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In this paper we study the security of the Bluetooth stream cipher E0 fromthe viewpoint it is a "difference stream cipher", that is, it is defined by asystem of explicit difference equations over the finite field GF(2). Thisapproach highlights some issues of the Bluetooth encryption such as theinvertibility of its state transition map, a special set of 14 bits of its132-bit state which when guessed implies linear equations among the other bitsand finally a small number of spurious keys, with 83 guessed bits, which arecompatible with a keystream of about 60 bits. Exploiting these issues, weimplement an algebraic attack using Gr\"obner bases, SAT solvers and BinaryDecision Diagrams. Testing activities suggest that the version based onGr\"obner bases is the best one and it is able to attack E0 in about 2^79seconds on an Intel i9 CPU. To the best of our knowledge, this work improvesany previous attack based on a short keystream, hence fitting with Bluetoothspecifications.

An algebraic attack to the Bluetooth stream cipher E0