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Hard mathematical problems are at the core of security arguments in cryptography. In this paper, the authors study mathematical generalizations of the famous Rubik's cube puzzle, namely the factorization, representation and balance problems in non-Abelian groups. These problems arise naturally when describing the security of Cayley hash functions, a class of cryptographic hash functions with very interesting properties. The factorization problem is also strongly related to a famous long-standing conjecture of Babai, at the intersection of group theory and graph theory. A constructive proof of Babai's conjecture would make all Cayley hash functions insecure, but on the other hand it would have many positive applications in graph theory and computer science.
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