Authenticable dynamic quantum multi-secret sharing based on the Chinese remainder theorem

Abstract

Quantum secret sharing is widely applied in the fields, such as communications and data transmission. In this paper, we propose an authenticable dynamic quantum multi-secret sharing scheme based on the Chinese remainder theorem. In our scheme, the dynamic update of the participants set is permissible without changing the shared secret. To share multi-secret, the distributor generates corresponding shares based on the Chinese remainder theorem and monotone span program, and the GHZ state acts as the information carrier traveling among the participants in the authorization set; the participants utilize the Hadamard operator and Pauli operators to embed their shares in the traveling particles. In this way, the participants will reconstruct multiple secrets. Furthermore, the proposed quantum digital signature algorithm based on entanglement swapping is utilized to realize the identity authentication between participants. The security analysis shows that the proposed scheme can resist intercept-resend attack, entanglement-measurement attack, internal dishonest participant attack, add or withdraw participant attack, and denial attack.

Appendix A Proof of Eq. (5)

Proof

$\left| \Psi (b,u_{1},\dots ,u_{n} )\right\rangle =\frac{1}{\sqrt{2}} \left( \left| 0u_{1}\dots u_{n}\right\rangle +\left( -1 \right) ^{b} \left| 1\overline{u_{1}\dots u_{n}}\right\rangle \right) =\frac{1}{\sqrt{2}}\sum _{j=0}^{1}$ $\left( -1 \right) ^{jb}\left| j,j+u_{1},\dots ,j+u_{n} \right\rangle$, the four Pauli operators in Eq.(4) can be rewritten to $U_{\alpha ,\beta }=\sum _{j=0}^{1} \left( -1 \right) ^{j\alpha } \mathinner {|{j}\rangle }\mathinner {\langle {j+\beta }|}$, $\alpha ,\beta \in \left\{ 0,1 \right\}$, thus,

(A1)

where, $\alpha _{i},\beta _{i}\in \left\{ 0,1 \right\} , i=0,1,\dots ,n$, and since

(A2)

removing the global phase, Eq.(5) holds,i.e.,

$$\begin{aligned} \begin{aligned}&U_{\alpha _{0},\beta _{0}}\otimes U_{\alpha _{1},\beta _{1}}\otimes U_{\alpha _{2},\beta _{2}}\otimes \dots \otimes U_{\alpha _{n},\beta _{n}} \left| \Psi (b,u_{1},\dots ,u_{n} ) \right\rangle \\=&\left( -1 \right) ^{( u_{1} +\beta _{1} )\alpha _{1}+(u_{2} +\beta _{2} )\alpha _{2}+\cdots +(u_{n} +\beta _{n} )\alpha _{n}}\left| \Psi \left( b+\alpha _{0}+\alpha _{1}+\alpha _{2}+\dots +\alpha _{n},u_{1}\right. \right. \\&\left. \left. +\beta _{0}+\beta _{1},\dots ,u_{n}+\beta _{0}+\beta _{n} \right) \right\rangle . \end{aligned} \end{aligned}$$

$\square$

Extension of Eq. (5)

$n+1$ generalized Pauli operator $U_{\alpha '_{i},\beta '_{i}} =\sum _{j=0}^{d-1} \omega ^{j\alpha '_{i} } \mathinner {|{j}\rangle }\mathinner {\langle {j+\beta '_{i}}|}\left( i=0,1,\cdots ,n \right)$ is applied to $n+1$-qudit GHZ state $\left| \Phi (b,u_{1},\dots ,u_{n} )\right\rangle =\frac{1}{\sqrt{d}}\sum _{j=0}^{d-1}\omega ^{jb}\left| j,j+u_{1},\right.$ $\left. \dots ,j+u_{n}\right\rangle$,where, $\omega =e^{\frac{2\pi \mathfrak {i}}{d}}$, we can obtain the equation as follows:

$$\begin{aligned} \begin{aligned}&U_{\alpha '_{0},\beta ' _{0}}\otimes U_{\alpha '_{1},\beta ' _{1}}\otimes U_{\alpha '_{2},\beta ' _{2}}\otimes \dots \otimes U_{\alpha '_{n},\beta ' _{n}} \left| \Phi (b,u_{1},\dots ,u_{n} ) \right\rangle \\ =&\omega ^{( u_{1} +\beta '_{1} )\alpha ' _{1}+(u_{2} +\beta '_{2} )\alpha '_{2}+\cdots +(u_{n} +\beta '_{n} )\alpha '_{n}}\left| \Phi \left( b+\alpha '_{0}+\alpha ' _{1}+\alpha ' _{2}+\dots +\alpha ' _{n},u_{1}+\beta ' _{0}\right. \right. \\&\left. \left. +\bar{\beta '_{1}},\dots ,u_{n}+\beta ' _{0}+\bar{\beta '_{n}} \right) \right\rangle , \end{aligned}\nonumber \\ \end{aligned}$$

(A3)

Proof

(A4)

since,

(A5)

where, $\alpha '_{i},\beta '_{i}\in \left\{ 0,1,\cdots ,d-1\right\}$, removing the global phase, from the Eq.(A4) and Eq.(A5), we know that the Eq.(A3) holds. $\square$