On the Modified Transparency Order of    n , m   -Functions

Yu Zhou,Yongzhuang Wei,Wenzheng Zhang,Hailong Zhang,Angel M Del Rey

doi:10.1155/2021/6640099

Abstract

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

Highlights

Differential power analysis (DPA) was introduced by Kocher et al in [1] and is a well-known and thoroughly studied threat for implementations of block ciphers, like DES and AES [2].Beierle et al [3] validated the correlation power analysis attack through the Hamming distance power model
In 2005, Prouff [5] gave the model of the DPA resilience of the S-boxes and proposed the definition of the transparency order based on the autocorrelation coefficients for (n, m)-functions
He obtained that S-boxes with smaller TO have higher DPA resilience and deduced the tightness of the upper bound and the lower bound on the TO

Summary

Introduction

Differential power analysis (DPA) was introduced by Kocher et al in [1] and is a well-known and thoroughly studied threat for implementations of block ciphers, like DES and AES [2]. In 2005, Prouff [5] gave the model of the DPA resilience of the S-boxes and proposed the definition of the transparency order (denoted by TO) based on the autocorrelation coefficients for (n, m)-functions. He obtained that S-boxes with smaller TO have higher DPA resilience and deduced the tightness of the upper bound and the lower bound on the TO. In 2020, Li et al [13] put up a flaw in original transparency order [10] and gave the modified transparency order denoted by MTO, obtained a lower bound on MTO based on the Walsh transform, and deduced the distribution of MTO values for sixteen optimal affine equivalent classes of (4,4) S-boxes.

Preliminaries

Bounds on the Modified Transparency Order

Findings

The Relationships between MTO and Some Cryptographic Properties