Connecting Cryptanalytic Distinguishers through Divisibility

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Integral distinguishers can reveal more than a zero-sum property: the number of ones in an output component may be divisible by a high power of $2$, rather than merely being even. To exploit this divisibility, we relate linear, integral, differential-linear, and restricted differential sums through two-way conversions with explicit divisibility guarantees and data complexity, recovering established links as special cases. We derive exact random-permutation probabilities and rigorous bounds to qu