Cryptanalysis of Acorn in Nonce-Reuse Setting

Acorn is a third-round candidate of the CAESAR competition. It is a lightweight authenticated stream cipher. In this paper, we show how to recover the initial state of Acorn when one pair of Key and IV is used to encrypt three messages. Our method contains two main steps: (1) gathering different states; (2) retrieving linear equations. At the first step, we demonstrate how to gather the relation between states when two different plaintexts are encrypted under the same nonce. And at the second step, we exploit how to retrieve a system of linear equations with respect to the initial state, and how to recover the initial state from this system of equations. We apply this method to both Acorn v2 and Acorn v3. The time complexity to recover the initial state of Acorn v2 is \(2^{78} c\), where c is the time complexity of solving linear equations. It is lower than that of the previous methods. For Acorn v3, we can recover the initial state with the time complexity of \(2^{120.6}c\), lower than that of the exhaustion attack. We also apply it on shrunk ciphers with similar structure and properties of Acorn v2 and Acorn v3 to prove the validity of our method. This paper is the first time to analyze Acorn v3 when a nonce is reused and our method provides some insights into the diffusion ability of such stream ciphers.