I'm new to cryptography. I want to ask about OTP-perfect secrecy diagram like figure below:

enter image description here

On the overlapping region (middle), notated by R(X;Y;Z). R can be calculated by I(X;Y) - I(X;Y|Z). That region is symmetric in X, Y, Z and can be negative. For example when X and Y are independent bits, and Z = X xor Y, then I(X;Y) = I(X;Z) = I(Y;Z) = 0, but I(X;Y|Z) > 0.

What is the "meaning/philosophy" of that region? I mean when the region is negative, when the region is positive, and how to describe that?

I have been trying to find the answer but no result.

  • 3
    $\begingroup$ Could you define what $H$, $I$, and $R$ mean? $\endgroup$
    – mephisto
    Sep 17 '15 at 10:53
  • 4
    $\begingroup$ I'm assuming H is entropy (or conditional entropy), I is mutual information (or conditional mutual information), R, not sure. Where did this image come from? A reverse image search turned up nothing. $\endgroup$
    – mikeazo
    Sep 17 '15 at 11:50
  • 2
    $\begingroup$ Give us some reference to help u better $\endgroup$
    – sashank
    Sep 17 '15 at 12:33
  • $\begingroup$ all : sorry for late reply, i just returned from a trip. Thanks for your response --- @mephisto : H entropy, I mutual information, R I still do know, but according to Chris's answer bellow, R is multivariate mutual information (???) --- mikeazo & sashank : I picked up the pic from lecture note in computer science, unconditional security in cryptography by stefan wolf $\endgroup$
    – stranger
    Sep 20 '15 at 22:38

Your diagram is a Venn diagram that illustrates the information measures between the correlated random variables $X,Y$ and $Z$.

  • $H(X)$ refers to a complete circle and is the entropy of $X$,
  • $H(X|YZ)$ is the entropy of $X$ under the observation of $Y$ and $Z$,
  • $I(X;Y|Z)$ is the mutual information between $X$ and $Y$ under the observation Z,
  • $R(X;Y;Z)$ seems to represent $I(X;Y;Z)$, the mutual information between X,Y and Z.

The region $R(X;Y;Z)$ is thus the amount of information that is shared by the random variables $X,Y$ and $Z$. If the random variables are completely independent, then this region is empty.

I don't know exactly how this is related to OTP. I guess that you typically want maximal entropy for OTP, and you don't want to have any mutual information between the variables.

PS: After writing most of this answer I found a wikipedia page about multivariate mutual information, which seems to correspond exactly to this question.

  • $\begingroup$ Thank you very much Chris.. I think i found the answer from you.. I will read it first.. $\endgroup$
    – stranger
    Sep 20 '15 at 22:40

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