On a set of impossible differences of Feistel ciphers with a non-bijective transform of a round function

In this paper, a family of $l$-round balanced Feistel ciphers with non-bijective combining functions is being considered. For any such cipher, the existence of impossible differentials for an arbitrary number of rounds $l$ is proved, and a lower estimate of the number of described impossible differentials is obtained. The GRANULE block cipher belongs to the family under consideration, for which a new approach for finding impossible differences is proposed. Its superiority, in comparison with other previously known approaches, is shown both in terms of the number of impossible differences found and in terms of the number of rounds. Experimental confirmation of the theoretical estimate of the number of impossible differences has been obtained.