Linearity Measures for MQ Cryptography

We propose a new general framework for the security of multivariate quadratic (MQ) schemes with respect to attacks that exploit the existence of linear subspaces. We adopt linearity measures that have been used traditionally to estimate the security of symmetric cryptographic primitives, namely the nonlinearity measure for vectorial functions introduced by Nyberg at Eurocrypt ’92, and the (s, t)–linearity measure introduced recently by Boura and Canteaut at FSE’13. We redefine some properties of MQ cryptosystems in terms of these known symmetric cryptography notions, and show that our new framework is a compact generalization of several known attacks in MQ cryptography against single field schemes. We use the framework to explain various pitfalls regarding the successfulness of these attacks. Finally, we argue that linearity can be used as a solid measure for the susceptibility of MQ schemes to these attacks, and also as a necessary tool for prudent design practice in MQ cryptography.