LowMC is a collection of block cipher families introduced at Eurocrypt 2015 by Albrecht et al. Its design is optimized for instantiations of multi-party computation, fully homomorphic encryption, and zero-knowledge proofs. A unique feature of LowMC is that its internal affine layers are chosen at random, and thus each block cipher family contains a huge number of instances. The Eurocrypt paper proposed two specific block cipher families of LowMC, having 80-bit and 128-bit keys. In this paper, we mount interpolation attacks (algebraic attacks introduced by Jakobsen and Knudsen) on LowMC, and show that a practically significant fraction of 2~(-38) of its 80-bit key instances could be broken 2~(23) times faster than exhaustive search. Moreover, essentially all instances that are claimed to provide 128-bit security could be broken about 1000 times faster. In order to obtain these results we optimize the interpolation attack using several new techniques. In particular, we present an algorithm that combines two main variants of the interpolation attack, and results in an attack which is more efficient than each one.
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