Twice-Hadamard-CNOT attack on Li et al.’s fault-tolerant quantum private comparison and the improved scheme

Recently, Li et al. presented a two-party quantum private comparison scheme using Greenberger–Horne–Zeilinger (GHZ) states and error-correcting code (ECC) [ Int. J. Theor. Phys. 52, 2818 (2013)], claiming it is fault-tolerant and could be performed in a non-ideal scenario. However, there exists a fatal loophole in their private comparison scheme under a special attack, namely the twice-Hadamard-CNOT attack. Specifically, a malicious party may intercept the other party’s particles and execute Hadamard operations on the intercepted particles as well as on his or her own particles. Then, the malicious party could sequentially perform a controlled-NOT (CNOT) operation between intercepted particles and the auxiliary particles, as well as between his or her own particles and the auxiliary particles prepared in advance. By measuring the auxiliary particles, the secret input will be revealed to the malicious party without being detected. For resisting this special attack, a feasible improved scheme is proposed by introducing a permutation operator before the third party (TP) sends the particle sequences to each participant.