Algebraic Properties of the Cube Attack

Cube attacks can be used to analyse and break cryptographic primitives that have an easy algebraic description. One example for such a primitive is the stream cipher Trivium. In this article we give a new framework for cubes that are useful in the cryptanalytic context. In addition, we show how algebraic modelling of a cipher can greatly be improved when taking both cubes and linear equivalences between variables into account. When taking many instances of Trivium, we empirically show a saturation effect, i.e. the number of variables to model an attack will become constant for a given number of rounds. Moreover, we show how to systematically find cubes both for general primitives and also specifically for Trivium. For the latter, we have found all cubes up to round 446 and draw some conclusions on their evolution between rounds. All techniques in this article are general and can be applied to any cipher.