Unlocking Mix-Basis Potential: Geometric Approach for Combined Attacks

This paper explores the possibility of using different bases in Beyne's geometric approach, a flexibility that was theoretically proposed in Beyne's doctoral thesis but has not been adopted in real cryptanalytic attacks despite its potential to unify multiple attack paradigms. We revisit three bases from previous geometric approach papers and extend them to four extra ones determined by simple rules. With the final seven bases, we can obtain $7^{2d}$ different basis-based attacks in the $d$-th-order spaces, where the order is defined as the number of messages used in one sample during the attack. All these attacks can be studied in unified automatic search methods.

We provide several demonstrative applications of this framework. First, we show that by choosing an alternative pair of bases, the divisibility property analyzed by Beyne and Verbauwhede with ultrametric integral cryptanalysis (ASIACRYPT 2024) can be interpreted as a single element rather than as a linear combination of elements of the transition matrix; thus, the property can be studied in a unified way as other geometric approach applications. Second, we revisit the multiple-of-$2^t$ property (EUROCRYPT 2017) under our new framework and present new multiple-of-$2^t$ distinguishers for -64 that surpass the state-of-the-art results, from the perspectives of both first-order and second-order attacks. Finally, we give a closed formula for differential-linear approximations without any assumptions, even confirming that the two differential-linear approximations of -32 and -48 found by Hadipour et al. are deterministic independently of concrete key values.