  In this paper, we propose an improved method based on Mixed-Integer Linear
Programming/Mixed-Integer Quadratic Constraint Programming (MILP/MIQCP) to
automatically find better differential-linear (DL) distinguishers for the all
members of Simon and Simeck block cipher families. To be specific, we first
give the completely precise MILP model to describe the linear part, and explain
how to utilize the general expressions of \textsf{Gurobi} solver to model the
propagation of continuous difference for the middle part in a quite easy way.
Secondly, in order to solve the MILP/MIQCP model in a reasonable time, we
propose two heuristic strategies based on the divide-and-conquer idea to speed
up the search process. Thirdly, we introduce the transforming technique, which
exploits the clustering effect on DL trails, to improve the estimated
correlation of the DL approximation.
  We apply our method to Simon and Simeck block cipher families. Consequently,
we find the 14/17/21/26-round theoretical DL distinguishers of
Simon32/48/64/96, which extend the previous longest ones of Simon32/48/96 by
one round and Simon64 by two rounds, respectively. For Simeck, we do not
explore longer distinguishers compared to the currently best results, but
refresh all the results of Zhou et al. (the first work to automate finding DL
distinguishers for Simon-like ciphers using MILP/MIQCP). Besides, in order to
validate the correctness of these distinguishers, the experimental
verifications are conducted on Simon32/Simeck32 and Simon48/Simeck48. The
results show that our theoretical estimations on correlations are very close to
the experimental ones, which can be regarded as a concrete support for the
effectiveness of our method.
