Am Freitag, 29. April 2016, 14:02:48 schrieb George Spelvin: Hi George,
> >> 1. It DOES depend on the attacker. Any statement about independence > >> > >> depends on the available knowledge. > >> > >> 2. XOR being entropy preserving depends on independence ONLY, it does > >> > >> NOT depend on identical distribution. The latter is a red herring. > >> (An English metaphor for "irrelevant distraction.") > >> > >> 3. Precisely because the bits are not independent, XOR is not > >> > >> guaranteed to be entropy-preserving (your sense) on real data. > > > > It seems we talk past each other. Your entire explanation refers to > > individual bits that come in sequentially where the attacker inbetween > > can analyze it and potentially modify his attack. Sure, in this case > > they are not independent and I am not claiming that. > > You can analyze it in two equivalent ways: > > 1. The attacker knows all previous timestamps, and we are tring to > quantify their uncertainty about the current timestamp word. > > In this case, some of the attacker's knowledge will be about > correlations between the bits of that word. > > I think this is the easier way to think about the problem and > the formulation I prefer. Especially because of the structure > of the work you do afterward, XORing those 32 bits. > > 2. An attacker knows all previous *bits*, and is presented with, and > tries to guess, the 32 bits of the timestamp one at a time. > > In this case, information gleaned from previously-seen bits which > would be called correlations in option 1 get incorporated into the > predicted distribution of the current bit. > > For measuring the source entropy, this is also a valid way to > proceed. It's like Shannon's early studies of the entropy of > English by letting readers read the first half of a novel and then > asking them to guess what follows, one letter at a time. > > I think this form is less convenient in general, and it's particularly > annoying when subsequently computing the parity of a word, as it's > hard to talk about cancellations due to non-independent bits. > > Entropy (Shannon, Renyi, and min-) is additive, meaning that the two > different ways of measuring will produce the same result. > > Either way, word or bit at a time, we are trying to quantify the *new* > additional entropy contributed by the current sample. That means > we assume for the analysis of each sample that all previous samples > are known to the attacker. > > > But one single time stamp is one value where the entire 32 bits are > > generated in an atomic fashion to any observer -- there is no sequential > > obtaining of its bits, analyzing it and then reacting on it. > > I never suggested doing it any other way, although as I explained above > it's possible to do so. > > I was only considering processing *words* sequentially. > > Knowledge of the previous *words* affect the predicted distribution > of individual bits in the current word. That is all correct what you write and I concur. But you still do not answer the point I am making. You always in all your descriptions compare two or more time stamps. And I always concured that they are dependent -- and XOR will do whatever to the entropy. What I am saying that the bits in one given time stamp are mutually independent. I.e. bit 0 of one time stamp does not depend on bit 1 of that very same time stamp. I totally agree with you that bit 0 from time stamp 1 may tell you something about bit 0 of time stamp 2. And therefore I am not considering multiple time stamps for this first step. Let me quote the definitions from SP800-90B: IID: A sequence of random variables for which each element of the sequence has the same probability distribution as the other values and all values are mutually independent. Independent: Two discrete random variables X and Y are (statistically) independent if the probability that an observation of X will have a certain value does not change, given knowledge of the value of an observation of Y (and vice versa). When this is the case, the probability that the observed values of X and Y will be x and y, respectively, is equal to the probability that the observed value of X will be x (determined without regard for the value of y) multiplied by the probability that the observed value of Y will be y (determined without regard for the value of x). All I am claiming and all I am saying is that the bits 0 through 31 in one given time stamp are mutually indpendent. And thus the XOR of those independent bits is appropriate. > > When woring word-at-a-time like this, we also have to consider > cross-correlations among the bits of a word. The most general way to Tell me where the correlations should be within one word. Where do you claim they come from? > express it is a set of 2^32 probabilities for each of the 2^32 possible > values. The awkwardness of this form is why it's sometimes useful to > think about smaller pieces. > > For example, if p(9) = 0.5, p(10) = 0.5, and p(x) = 0 for all other x, > then we have 1 bit of entropy in the word. > > We can analyze it bit a time, and proceed in oe of two ways: > > - If we go lsbit first, then we have 1 bit of entropy in bit 0, but > then zero entropy in bit 1. > - If we go msbit first, we have 1 bit of entropy in bit 1, but then > zero entropy in bit 0. This description makes no sense. > > Either way, there's only 1 bit of entropy total because of the > correlation between the bits. Once we have seen one of the bits, the > entropy of the second one collapses to 0. Again, where does the correlation is supposed to come from? > > And either way, we have 1 bit of entropy in the word, but 0 bits of entropy > in the parity of the word. Ciao Stephan
