The Notion of Transparency Order, Revisited

We revisit the definition of transparency order (TO) and that of modified transparency order (MTO) as well, which were proposed to measure the resistance of substitution boxes (S-boxes) against differential power analysis (DPA). We spot a definitional flaw in original TO, which is proved to significantly affect the soundness of TO. Regretfully, MTO overlooks this flaw, yet it happens to incur no bad effects on the correctness of MTO, even though the start point of this formulation is highly questionable. It is also this neglect that made MTO consider a variant of multi-bit DPA attack, which was mistakenly thought to appropriately serve as an alternative powerful attack. This implies the soundness of MTO is also more or less arguable. Therefore, we fix this definitional flaw and provide a revised definition named reVisited TO (VTO). For demonstrating validity and soundness of VTO, we present simulated and practical DPA attacks on implementations of $4\times 4$ and $8\times 8$ S-boxes. In addition, we also illustrate the soundness of VTO in masked S-boxes. Furthermore, as a concrete application of VTO, we present the distribution of VTO values of optimal affine equivalence classes of $4\times 4$ S-boxes and give some recommended guidelines on how to select $4\times 4$ S-boxes with higher DPA resistance at the identical level of implementation cost.