CSSM: A Combined Structure of SM4-Like Structure and MARS-Like Structure

SM4-like structure and MARS-like structure are generalized Feistel structures with the property that the decryption is similar to the encryption. These two structures are the same up to affine equivalence. In this paper, we propose a new generalized Feistel structure, CSSM, which combines the design philosophies of both SM4-like and MARS-like structures. We further investigate the refined full-diffusion round and analyze the number of rounds of impossible differentials, zero correlation linear hulls, and integral distinguishers for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex>-branch CSSM with bijective <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f$</tex>-functions. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d / 2$</tex> is an odd number, then the refined full-diffusion round of CSSM is at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex>, and there exist <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$3 d / 2$</tex> round impossible differentials, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$3d / 2$</tex>-round zero correlation linear hulls, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$3 d / 2$</tex>-round integral distinguishers. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d / 2$</tex> is an even number, then the refined full-diffusion round of CSSM is at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d-1$</tex>, and there exist <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(3 d / 2-1)$</tex> -round impossible differentials, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(3 d / 2-1)$</tex>-round zero correlation linear hulls, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(3 d / 2-1)$</tex> round integral distinguishers. The structure proposed in this paper may contribute to the design of block ciphers.