Paper 2024/1278 Original: eprint.iacr.org/2024/1278 ↗

Quantum Key Recovery Attacks on 4-round Iterated Even-Mansour with Two Keys

Abstract

In this paper, we propose quantum key recovery attacks on 4-round iterated Even-Mansour (IEM) with a key schedule that applies two keys alternately. We first show that a conditional periodic function such that one of the secret keys appears as a period conditionally can be constructed using the encryption function and internal permutations. By applying the offline Simon's algorithm to this function, we construct a key recovery attack with a complexity of $O(\sqrt{N} \log N)$ for $N = 2^n$, where $n$ is the block size and one secret key size. Using quantum queries, this attack outperforms the generic quantum attack, i.e., Grover's search which takes the time complexity of $O(N)$. Moreover, we propose the quantum version of the multibridge attack proposed by Dinur et al. in ASIACRYPT 2014 to analyze the 4-round IEM. As a result, we show that the quantum multibridge attack can achieve the optimal complexity of $O(N)$ even if we have only $O(1)$ data without quantum queries, while the classical attack requires $O(N)$ data to achieve the same time complexity. Furthermore, we show that the quantum multibridge attack slightly outperforms Grover's search when considering the quantum circuit depth for these attacks.