Splittable random numbers

[Simon Peyton-Jones]

Hi Cafe

A while back there was a thread<http://www.mail-archive.com/haskell-cafe@haskell.org/msg79633.html> about a good implementation of a (pseudo) random number generator with a good "split" operation.  There's lots of material on generators that generate a linear sequence of random numbers, but much less on how to generate a tree of random numbers, which is what Haskell's System.Random API requires.

I happened to meet Burton Smith recently, who wrote some early papers about this stuff (eg "Pseudo random trees in Monte-Carlo<http://portal.acm.org/citation.cfm?id=1746034>"), so I asked him.

His reply is below, along with some follow-up comments from his colleagues Tolga Acar and Gideon Yuval.   The generator uses crypto functions, so it's probably more computationally expensive than common linear-sequence generators, but in exchange you get robust splitting.

Does anyone feel like taking the idea and turning it into a Haskell library?   (Or even a Haskell Wiki page?)  I'm taking the liberty of cross-posting to the libraries list.

Simon


From: Burton Smith
Sent: Tuesday, November 02, 2010 3:58 PM
To: Simon Peyton-Jones
Cc: Gideon Yuval (Gideon Yuval); Tolga Acar
Subject: Random number generation

With some help from Gideon and Tolga, I think the solution to the "arbitrary tree of random numbers problem" is as follows:

The generator G is a pair comprising a crypto key G.k and an integer counter (the "message") G.c.  The (next G) operation returns a pair: 1. a random integer r obtained by encrypting G.c with G.k, and 2. a new generator G' such that G'.k = G.k and G'.c = G.c + 1.  The (split G) operation is similar, returning the same G', except that instead of returning a random integer r it returns a third generator G'' such that G''.k = r and G''.c = 0.

A suitable block cipher system might be 128-bit AES (Rijndael).  Unencumbered implementations exist in a variety of languages, and performance is pretty good and will improve dramatically as hardware support improves.  I'd pick both crypto key size and the size of the result r to be 128 bits, and employ a  64 bit counter c.  Other crypto options exist.

From: Simon Peyton-Jones
Sent: Wednesday, November 03, 2010 3:11 AM
To: Burton Smith; Gideon Yuval (Gideon Yuval)
Cc: Tolga Acar; Simon Peyton-Jones
Subject: RE: Random number generation

Burton, Gideon, Tolga

Aha, that's interesting.   I'd never seen a random number generator based on crypto, but it seems like an attractive idea.  As I understand it, successive calls to 'next' will give you
          encrypt(0), encrypt(1), encrypt(2), encrypt(3),....

Is this standard?  Does it have provably good randomness properties, (cycle length, what else?) like other RNGs?  Or does it simply seem very plausible?

Can I send it round to the Haskell mailing list, in the hope that someone will turn the idea into a library?   (Ideally I'd like to make claims about the randomness properties in doing so, hence my qns above.)


From: Gideon Yuval (Gideon Yuval)
Sent: Wednesday, November 03, 2010 7:15 AM
To: Simon Peyton-Jones; Burton Smith
Cc: Tolga Acar
Subject: RE: Random number generation

As long as the key, and the non-counting part of the counter, are kept" secret", anyone who can distinguish these pseudorandoms from real random, in less than 2^128 steps, has a nice paper for crypto-2011 (this is known as "provable security") concerning a weakness in AES128.

One exception: real randoms have a birthday paradox; the pseudorandoms suggested do not. If you care, you can:

(1)    Limit the counter to 2^32 steps (paradox has 2^-64 probability) or even 2^16 (2^-96), then rekey; or

(2)    XOR 2 such encrypted counters, with different keys; or

(3)    XOR 3 successive values for the same counter (just possibly cheaper; top-of-head idea).

More hard-core: swap the position of key & message: encrypting a constant "secret" with 1,2,3,4.... Gives pseudorandoms with no birthday paradox.


From: Tolga Acar
Sent: 03 November 2010 15:50
To: Gideon Yuval (Gideon Yuval); Simon Peyton-Jones; Burton Smith
Subject: RE: Random number generation

Simon,

The general idea is not really that new in the crypto area with constraints Gideon describes, of course. That is typically called a PRNG - Pseudo Random Number Generator, or in another parlance, Deterministic Random Bit Generators (DRBG). The DRBG constructions based on hash functions and block ciphers are even standardized in NIST publication SP800-90 (even though I may not recommend every one of them).

As for the construction below, that is based on the AES block cipher, that essentially takes advantage of the PRP (Pseudo Random Permutation) property of the AES block cipher, as each block cipher ought to be. So, as Gideon outlines below, if you fix the key, the PRP gives you a random-looking (or, in other terms, indistinguishable from random) output that no one without the secret key and the "state" can generate or easily predict. Assuming an ideal cipher (and AES is a close approximation to it), the probability is believed to be the birthday paradox - no counterexample or a proof exists without assumptions; so we stick to the birthday bound.

There ought to be papers on this somewhere. If not, that condensed information is spread across many papers and is part of the crypto folklore, I'd say.


From: Burton Smith
Sent: 03 November 2010 19:03
To: Simon Peyton-Jones
Cc: Gideon Yuval (Gideon Yuval); Tolga Acar
Subject: RE: Random number generation

Just two points of further clarification:

1.        All PRNGs used in the technical computing space fail the birthday paradox criterion (i.e. have full period), so we really need not worry about this.  Also, there are other mitigating factors, e.g. suppose we are using the PRNG to generate pseudorandom residues mod n << 2^128; the paradox is happily present there.

2.       The big innovation in this scheme is that the rekeying operation done by split creates a new generator with  independence guaranteed by "provable security" in the sense Gideon mentioned - if someone can find something nonrandom in the correlation between G' and G'', say, then it amounts to a weakness in AES128 and is publishable.  So it's yet another example of reducibility, common in our field: "if you can easily transform a known/famous hard problem P into this other problem Q, Q must be hard".

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