An algebraic attack to the Bluetooth stream cipher E0

In this paper we study the security of the Bluetooth stream cipher E0 from the viewpoint it is a difference stream cipher, that is, it is defined by a system of explicit difference equations over the finite field GF(2). This approach highlights some issues of the Bluetooth encryption as the invertibility of its state transition map, a special set of 14 bits of its 132-bit state which when guessed imply linear equations among the other bits and finally a very small number of spurious keys compatible with a keystream of about 60 bits. Exploiting such issues, we implement an algebraic attack using Grobner bases, SAT solvers and Binary Decision Diagrams. Testing activities suggest that the version based on Grobner bases is the best one and it is able to attack E0 in about 2^79 seconds on an Intel i9 CPU. To the best of our knowledge, this work improves any previous attack based on a short keystream, hence fitting with Bluetooth specifications.