The ‐protocols for homomorphism relations are one of the cryptographic protocols which are used to prove knowledge of homomorphism relations. The Schnorr protocol is one of the most famous ‐protocols used for proving knowledge of discrete logarithm (DL) relation in which the verifier essentially performs one double‐exponentiation (i.e. a group computation of the form a x b y ). A direct application of the Schnorr protocol for proving simultaneous knowledge of n DLs with a common base leads to a ‐protocol in which the verifier performs n double‐exponentiations. In this study, the authors propose another ‐protocol for homomorphism relations. The proposed ‐protocol has fast verification when is used to prove the simultaneous homomorphism relations with a common homomorphism. Also, when the DL instantiation (DL‐instantiation) of the proposed ‐protocol is used to prove simultaneous knowledge of n DLs with a common base, it leads to a ‐protocol in which the verifier performs n +1 single‐exponentiations.

One of the most important structures used in modern block ciphers is the substitution–permutation network (SPN) structure. Many block ciphers with this structure widely use Maximun Distance Separable (MDS) matrices over finite fields as their diffusion layers, for example, advanced encryption standard (AES) uses a 4×4 MDS matrix as the main part of its diffusion layer and the block cipher Khazad has an involutory 8×8 matrix. In this study, first a construction is proposed for a 4×4 linear diffusion layer that can intermix four words of arbitrary size with branch number 5. Then extend this idea for 8×8 diffusion layer using low-cost linear functions. In this construction, first, certain binary linear combinations of inputs are fed into two or three different invertible linear functions and then combined using XOR operation. In order to show the efficiency of the proposed diffusion layer, the authors exploit it in a nested SPN structure and compare its efficiency with some well-known diffusion layers such as the diffusion layer of Hierocrypt.