Exact Formula for RX-Differential Probability Through Modular Addition for All Rotations

Exact Formula for RX-Differential Probability Through Modular Addition for All Rotations

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This work presents an exact and compact formula for the probability of rotation-xor differentials (RX-differentials) through modular addition, for arbitrary rotation amounts, which has been a long-standing open jake wire tighteners problem.The formula comes with a rigorous proof and is also verified by extensive experiments.Our formula uncovers error in a recent work from 2022 proposing a formula for rotation amounts bigger than 1.Surprisingly, it also affects correctness of the more studied and used formula for the rotation amount equal to 1 (from TOSC 2016).Specifically, it uncovers rare cases where the assumptions of this formula do not hold.

Correct formula for arbitrary rotations now opens up a larger search space where one can often find better trails.For applications, we propose automated mixed integer linear programming (MILP) modeling techniques for searching optimal RX-trails based on our exact formula.They are consequently applied to several ARX designs, including Salsa, Alzette and a small-key variant of Speck, and yield many new RX-differential distinguishers, some cabin still decanter of them based on provably optimal trails.In order to showcase the relevance of the RX-differential analysis, we also design Malzette, a 12-round Alzette-based permutation with maliciously chosen constants, which has a practical RX-differential distinguisher, while standard differential/linear security arguments suggest sufficient security.