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I'm currently reading the tutorial on linear/differential crypta here: https://www.engr.mun.ca/~howard/PAPERS/ldc_tutorial.pdf

End of page 14/Beginning of page 15: all the bits from the subkeys are summed up to a sum(k) variable. It is then written as 0 in the equation (5) and the probability stays the same.

I don't understand this step, why is the probability not changing if sum(k) is zero, and why is it 1-p if sum(k) is 1?

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So after re-reading a few times it becomes clear. The key bits are static, so they won't change the probabilities.

Now suppose that $\sum_K = 0$, then the equation in (5) will be true, with probability $\frac{15}{32}$ since it is the same equation.

Suppose now that $\sum_K = 1$, the equation in (5) is now the inverse, and it will thus happen with probability $1-(15/32).$

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