Mathematical Aspects of Division Property

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 weak-ness 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. 1 X 2 is 3, while the number of trails is only 1. This shows that analysis of a decomposition of the given function into basic operations can increase the number of trails.