On Characterization of Transparency Order for (n, m)-functions.

The transparency order (denoted by TO ) is a useful measure of the robustness of ( n , m )-functions (cryptographic S-boxes as mappings from G F ( 2 ) n to G F ( 2 ) m ) to multi-bits Differential Power Analysis (DPA). An improved version of transparency order (denoted by RTO ), based on the use of cross-correlation coefficients, was also introduced recently. For the first time, we resolve this open problem which ( n , m )-functions reach the upper bound on TO for odd n ( m is a power of 2). We also investigate the tightness of upper and lower bounds related to RTO and derive its relationship to main cryptographic characterizations of ( n , m )-functions (such as nonlinearity, the sum-of-square indicator and algebraic immunity). Finally, concerning S-boxes of size 4 × 4 , the distributions of RTO for all 302 balanced S-boxes (up to affine equivalence) and 16 equivalence classes of optimal S-boxes are given.