Maximums of the Additive Differential Probability of Exclusive-Or

• Main Authors: Nicky Mouha, Nikolay Kolomeec, Danil Akhtiamov, Ivan Sutormin, Matvey Panferov, Kseniya Titova, Tatiana Bonich, Evgeniya Ishchukova, Natalia Tokareva, Bulat Zhantulikov
• Format: Article
• Language: English
• Published: Ruhr-Universität Bochum 2021-06-01
• Series: IACR Transactions on Symmetric Cryptology
• Subjects: Differential cryptanalysis; ARX; XOR; modular addition
• Online Access: https://tosc.iacr.org/index.php/ToSC/article/view/8912

At FSE 2004, Lipmaa et al. studied the additive differential probability adp⊕(α,β → γ) of exclusive-or where differences α,β,γ ∈ Fn2 are expressed using addition modulo 2n. This probability is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as the increasingly popular Addition-Rotation-XOR (ARX) constructions. The focus of this paper is on maximal differentials, which are helpful when constructing differential trails. We provide the missing proof for Theorem 3 of the FSE 2004 paper, which states that maxα,βadp⊕(α,β → γ) = adp⊕(0,γ → γ) for all γ. Furthermore, we prove that there always exist either two or eight distinct pairs α,β such that adp⊕( α,β → γ) = adp⊕(0,γ → γ), and we obtain recurrence formulas for calculating adp⊕. To gain insight into the range of possible differential probabilities, we also study other properties such as the minimum value of adp⊕(0,γ → γ), and we find all γ that satisfy this minimum value.