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I'm reading and implementing this tutorial, the author explains everything pretty clearly, the only thing I'm missing is how he decides which trail to use (pg. 12). I understand that one should prefer trails with the least amount of active S-Box and maximize the bias of the trail (in fact, finding the optimal trail seems to be the most important step when trying to apply linear cryptanalysis to a symmetric encryption algorithm).

It seems to me like some sort of dynamic programming problem, but I wonder if there is a general algorithm to solve the problem of finding the optimal trail (or at least, a series of candidate trails).

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People use Matsui's "Best Path Search Algorithm" which is indeed a dynamic programming approach. The resource costs hard to anticipate and so branch and bound variations are employed in practice.

A skeletal outline can be found on slides 14-17 of Matsui's retrospective talk on linear cryptanalysis at Asicrypt 2018.

A more formal description and some variants can be found in Ji, Zhang and Ding's 2019 paper "Improving Matsui’s Search Algorithm for the Best Differential/Linear Trails and its Applications for DES, DESL and GIFT"

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Matsui's method is the classical one, but nowadays the trend is towards MILP/SAT/SMT-based search. The correctness of the trail and its link with the transition probability is encoded into the system and is then optimized to minimize the total probability (for SAT/SMT binary search can be used, for MILP the optimization is native).

Some examples (most of the research is for differential attacks though, but it typically translated to linear attacks):

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    $\begingroup$ I thinks this answer needs some links to those... $\endgroup$
    – kelalaka
    Commented Feb 27, 2022 at 12:23
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    $\begingroup$ SAT=satisfiability, MILP = mixed integer linear programming, SMT? Please don't use undefined acryonyms that are not so common and reduce an answer's usefullness $\endgroup$
    – kodlu
    Commented Feb 28, 2022 at 0:42

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