On the Modified Transparency Order of n , m -Functions

Abstract

The concept of transparency order is introduced to measure the resistance of $\left(n,m\right)$ -functions against multi-bit differential power analysis in the Hamming weight model, including the original transparency order (denoted by $\mathcal{TO}$ ), redefined transparency order (denoted by $\mathcal{RTO}$ ), and modified transparency order (denoted by $\mathcal{MTO}$ ). In this paper, we firstly give a relationship between $\mathcal{MTO}$ and $\mathcal{RTO}$ and show that $\mathcal{RTO}$ is less than or equal to $\mathcal{MTO}$ for any $\left(n,m\right)$ -functions. We also give a tight upper bound and a tight lower bound on $\mathcal{MTO}$ for balanced $\left(n,m\right)$ -functions. Secondly, some relationships between $\mathcal{MTO}$ and the maximal absolute value of the Walsh transform (or the sum-of-squares indicator, algebraic immunity, and the nonlinearity of its coordinates) for $\left(n,m\right)$ -functions are obtained, respectively. Finally, we give $\mathcal{MTO}$ and $\mathcal{RTO}$ for (4,4) S-boxes which are commonly used in the design of lightweight block ciphers, respectively.