Quantum cryptanalysis of reduced-round E2 algorithm

Abstract

E2 algorithm is one of the 15 candidate algorithms in the first round of AES collection. In this paper, taking E2-128 as an example, the quantum security analysis on E2 algorithm is proposed for the first time in quantum chosen-plaintext attack setting. First, a polynomial-time distinguisher on 4-round E2-128 is constructed with \(2^{12.1}\) quantum queries by taking the properties of the internal round function into consideration. Then, by extending the distinguisher 2 rounds backward, a 6-round quantum key recovery attack is achieved with the help of Grover-meet-Simon algorithm, whose time complexities gain a factor of \(2^{76}\), where the subkey length that can be recovered is 152 bits with the occupation of 560 qubits. Furthermore, when attacking \(r>6\) rounds, \(152+(r-6)\times 128\)-bit subkey needs to be guessed in time \(2^{76+(r-6)\times 64}\), which is \(\frac{1}{2^{52}}\) of Grover’s quantum brute force search. Finally, we present a quantum attack against E2-128 with \({2^{88.1}}\) quantum queries by taking initial transformation and terminal transformation into consideration. The result shows that the time complexity of the quantum attack is significantly reduced, and E2 algorithm is safe enough to resist quantum attack.