boomerang attacks on reduced-round midori64

Midori64 is a lightweight spn block cipher introduced by banik et al. at asiacrypt 2015 which operates on 64-bit states through 16 rounds using a 128-bit key. in the last decade, midori64 has been exposed to several attacks intensely. in this paper, we provide the first boomerang attack on midori64 in the literature, to the best of our knowledge. for this purpose, firstly we present a practical single key 7-round boomerang attack on midori64 improving the mixture idea of biryukov by a new technique which we call ``mixture pool&, and then extend our attack up to 9 rounds with time complexity $2^{122.3}$, and memory and data complexity $2^{36}$. (the authors of midori stated that they expect much smaller rounds than 8 rounds of midori64 are secure against boomerang-type attacks.) we also emphasize that the mixture pool idea provides a kind of data-memory tradeoff and hence presents more usefulness for boomerang-type attacks.