Rotational-XOR Differential Rectangle Cryptanalysis on Simon-Like Ciphers

Abstract

In this paper, we propose a rectangle-like method called rotational-XOR differential rectangle attack to search for better distinguishers. It is a combination of the rotational-XOR cryptanalysis and differential cryptanalysis in the rectangle-based way. In particular, we put a rotational-XOR characteristic before a differential characteristic to construct a rectangle structure. By choosing some appropriate rotational-XOR and differential characteristics as well as considering multiple differentials, some longer distinguishers that have the probability greater than $$2^{-2n}$$ can be constructed effectively where n is the block size of a block cipher. We apply this new method to some versions of Simon and Simeck block ciphers. As a result, we obtain rotational-XOR differential rectangle distinguishers up to 16, 16, 17, 16 and 21 rounds for Simon32/64, Simon48/72, Simon48/96, Simeck32 and Simeck48, respectively. Our distinguishers for Simon32/64 and Simon48/96 are both longer than the best differential and rotational-XOR distinguishers. Also, our distinguisher for Simeck32 is longer than the best differential distinguisher (14 rounds) and has the full weak key space (i.e., $$2^{64}$$ ) whereas the 16-round rotational-XOR distinguisher has a weak key class of $$2^{36}$$ . In addition, our distinguisher for Simeck48 has a better weak key class ( $$2^{72}$$ weak keys) than the 21-round rotational-XOR distinguisher ( $$2^{60}$$ weak keys). To the best of our knowledge, this is the first time to consider the combinational cryptanalysis based on rotational-XOR and differential cryptanalysis using the rectangle structure.