Multiset-Algebraic Cryptanalysis of Reduced Kuznyechik, Khazad, and secret SPNs

We devise the first closed formula for the number of rounds of a blockcipher with secret components so that these components can be revealed using multiset, algebraic-degree, or division-integral properties, which in this case are equivalent. Using the new result, we attack 7 (out of 9) rounds of Kuznyechik, the recent Russian blockcipher standard, thus halving its security margin. With the same technique we attack 6 (out of 8) rounds of Khazad, the legacy 64-bit blockcipher. Finally, we show how to cryptanalyze and find a decomposition of generic SPN construction for which the inner-components are secret. All the attacks are the best to date.