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See detailMathematical Aspects of Division Property
Hebborn, Phil; Leander, Gregor; Udovenko, Aleksei UL

in Cryptography and Communications (2023)

This work surveys mathematical aspects of division property, which is a state of the art technique in cryptanalysis of symmetric-key algorithms, such as authenticated encryption, block ciphers and stream ... [more β–Ό]

This work surveys mathematical aspects of division property, which is a state of the art technique in cryptanalysis of symmetric-key algorithms, such as authenticated encryption, block ciphers and stream ciphers. It aims to find integral distinguishers and cube attacks, which exploit weakness in the algebraic normal forms of the output coordinates of the involved vectorial Boolean functions. Division property can also be used to provide arguments for security of primitives against these attacks. The focus of this work is a formal presentation of the theory behind the division property, including rigorous proofs, which were often omitted in the existing literature. This survey covers the two major variants of division property, namely conventional and perfect division property. In addition, we explore relationships of the technique with classic degree bounds [less β–²]

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See detailOn degree-d zero-sum sets of full rank
Beierle, Christof UL; Biryukov, Alex UL; Udovenko, Aleksei UL

in Cryptography and Communications (2019)

A set π‘†βŠ†π”½π‘›2 is called degree-d zero-sum if the sum βˆ‘π‘ βˆˆπ‘†π‘“(𝑠) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean ... [more β–Ό]

A set π‘†βŠ†π”½π‘›2 is called degree-d zero-sum if the sum βˆ‘π‘ βˆˆπ‘†π‘“(𝑠) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n βˆ’ d βˆ’β€‰1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independent elements, and show relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices. We are particularly interested in the smallest of such sets and prove bounds on the minimum number of elements in a degree-d zero-sum set of rank n. The motivation for studying those objects comes from the fact that degree-d zero-sum sets of full rank can be used to build linear mappings that preserve special kinds of nonlinear invariants, similar to those obtained from orthogonal matrices and exploited by Todo, Leander and Sasaki for breaking the block ciphers Midori, Scream and iScream. [less β–²]

Detailed reference viewed: 142 (5 UL)