Comment on “Quantum string seal is insecure”

Abstract

In a recent paper [1], Chau claimed that all quantum string seals are insecure. The core of Chau's attack strategy is the measurement(see Eq. (29) of that reference). It was claimed that with this measurement, the attacker can obtain non-trivial information on the sealed string while escapes the verifier's detection with at least 50% chance. However, the paper concentrated only on the fidelity of the sealed state corresponding to the attacker's measurement, without providing a detailed evaluation on the amount of information obtained by the attacker. Here it will be shown that for a class of quantum string seal protocols including the one proposed by He [2], this amount of information is only trivial. Therefore in contrast to Chau's claim, quantum string seal can be unconditionally secure. In fact, the general proof on why Chau's attack strategy fails had already been well addressed in Ref. [3]. Briefly, consider a simple model of imperfect quantum string seal, in which the sealed state for the message i ′ is taken aswhere the notation is the same as that in Eq. (1) of Ref.[1]. Applying the measurement Q i0 on it yieldsThus the probability for the message i ′ to be decoded as i by the attacker iswhere ν is defined by Eq. (12) It means that any one of the N possible choices of the message i ′ has at least the probability 1/(2N ) to be decoded as message i, even if its content is completely irrelevant with i. In other words, whenever the attacker obtains a message i via the measurement strategy, there is at less a probability p i = i ′ 1/(2N ) = 1/2 that the original message can be anything, i. e., the amount of information he obtained is zero. Thus it can be seen that the attack strategy is useless. Though at half of the time it can escape the verifier's detection, the amount of information obtained on the sealed message is only trivial. Therefore Chau's claim that all quantum seals are insecure is wrong. Now it will be shown that the protocol proposed in Ref.[2] is indeed such a secure quantum string seal. In this protocol, to seal a stringThus by takingwhere f m (θ m ) is cos θ m (sin θ m ) if the m-th bit of the string j ′ equals to (does not equal to) that of the string i ′ , we can see that the protocol belongs to the class of quantum string seal described by Eq. (2). Therefore as shown above, it cannot be broken by Chau's attack strategy. In Sec. IV of Ref.[1], it was claimed that "the major loophole in He's proof of the security of his quantum string seal in Ref.[2] is that he incorrectly assumed that measuring all the qubits is the only method to obtain a significant portion of information of the sealed message". But this is obviously incorrect. In the paragraph before Eq. (5) of Ref. [2], it was clearly written that the general security proof starts as follows. Let H denotes the 2 n dimensional Hilbert space where the sealed state lives

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