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Best ways to estimate average bits of entropy in a byte stream.

I know there are different statistical tests out there (NIST, Dieharder, etc), which all do different ways of analyzing entropy.

What I'm having a hard time finding is any particular literature which describes how to go from those tests to actual bits of entropy in the byte stream.

How do you go from p value of a byte stream (say a 100 MB long) to bits of entropy? And as an aside and very related — what are the "top 3" best ways?

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The common notion of entropy is the notion of Shannon entropy. The Shannon entropy H(x) of a value x that occurs with probability Pr[x] is H(x) = -log_2(Pr[x]). Related questions: crypto.stackexchange.com/q/378/6961 and crypto.stackexchange.com/q/700/6961 –  e-sushi Sep 18 '13 at 3:32
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There are two issues with Shannon entropy: 1) It's only defined for a probability distribution, not for an individual string 2) Shannon entropy and average key-strength aren't exactly the same thing if the probability distribution isn't uniform. –  CodesInChaos Sep 18 '13 at 11:32
    
I've made some progress here. When I have a full answer, I will post it (especially given all the great help I've got it). I'm currently investigating Ping Li's work here: stanford.edu/group/mmds/slides2010/Li.pdf Ideally though I'd do something based off of NIST's work here: csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html –  Blaze Sep 23 '13 at 22:16
    
Can you tell us more? Why do you want to measure the amount of entropy? What do you know about the source of the byte stream? The answer is going to depend heavily on the answers to these questions and on other details, so if you can give us more details, we might be more likely to be able to give you a good answer. This is not a simple subject with a simple one-line answer... –  D.W. Oct 1 '13 at 22:49
    
No, I can't. I really want to simply estimate entropy. –  Blaze Oct 3 '13 at 17:25

2 Answers 2

Entropy is a function of the distribution. That is, the process used to generate a byte stream is what has entropy, not the byte stream itself. If I give you the bits 1011, that could have four bits of entropy or it could have zero. In fact, it only has one bit of entropy, but you have no way of verifying that.

Here is the definition of Shannon entropy. Let $X$ be a random variable that takes on the values $x_1,x_2,x_3,\dots,x_n$. Then the Shannon entropy is defined as

$$H(X) = -\sum_{i=1}^{n} \operatorname{Pr}[x_i] \cdot \log_2\left(\operatorname{Pr}[x_i]\right)$$

where $\operatorname{Pr}[\cdot]$ represents probability. Note that the definition is a function of a random variable (i.e., a distribution), not a particular value!

So what is the entropy in a single flip of a coin? Let $F$ be a random variable representing such. There are two events, heads and tails, each with probability $0.5$. So, the Shannon entropy of $F$ is:

$$H(F) = -(0.5\cdot\log_2 0.5 + 0.5\cdot\log_2 0.5) = -(-0.5 + -0.5) = 1.$$

Thus, $F$ has exactly one bit of entropy, what we expected.

So, to find how much entropy is present in a byte stream, you need to know how the byte stream is generated and the entropy of any inputs (in the case of PRNGs). Recall that a deterministic algorithm cannot add entropy to an input, only take it away, so the entropy of all inputs to a deterministic algorithm is the maximum entropy possible in the output.

If you're using a hardware RNG, then you need to know the probabilities associated with the data it gives you, else you cannot formally find the Shannon entropy (though you could give it a lower bound if you know the probabilities of some, but not all, events).

But note that in any case, you are dependent on the knowledge of the distribution associated with the byte stream. You can do statistical tests, like you mention, to verify that the output "looks random" (from a certain perspective). But you'll never be able to say any more than "it looks pretty uniformly distributed to me!". You'll never be able to look at a bitstream without knowing the distribution and say "there are X bits of entropy here."

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Hmm, that doesn't seem to jive with what I've been asked unfortunately. I think if you have enough data and enough tests, you should be able to come up with a reasonable estimate of entropy. –  Blaze Sep 18 '13 at 0:33
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I think what people are looking for is heuristic analysis + statistical analysis. For example, if I flip a coin, I can theoretically say 1 bit of entropy. Now, let's flip that coin 10 thousand times and do a statistical analysis to see if that measures up. If I get all heads, then so much for my heuristical analysis... I'm pretty sure this is the motivation. The part I'm missing is how to go from statistical analysis to 'bits per byte' of entropy. Note this request comes from experts. –  Blaze Sep 18 '13 at 2:53
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@Blaze: My whole point is that you can't go from a statistical analysis to a measurement of entropy. If you happen to know how the bitstream is generated, then you can calculate the entropy directly (via the above formula) --- but if you don't know how it's calculated, then it simply can't be done. –  Reid Sep 18 '13 at 3:45
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I would think we could pragmatically estimate entropy via a combination of statistics and good analysis. Empirically measure a statistical distribution on the lowest level of the entropy source that you can. Calculate $H$ where $Pr[x_i]$ is calculated from that distribution. How good an estimate of entropy this produces will rely on how good of a choice was made for "lowest level". Eg, if you chose the output of a PRG seeded by 0 you will have a false high entropy due to a poor decision. But if you choose a level the attacker won't have deeper insight to, the estimate should be good. –  B-Con Sep 18 '13 at 4:58
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Do we have anything better than that for practical entropy estimates in real life? We don't (well, rarely) deal with truly random events, just events where we have a distribution and very little additional insight beyond that distribution. (Top of my head examples: hard drive seek times, mouse coordinates, etc. They aren't truly random, but at some level of detail you're stuck with a distribution of behavior and no way to analyze the source any more finely.) –  B-Con Sep 18 '13 at 5:03
up vote 0 down vote accepted

There are some great tests out there: Draft Special Publication 800-90B - National Institute of Standards

In particular the min-entropy, partial collections, Markov (useful for non-IID sources), collisions, and compression tests.

The issue with the Markov test is the constraint on bit size of the sample.

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