A deeper understanding of the XOR count distribution in the context of lightweight cryptography

In this paper, we study the behavior of the XOR count distributions under different bases of finite field. XOR count of a field element is a simplified metric to estimate the hardware implementation cost to compute the finite field multiplication of an element. It is an important criterion in the design of lightweight cryptographic primitives, typically to estimate the efficiency of the diffusion layer in a block cipher. Although several works have been done to find lightweight MDS diffusion matrices, to the best of our knowledge, none has considered finding lightweight diffusion matrices under other bases of finite field apart from the conventional polynomial basis. The main challenge for considering different bases for lightweight diffusion matrix is that the number of bases grows exponentially as the dimension of a finite field increases, causing it to be infeasible to check all possible bases. Through analyzing the XOR count distributions and the relationship between the XOR count distributions under different bases, we find that when all possible bases for a finite field are considered, the collection of the XOR count distribution is invariant to the choice of the irreducible polynomial of the same degree. In addition, we can partition the set of bases into equivalence classes, where the XOR count distribution is invariant in an equivalence class, thus when changing bases within an equivalence class, the XOR count of a diffusion matrix will be the same. This significantly reduces the number of bases to check as we only need to check one representative from each equivalence class for lightweight diffusion matrices. The empirical evidence from our investigation says that the bases which are in the equivalence class of the polynomial basis are the recommended choices for constructing lightweight MDS diffusion matrices

Some methods for constructing mds-matrices over finite field

Предлагаются новые методы построения MDS-матриц с использованием возведения в степень сопровождающих матриц многочленов над конечным полем. Изучается ряд неприводимых многочленов степени t = 4 и 6, сопровождающая матрица которых при возведении в соответствующую степень t является MDS-матрицей. Представлен новый метод построения MDS-матриц, ориентированных на низкоресурсную программную и аппаратную реализации

Optimizing Implementations of Lightweight Building Blocks

We study the synthesis of small functions used as building blocks in lightweight cryptographic designs in terms of hardware implementations. This phase most notably appears during the ASIC implementation of cryptographic primitives. The quality of this step directly affects the output circuit, and while general tools exist to carry out this task, most of them belong to proprietary software suites and apply heuristics to any size of functions. In this work, we focus on small functions (4- and 8-bit mappings) and look for their optimal implementations on a specific weighted instructions set which allows fine tuning of the technology. We propose a tool named LIGHTER, based on two related algorithms, that produces optimized implementations of small functions. To demonstrate the validity and usefulness of our tool, we applied it to two practical cases: first, linear permutations that define diffusion in most of SPN ciphers; second, non-linear 4-bit permutations that are used in many lightweight block ciphers. For linear permutations, we exhibit several new MDS diffusion matrices lighter than the state-of-the-art, and we also decrease the implementation cost of several already known MDS matrices. As for non-linear permutations, LIGHTER outperforms the area-optimized synthesis of the state-of-the-art academic tool ABC. Smaller circuits can also be reached when ABC and LIGHTER are used jointly