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

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 nn and mm. We find that any (n,1)\left(n,1)-function cannot reach TO=m{\mathcal{TO}}=m for odd nn. 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}}).