# Measuring Shannon's diffusion

Shannon's idea of diffusion is fundamental to cryptography.

Besides being a descriptive idea, is there any work on measuring or expressing it? Saying something like "System A has more diffusion than B" or "The information in bit x is diffused 1/3 into bit y, 1/3 into bit z, and the remaining third among 11 more bits".

Is there a way to evaluate the diffusion of a transform? XOR does no diffusion, Add with carry does small diffusion, and a permutation may do a lot of diffusion.

In short: Diffusion is a great idea - how can we approach it mathematically?

UPDATE: For instance, we should be able to quantify the diffusion done by adding with carry. This breaks both of the suggested answers: Adding may update the value of every bit. Modifying one bit of the input may modify several bits of the output. But adding still doesn't diffuse very well.

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Measuring the number of rounds a full diffusion takes is common. – CodesInChaos Feb 12 '13 at 17:45

One way is to look at how many bits of the outputs are modified if you modify, say $t$ bits in the input. Actually, a function from $\{0,1\}^n$ to $\{0,1\}^m$ such that modifying $t$ bits in input modifies $m-t+1$ bits of output is said to achieve perfect diffusion.
It would be nice to disambiguate the definition or/and link to a reference. A small example could help. As is, we are left to guess if "modify $j$ bits" is to be taken as exactly $j$ bits, or has some statistical meaning as in other usages of diffusion; the range of $t$; and for which $(m,n)$ perfect diffusion can hold. – fgrieu Feb 13 '13 at 7:29