Characterizing the upper bound on the transparency order of (n, m)-functions

Abstract

Abstract Transparency order ( TO {\mathcal{TO}} ) is one of the indicators used to measure the resistance of ( n , m ) \left(n,m) -function to differential power analysis. At present, there are three definitions: TO {\mathcal{TO}} , redefined transparency order ( ℛTO {\mathcal{ {\mathcal R} TO}} ), and modified transparency order ( ℳTO {\mathcal{ {\mathcal M} TO}} ). For the first time, we give one necessary and sufficient condition for ( n , m ) \left(n,m) -function reaching TO = m {\mathcal{TO}}=m and completely characterize ( n , m ) \left(n,m) -functions reaching TO = m {\mathcal{TO}}=m for any n n and m m . We find that any ( n , 1 ) \left(n,1) -function cannot reach TO = m {\mathcal{TO}}=m for odd n n . Based on the matrix product, the necessary conditions for ( n , m ) \left(n,m) -function reaching ℳTO = m {\mathcal{ {\mathcal M} TO}}=m or ℛTO = m {\mathcal{ {\mathcal R} TO}}=m are given, respectively. Finally, it is proved that any balanced ( n , m ) \left(n,m) -function cannot reach the upper bound on TO {\mathcal{TO}} (or ℛTO {\mathcal{ {\mathcal R} TO}} , ℳTO {\mathcal{ {\mathcal M} TO}} ).