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From what I could find it relates input differences to output differences usually across multiple rounds.

But is it the entire probability distribution over all output differences for one input difference? Or is it the list of differences that produce the output difference with the highest probability?

What is the exact definition of differential trail?

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Imagine you have a function like this: $$f(x) = p_3(p_2(p_1(x))).$$ Now imagine that you find a pair $\Delta_0, \Delta_1$ such that $p_1(x \oplus \Delta_0) = p_1(x) \oplus \Delta_1$ with probability $2^{-n_1}$, $\Delta_2$ such that $p_2(x \oplus \Delta_1) = p_2(x) \oplus \Delta_2$ with probability $2^{-n_2}$, and $\Delta_3$ such that $p_3(x \oplus \Delta_2) = p_3(x) \oplus \Delta_3$ with probability $2^{-n_3}$. Then:

  • $\Delta_0 \xrightarrow{f} \Delta_3$ is a differential for $f$ (with probability at least $2^{-n_1 n_2 n_3}$ if we can assume independence);
  • $\Delta_0 \xrightarrow{p_1} \Delta_1 \xrightarrow{p_2} \Delta_2 \xrightarrow{p_3} \Delta_3$ is a differential trail, also known as differential characteristic, for $f$ (with probability $2^{-n_1 n_2 n_3}$ if we can assume independence).

So, a differential trail contains not only the input and output differences going into your function, but also all of the intermediate values (and respective probabilities). You can also say that a differential is the combination of all the differential trails with the same input and output values.

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