Re: [PATCH] (0/4) Entropy accounting fixes

From: Tommi Kyntola (
Date: Tue Aug 20 2002 - 11:19:26 EST

On Tue, 20 Aug 2002, Oliver Xymoron wrote:
> On Tue, Aug 20, 2002 at 11:59:49AM +0300, Tommi Kyntola wrote:
> > On Mon, 19 Aug 2002, Oliver Xymoron wrote:
> > > On Mon, Aug 19, 2002 at 01:43:59AM -0400, Theodore Ts'o wrote:
> > > > On Sat, Aug 17, 2002 at 09:15:22PM -0500, Oliver Xymoron wrote:
> > > >
> > > > > Assuming the interrupt actually has a nice gamma-like distribution
> > > > > (which is unlikely in practice), then this is indeed true. The
> > > > > trouble is that Linux assumes that if a delta is 13 bits, it contains
> > > > > 12 bits of actual entropy. A moment of thought will reveal that
> > > > > binary numbers of the form 1xxxx can contain at most 4 bits of
> > > > > entropy - it's a tautology that all binary numbers start with 1 when
> > > > > you take off the leading zeros. This is actually a degenerate case of
> > > > > Benford's Law (, which
> > > > > governs the distribution of leading digits in scale invariant
> > > > > distributions.
> > > > >
> > > > > What we're concerned with is the entropy contained in digits
> > > > > following the leading 1, which we can derive with a simple extension
> > > > > of Benford's Law (and some Python):
> > I think you have it slightly wrong there. By snipping away the first digit
> > from a number leaves you with, not Benford's distribution, but
> > uniform distribution, for which the Shannon entropy is naturally roughly
> > the bitcount.
> No, it's much more complicated than that - that doesn't give us scale
> invariance. Observe that the distribution of the first and second
> digits in base n is the same as the distribution of the first digit in
> base n*2. The probability of finding a 0 as the second digit
> base 10 is simply the sum of the probabilities of finding 10, 20,
> 30,..90 as the first digit, base 100, see? It's .1197, rather than the
> expected .1. In base 2, the odds of the second digit being 0 is .58.

Oh now I see, you're relying on the given times to be gamma distributed
(1/x, and thus naturally scale invariant). I was constantly thinking about
the jiffie count that got masked that I took as uniform, but naturally
within the delta context it's not uniform, since the first order delta
naturally defines the timestamp when the previous timestamp is known.
And thus yes, you are indeed right. Furthermore it wouldn't even have to
be gamma and as such scale invariant. Even mere nonuniformness (which I'm
sure the timings are) guarantees that mere first bit snipping is not
enough to ensure entropy inside that full range.

The current the delta bitcount - 1 is not the correct entropy amount in
any said gamma (or likes) distributed number. Does strict gamma assumption
really lead to so strict figures as you showed in your patch :
static int benford[16]={0,0,0,1,2,3,4,5,5,6,7,7,8,9,9,10};

Numbers below 0..7, don't have a single bit of entropy?

I'm more inclined to think that where as this may be sound for the larger
delta bit counts, I don't think it applies that well for the smaller
deltas. It's unlikely that the timing distributions stay scale invariant
for the lower end bits.

Not mention that if the actual time separation is something like 12 bits,
but the second order delta drops the entropy bit count down to 4,
couldn't those four be considered uniformly distributed, atleast roughly
enough, so that the benford array reduction could be skipped.
Because for a 12 bit numbers the 4 lower bits are by far not scale
invariant. Atleast that part of the patch might need tweaking.
Or did I miss something?
I can't say for sure though, gotta sleep on it first.

Although Linus was not so hot about your "draconian measures" patch set
this part of it would atleast seem worth the while. Atleast when the
limiting delta is the first order, it is indeed unreasonable to think
that it's bit count - 1 would show the entropy that's present.

> > Wether the bit count of the smallest of the three deltas is
> > actually sufficient to guarantee us that amount of randomness in the
> > choice is another question. Like stated here already, it can be easily
> > fooled, and there's a strong possibility that it gets "fooled" already.
> That's why my code makes a distinction between trusted and untrusted
> sources. We will only trust sources that can't be used to spoof us.
> Detecting spoofing is impossible.

Fully agreed.
I was merely suggesting that it might be if not common, atleast not
totally out of the question, that certain interrupt or interrupt bursts
kick in with predictable intervals (tcp handshake and other protocol
related stuff, that might lead to predictable network traffic), e.g. 1 13
2 50, which would completely fool the delta analysis. But yes, such cases
should be ruled out by source selection. As we have, or atleast last I
checked the network irqs didnt contribute entropy, although IMHO even in
network traffic there could be some level of entropy that could be gotten,
just not as simply as presently from user input timings to be fully
> > Some level of fourier analysis would be necessary to go further than what
> > we can with the deltas.
> There's no point in going further. If an attacker is trusted, he can
> send timing streams that would fool _any_ filter. An example is some
> subset of the digits of pi, which appear perfectly evenly distributed
> but are of course completely deterministic.

I couldn't agree more.

Tommi Kynde Kyntola
      "A man alone in the forest talking to himself and
       no women around to hear him. Is he still wrong?"

- To unsubscribe from this list: send the line "unsubscribe linux-kernel" in the body of a message to More majordomo info at Please read the FAQ at

This archive was generated by hypermail 2b29 : Fri Aug 23 2002 - 22:00:20 EST