$\mathsf{XS}$-circuits in block ciphers

$\mathsf{XS}$-circuits describe block ciphers that utilize $2$ operations: $\mathsf{X}$ (bitwise modulo $2$ addition of binary words) and $\mathsf{S}$ (substitution of words using keydependent $S$-boxes). We propose a model of $\mathsf{XS}$-circuits which covers a rather wide range of block ciphers: several one-round circuits having only one operation $\mathsf{S}$ each are linked together to form a cascade. Operations $\mathsf{S}$ in rounds are interpreted as independent round oracles. We deal with diffusion characteristics which are related to the cryptographic strength of cascades.