MDS Diffusion Layers for Arithmetization-Oriented Symmetric Ciphers: The Rotational-Add Construction

We introduce the rotational-add diffusion layers aimed for applications in the design of arithmetization-oriented (AO) symmetric ciphers, such as fully homomorphic encryption (FHE)-friendly symmetric ciphers. This generalizes the rotational-XOR diffusion layers which have been utilized in the design of many important conventional symmetric ciphers like SHA-256, SM4, ZUC and Ascon. A rotational-add diffusion layer is defined over the finite field Fp for arbitrary prime p, enabling implementations using only rotations and modular additions/subtractions. The advantage of using such diffusion layers in AO ciphers is that, the costs of scalar multiplications can be reduced since the appearing scalars include only ±1, thus the total costs depend on sizes of the rotation offsets. In this paper, we investigate characterizations and constructions of lightest rotational-add diffusion layers over (Fmp)n that are maximum distance separable (MDS) with a focus on the case n = 4. It turns out that the minimum achievable size of the rotation offsets is 5 subject to the MDS property constraint. We specify a large class of rotational-add diffusion layers with 5 rotations and traverse all possible patterns of appearance of the scalars ±1. In four cases we can derive computationally tractable necessary and sufficient conditions for the rotational-add diffusion layers to attain the MDS property. These conditions enable explicit characterization of suitable primes p for given parameters. Leveraging these results, we construct three distinct families of rotational-add MDS diffusion layers applicable to AO ciphers. Although a rotational-add diffusion layer with 7 rotations and only additions has already been used in the design of the FHEfriendly block cipher YuX recently, to our knowledge, our work presents the first systematic theoretical characterization of rotational-add MDS diffusion layers and provides explicit constructions of them.