BIZness: Bit Invariant Zero-Sum Property Based on Division Trail

The introduction of the division property by Todo in 2015 gave cryptanalysts a new tool to probe further into the inner workings of symmetric-key primitives. The subsequent works investigating the property at the bit-level lead to better insights with new distinguishers improving our understanding of cipher design. Contemporary improvements in the automated cryptanalysis domain have only accelerated this direction of research. In ACNS 2023, Kundu et al. exploited a very interesting property based on division trails to mount a fault attack on PRESENT and GIFT64. However, they did not provide any justification for existence of such a property. The current work makes a systematic attempt to dissect this property by making some interesting observation for Substitution-Permutation Network (SPN) structures with bitwise permutation (S-bP) based lightweight ciphers. We reintroduce this property as the Bit Invariant Zero-Sum (BIZ) which can be witnessed by analyzing the division trails (returned by any solver like Gurobi) corresponding to different input division properties. BIZ essentially captures division trails that are invariant with respect to cardinality of the input multiset with certain restrictions. For instance, we find that for all the 24 permutation-equivalent SBoxes of the PRESENT SBox, there exist only four distinct BIZ properties. Our research justifies the existence of such a property by minutely analyzing the diffusion of the permutation layer and the Algebraic Normal Form of the SBox. Similar, observations are made for GIFT as well. To further understand the impact of this property on cipher design, we furnish a characterization of the BIZ property by combining prominent 4-bit SBoxes with the PRESENT and GIFT permutations to identify the circumstances that would lead to such a property. Moreover, we observed that the BIZ property is preserved even when the original permutation of the GIFT cipher is replaced with any of the 24 classes (covering 41,472) of 16-bit BOGI permutations. Finally, our investigation reveals that BIZ is a stronger property that emerges out of the interactions of diffusion and confusion layers provides a better understanding of cipher design, and may be further exploited as an aid to future cryptanalytic attacks.