
> Don’t worry about all the errors. I doubt this one is fixable.
There is no error. Again, read Bob Jenkins' article for a clear discussion of cycle lengths.
Let's not discuss the topic of permutations and cycle lengths any further; it won't be productive.
Instead: Please post your list of perceived errors in the paper, and let's talk about those. It'll only take a few minutes to write, and will benefit many people because I will clarify unclear parts of the paper.
I have a PhD in math. I've worked on this stuff for decades. Among other work, I wrote a decently cited article on the Hidden Subgroup Problem for quantum computation on arxiv, and this problem requires great understanding of the group structure of the permutation groups as well as a solid grounding in probabilty. It's not likely I am the one who misunderstands probability (or, as I will show you, permutations).
>A Romu generator uses one permutation. But you don't know the cycle-lengths. What do you know? You know the probabilities of various lengths.
Take your page 2 generator, switch to a 32 bit state so you can empirically check everything with code. The largest cycle length is 1499760802, or 34% of the state space. Check each cycle. The results contradict your "proof" result (2) that the probability P(|cycle with x|<= 2^k) = 2^k/(2^s-1). You simply don't get the bounds you claim.
The next two largest cycles are 731971696(17%), 1137668726(26%) to aid you in your checking. They fall off pretty quickly after that.
Maybe we were just unlucky, and the original constant caused the failure. Pick another multiplicand. Whoops. Same result. Try again. Same result. In fact, you will fail to meet the bounds of your "proof" for any multiplicand value.
So, go ahead and tell me how to use a 32 bit state version of the page 2 generator for which we can compute exact cycle lengths that also meets your bounds. I'll wait.....
Thus your "proof" fails to give correct bounds, for precisely the reasons I stated earlier. And thus your table 2 is incorrect - you don't understand the structure of your PRNG, or that it does not satisfy the theorems you are applying to analyze it. Otherwise the "proof" would be a proof. Math doesn't lie.
And before you next make the error of claiming the quality increases with bit size, that is usually false. So many people have fallen into this trap in both PRNG creation and crypto. When each choice in your "proofs" is smaller than you claim, often these accumulate faster than the space increases, and you can have catastrophic failure in behavior.
Another way to see it: using your page 2 generator again as an example (the others have the same flaws): take the state, mult by your odd, then the last bit is still the same. Rotate by half the bits. Now a middle bit of the next state is always the same as the last bit of the previous state. This is not random, and your proof requires that each choice in it is allowed to choose from the entire remaining possible state values. But we just cut that in half. This problem happens at every bit - there are too many relations between them due to the simplicity of your method to use that proof.
Another way to see it: take 64 bit numbers and consider cycles from some permutation. There are (2^64)! such permutations, a number which has ~10^20 digits. It's an astounding size. The proofs apply when you choose your permutation unifrmly from this space. The word "uniformly" has precise mathematical meaning, and you no where near satisfy it. If you don't have a clear understand of what the word means and why you don't meet this requirement then you need to learn what it means. Your simple ROMA has a 64 bit constant (2^64 choices) and a rotation (a generous 64 choices). The number of such choices has about 21 digits, off by around 20 orders of magnitude. You are choosing from an infinitesimally tiny, well-strucutred subset of all the permutations. The theorems do not apply. The "well-structured" part will kill you in practice, just like here.
Another way to see it: your "proof" doesn't mention your choices, thus your "proof "should hold for any constant. But it doesn't, which is easy to check.
Hopefully that's enough different directions on why your cannot conclude what you did that you see one. Of course the code example shows it without any question.
Finally, your entire method is a weaker version of PCG. Your first step is a MCG, which the PCG paper uses in the same manner, then you do a fixed rotation, wheras they use a varying rotation. You cannot prove anything, like cycle lengths or any equidistribution, but they, using the superior structure of PCG, can prove both.
You claim yours is faster, but you don't compare against the best PCG methods - choice or accident? Without timing, PCG is likely as fast as yours, with demonstrable quality RNGs.
And here's the kicker - the main reason people want PRNGs this fast are things like Monte Carlo, where large cycles and k-equidistribution is of serious concern, and you cannot provide one and make demonstrably incorrect claims on the other.
>Bob Jenkins ... which is similar to Romu except it uses additions instead of a multiplication
Bob does not make the errors you make; he carefully avoids them. His uses an XOR in the middle of some add, subtract, and rotates. This is fundamentally different. He even states this in the description you recommended I read: "For example, + is linear mod 2^32 and XOR is linear in GF(2^32), but the combination of + and XOR isn't linear. You combine + and XOR, and linearity goes away."
You should read it and understand it too. That you see a plus in his and claim it as a defense of yours without understanding what he did and why doesn't create much faith in your analysis.
Your sequence overlap "proofs" suffer from the same errors - and you can check them with the 32-bit versions of the code where you can empirically compute all cycles and check odds. Part of this follows from (2) being wrong, as demonstrated with code empirically, and being used as a basis in section 3.2.
Without proper proofs I'd recommend no one use this for serious work. There is no real benefit to using it and so many wrong things with it as demonstrated.
>Let's not discuss the topic of permutations and cycle lengths any further; it won't be productive.
Agreed. Maybe you can either provide a 32 bit ROMU that meets your claimed bounds in (2) or replace your "proofs" with correct ones that agree with exhaustive experiment, then we have progress.
Both use a mult to scramble, and then retain a subset of the bits. Adding more rotates and mults does not change the underlying issues.
This type of PRNG has been pretty thoroughly mined over the past 70+ years. You should dig through this work and the following literature on why PRNGs of this structure fell out of favor. I suspect exactly yours appears somewhere following this work.
I don't think there is any reason to use non-cryptographic PRNGs. (CTR_DRBG is deterministic if seeded with a fixed seed, so that's not an excuse either).
I guess my point was there are often legitimate uses for things that are random-like but far from crypto random.
// Romu generators, by Mark Overton, 2020-2-7.
//
// This code is not copyrighted and comes with no warranty of any kind, so it is as-is.
// You are free to modify and/or distribute it as you wish. You are only required to give
// credit where credit is due by:
// (1) not renaming a generator having an unmodified algorithm and constants;
// (2) prefixing the name of a generator having a modified algorithm or constants with "Romu";
// (3) attributing the original invention to Mark Overton.
The more likely outcome though is that it's copyrighted, because things just are copyrighted unless very clearly stating otherwise. CC0 needs three pages to explain just how "not copyrighted" the associated work is.
Personally I would never use or depend on code that has a license its author wrote themselves. It's just too risky and other people will be less inclined to use whatever project I wrote that uses the weirdly-licensed code.
Boost licence
Unlicense
WTFPL
> The copyright notices in the Software and this entire statement, including the above license grant, this restriction and the following disclaimer, must be included in all copies of the Software, in whole or in part, and all derivative works of the Software, unless such copies or derivative works are solely in the form of machine-executable object code generated by a source language processor.
It's similar to MIT or Apache-2.0 in that respect. As for Unlicensed and WTFPL, sure they're both effectively the same as CC0 (though I would personally be cautious about removing copyright notices from a WTFPL codebase).
... for the seeds that were tested.
In theory, you can get unlucky and end up in a short cycle, and it will not pass these tests.
That's why I'm not comfortable with it, when there are generators with a known, guaranteed period. They simply do not have this failure mode.
Again, the application determines what the most suitable PRNG is.
For non-simulation purposes, however, using a fast-key-erasure[2] ChaCha20 or AES-256-CTR RNG has served me well enough, and doesn't leave me worrying if maybe someone could abuse the RNG for whatever purposes (e.g. cheating in a game by using RNG manipulation).
Like this website, the PCG website also makes rather bold claims and has no peer review to back it up. What's the deal with this?
I wish, as in the romantic ideal of science, a consensus would emerge of what “the best” PRNG are, and how to choose among them. Instead, you seem to have many academics promoting their own ones (Vigna with the xoroshiro family, O’Neill with PCG, now this guy Overton with Romu).
And in effect every programming language has to go and make sensible default choices, while the experts in PRNG refuse to make unambiguous recommendations.
(FWIW, I’d advocate to have, by default, a cryptographically secure PRNG, and offer a choice of faster ones (like the three mentioned above) that have to be chosen explicitly, maybe naming them InsecureXYZ, for applications where speed is paramount.)
I would like to see a systematic comparison, some way of arriving at an answer to the question of which of these is "best."
I suspect that large part of the problem is that there are not all that many academics in the field, promoting their own or otherwise. Ultra high speed PRNGs are just a bit of a niche, and the widely used stuff is already good enough for many industry uses.
O'Neill also uses the same methods to study PCG.
There are many bold claims on this website. But you don't screw around with RNGs: they need to be right. Until this author's claims have been verified, I would not use this algorithm.
> In effect, Romu generators are infinitely fast when inlined.
What the hell does "infinitely fast" even MEAN in this context? They can be parallelized easily? The computational cost amortizes well? That it literally requires zero operations to generate output?
This may be something interesting if it packs a lot of the features claimed. But there are a LOT of ill-defined claims on the website, and a fair bit of time spent pointing to the state of current research in OTHER algorithms/proposals as if open questions are an automatic disqualification ("[H]ow can you know whether such a generator has enough capacity for your large job? You don’t know.").
I don't think the guy is a crank, per se-- just excited. But this is the sort of stuff that makes me think that he hasn't done the requisite level of research required before claiming a breakthrough.
It's nonsense, but their argument seems to be that instruction level parallelism allows the RNG's user to keep executing simultaneously while the RNG pre-computes its next output. So the application doesn't have to delay execution to wait for the RNG to spit something out.
It doesn't mean that it is infinitely fast. It does however mean that in this particular case it adds no latency for adding random number generation; an infinitely fast algorithm would also add no latency for random number generation. Doesn't mean one implies the other.
static uint128_t state = 1; // can be seeded to any odd number
static inline uint64_t next(){
return (state *= 0xda942042e4dd58b5ULL) >> 64;
}
GCC and clang do have a 128-bit integral type, __int128, but arithmetic operations are synthesized inline, and judging by the generated assembly don't even make use of SIMD at all.
Note that an unintended consequence of C99's uintmax_t made the introduction of a standard 128-bit integral type impossible on platforms that cared about ABI compatibility, an effect somewhat contradictory to the original intention of improving portability and code integration. A future C standard will loosen some language and introduce some new facilities to help promote the introduction of larger integral types. At that point Unix platforms may begin typedef'ing __int128 to int128_t.
Witness the security researcher's mindset. RNGs aren't just used in security applications.
A large chunk of RNG needs are in simulation. Simulation has no need for security guarantees. It does need to be as fast as possible. A great many secure generators are pokey.
state_t s;
uint rng_next() {
X;
return Y;
}
state_t s;
uint next;
uint rng_next() {
uint n = next;
X;
next = Y;
return n;
}Only minor nit: would be nice if they had ready to compile libraries. The ROTL macro is missing, and useful wrappers are missing too (like correct limiting to a smaller range) Of course it's trivial to implement, and I like the simplicity of the algorithm to copy it into programs.
And both vectorization and inlining works fine with a function with modern tool chains that do LTO.
It's missing in the landing page, but the paper [1] linked at the bottom of it provides two implementations:
// C
#define ROTL(d,lrot) ((d<<(lrot)) | (d>>(8*sizeof(d)-(lrot))))
// C++
template <class uDataT>
inline uDataT rotl (uDataT d, unsigned lrot) {
return (d<<lrot) | (d>>(8*sizeof(d)-lrot));
}