An algebraic attack to the Bluetooth stream cipher E0

TL;DR

This paper presents an algebraic attack on the Bluetooth E0 stream cipher using advanced algebraic tools, significantly improving previous attack methods and demonstrating vulnerabilities within Bluetooth security standards.

Contribution

The paper introduces a novel algebraic attack on E0 using Gr"obner bases, SAT solvers, and BDDs, achieving a more efficient attack than prior work.

Findings

Attack time of approximately 2^79 seconds on an Intel i9 CPU.

The algebraic attack exploits specific structural properties of E0.

The attack is effective against the version compliant with Bluetooth specifications.

Abstract

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 such as the invertibility of its state transition map, a special set of 14 bits of its 132-bit state which when guessed implies linear equations among the other bits and finally a small number of spurious keys, with 83 guessed bits, which are compatible with a keystream of about 60 bits. Exploiting these issues, we implement an algebraic attack using Gr\"obner bases, SAT solvers and Binary Decision Diagrams. Testing activities suggest that the version based on Gr\"obner 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.