Cryptanalysis and improvement of the novel quantum scheme for secure two-party distance computation

Abstract

Secure multiparty computational geometry is a vital field of secure multiparty computation, which computes a computation geometric problem without revealing any private information of each party. A recent paper proposed a scheme about a novel quantum scheme for secure two-party distance computation. We cryptanalyze the scheme in the following three aspects: (1) There exists an entangle-and-measure attack method for Bob to detect Alice’s location with a probability of 50% and the attack cannot be detected whether this attack is successful or not. (2) There is a loophole for Alice to get more information if she submits a different point in the second chance. The amount of information exposed by Bob is unacceptable. (3) In the definition of S2PDC, only Alice can get the distance between both positions while Bob gets nothing. However, under some circumstances, as a participant in the scheme, Bob has right to get the distance. Above all, we have improved the agreement from different items: (1) Security: the improved scheme can defend our new type attack based on the original security. (2) Fairness: The amount of information leaked by Bob is minimum in the new scheme. (3) Symmetric: Our scheme allows that both parties can get the distance from the scheme directly. (4) Efficiency: The information complexity of the new scheme is no more than the former one.

Appendices

A Process of the original scheme

$$\begin{aligned}&\begin{aligned}&|\psi _1 \rangle _{Q_1,R_1}|\psi _2 \rangle _{Q_2,R_2}\\&\quad =\frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle + |0\rangle |k\rangle |0\rangle |k \rangle \\&\qquad + |k\rangle |0\rangle |k\rangle |0 \rangle + |k\rangle |k\rangle |k\rangle |k \rangle )_{Q_1,Q_2,R_1,R_2}\\&\quad \xrightarrow []{Oracle} \frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle +(-1)^{d(k)}|0\rangle |k\rangle |0\rangle |k\rangle \\&\qquad +(-1)^{d(k)}|k\rangle |0\rangle |k\rangle |0\rangle +|k\rangle |k\rangle |k\rangle |k\rangle ) _{Q_1,Q_2,R_1,R_2}\\&\quad \xrightarrow []{CNOT}\frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle +(-1)^{d(k)}|0\rangle |0\rangle |0\rangle |k\rangle \\&\qquad +(-1)^{d(k)} |0\rangle |0\rangle |k\rangle |0\rangle +|0\rangle |0\rangle |k\rangle |k\rangle )_{Q_1,Q_2,R_1,R_2}\\&\quad =\frac{1}{2}|0\rangle |0\rangle _{Q_1,Q_2}((|0 \rangle |0\rangle +|k\rangle |k\rangle )_{R_1,R_2} +(-1)^{d(k)}(|0\rangle |k\rangle +|k\rangle |0\rangle )_{R_1,R_2}\\&\quad =|0\rangle |0\rangle _{Q_1,Q_2}|x\rangle |x\rangle _{R_1,R_2}\\ \end{aligned} \end{aligned}$$

(7)

$$\begin{aligned}&|x\rangle =\frac{|0\rangle +(-1)^{d(k)}|k\rangle }{\root \of {2}} \end{aligned}$$

(8)

$$\begin{aligned}&|y\rangle =\frac{|0\rangle -(-1)^{d(k)}|k\rangle }{\root \of {2}} \end{aligned}$$

(9)

B Process of the attack

The process of the attack is as below. And the definitions of $|x\rangle$ and $|y\rangle$ in Eq. 10 are noted in Eqs. 8 and 9

$$\begin{aligned}&|\psi _1 \rangle _{Q_1,R_1}|\psi _2 \rangle _{Q_2,R_2}\nonumber \\&\quad =\frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle + |0\rangle |k\rangle |0\rangle |k\rangle \nonumber \\&\qquad + |k\rangle |0\rangle |k\rangle |0\rangle + |k\rangle |k\rangle |k\rangle |k\rangle )_{Q_1,Q_2,R_1,R_2}\nonumber \\&\quad \xrightarrow []{Oracle} \frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle +(-1)^{d(k)}|0\rangle |k\rangle |0\rangle |k\rangle \nonumber \\&\qquad +(-1)^{d(k)}|k\rangle |0\rangle |k\rangle |0\rangle +|k\rangle |k\rangle |k\rangle |k\rangle )_{Q_1,Q_2,R_1,R_2}\nonumber \\&\quad \xrightarrow []{Entangle} \frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle +(-1)^{d(k)}|0\rangle |k\rangle |0\rangle |k\rangle \nonumber \\&\qquad +(-1)^{d(k)}|k\rangle |0\rangle |k\rangle |0\rangle +|k\rangle |k\rangle |k\rangle |k\rangle )_{Q_1,Q_2,R_1,R_2}|0\rangle _P\nonumber \\&\quad \xrightarrow []{Double-CNOT} \frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle |0\rangle +(-1)^{d(k)}|0\rangle |k\rangle |0\rangle |k\rangle |k\rangle \nonumber \\&\qquad +(-1)^{d(k)}|k\rangle |0\rangle |k\rangle |0\rangle |k\rangle +|k\rangle |k\rangle |k\rangle |k\rangle |0\rangle )_{Q_1,Q_2,R_1,R_2,P}\nonumber \\&\quad \xrightarrow []{CNOT}\frac{1}{2}(|0\rangle |0\rangle |0\rangle |0\rangle |0\rangle +(-1)^{d(k)}|0\rangle |0\rangle |0\rangle |k\rangle |k\rangle \nonumber \\&\qquad +(-1)^{d(k)}|0\rangle |0\rangle |k\rangle |0\rangle |k\rangle +|0\rangle |0\rangle |k\rangle |k\rangle |0\rangle )_{Q_1,Q_2,R_1,R_2,P}\nonumber \\&\quad =\frac{1}{2}|0\rangle |0\rangle _{Q_1,Q_2}(|0\rangle |0\rangle +|k\rangle |k\rangle )_{R_1,R_2}|0\rangle _{P}+(-1)^{d(k)}(|0\rangle |k\rangle +|k\rangle |0\rangle )_{R_1,R_2}|k\rangle _{P})\nonumber \\&\quad =\frac{1}{2}|0\rangle |0\rangle _{Q_1,Q_2}(|x\rangle |x\rangle +|y\rangle |y\rangle )_{R_1,R_2}|0\rangle _{P}+(|x\rangle |x\rangle -|y\rangle |y\rangle )_{R_1,R_2}|k\rangle _{P})\nonumber \\&\quad =|0\rangle |0\rangle _{Q_1,Q_2}\left( \frac{1}{2}|x\rangle |x\rangle _{R_1,R_2}(|0\rangle +|k\rangle )_P+ \frac{1}{2}|y\rangle |y\rangle _{R_1,R_2}(|0\rangle -|k\rangle )_P\right. \end{aligned}$$

(10)