Ternary quantum public-key cryptography based on qubit rotation

Abstract

Quantum public-key cryptography (QPKC) can perform the management and distribution of keys in a large-scale quantum communication network, thus being the foundation for many cryptographic applications. The QPKC system, which uses qubit rotation as a quantum one-way function, is one of the most practical public key systems and has found many valuable extensions. In this paper, we analyze and provide supplementary properties of binary qubit rotation. On the basis of Nikolopoulos’work (Phys Rev A 77(3):032348, 2008), we have extended the qubit rotation to three dimensions and propose a ternary QPKC protocol. The protocol can resist forward search attack and achieve higher security than binary QPKC protocol. Theoretically, the protocol can be extended to a higher dimension.

Appendix A The proof of the properties of generalized qubit rotation

Similar to the proof of the superposition of binary qubit rotation, we first obtain ${\widetilde{R}}(\theta )\vert \psi _2\rangle$ by changing the phase and bit flip.

${\widetilde{X}}$ and ${\widetilde{S}}$ are two N-dimensional quantum gates, the effect of ${\widetilde{X}}$ is to make $\vert \psi _2\rangle \Rightarrow \vert \psi _1\rangle$ and $\vert \psi _1\rangle \Rightarrow \vert \psi _2\rangle$, the effect of ${\widetilde{S}}$ is to change the phase of the quantum state by $\frac{\pi }{2}$. Then we have

$$\begin{aligned} \begin{aligned} {\widetilde{R}}(\theta )_{({\hat{\alpha }})}\vert \psi _2\rangle&={\widetilde{X}}{{\widetilde{S}}}^2{\widetilde{R}}(\pi -\theta )_{({\hat{\alpha }})}\vert \psi _1\rangle \\&={\widetilde{X}}{{\widetilde{S}}}^2\left\{ {\cos \left( \frac{\theta }{2}\right) }\vert \psi _1\rangle +{\sin \left( \frac{\theta }{2}\right) }\vert \psi _2\rangle \right\} \\&={\widetilde{X}}\left\{ {\sin \left( \frac{\theta }{2}\right) \vert \psi _1\rangle -\cos \left( \frac{\theta }{2}\right) \vert \psi _2\rangle }\right\} \\&=-\sin \left( \frac{\theta }{2}\right) \vert \psi _1\rangle +\cos \left( \frac{\theta }{2}\right) \vert \psi _2\rangle .\\ \end{aligned} \end{aligned}$$

(A1)

Then it can be concluded that

$$\begin{aligned} \begin{aligned} {\widetilde{R}}(\theta _1+\theta _2)_{({\hat{\alpha }})}\vert \psi _1\rangle&=\cos \left( \frac{\theta _1}{2}+\frac{\theta _2}{2}\right) \vert \psi _1\rangle +\sin \left( \frac{\theta _1}{2}+\frac{\theta _2}{2}\right) \vert \psi _2\rangle \\&=\cos \left( \frac{\theta _1}{2}\right) \left\{ \cos \left( \frac{\theta _2}{2}\right) \vert \psi _1\rangle +\sin \left( \frac{\theta _2}{2}\right) \vert \psi _2\rangle \right\} \\&\quad +\sin \left( \frac{\theta _1}{2}\right) \left\{ \cos \left( \frac{\theta _2}{2}\right) \vert \psi _2\rangle -\sin \left( \frac{\theta _2}{2}\right) \vert \psi _1\rangle \right\} \\&={\widetilde{R}}(\theta _2)_{({\hat{\alpha }})}\left\{ \cos \left( \frac{\theta _1}{2}\right) \vert \psi _1\rangle +\sin \left( \frac{\theta _1}{2}\right) \vert \psi _2\rangle \right\} \\&={\widetilde{R}}(\theta _2)_{({\hat{\alpha }})}{\widetilde{R}}(\theta _1)_{({\hat{\alpha }})}\vert \psi _1\rangle .\\ \end{aligned} \end{aligned}$$

(A2)

It is proved that the generalized qubit rotation is also have superposition. The above conclusion can also be expressed as

$$\begin{aligned} \begin{aligned} {\widetilde{R}}(\theta _1)_{({\hat{\alpha }})}{\widetilde{R}}(\theta _2)_{({\hat{\alpha }})}\vert \psi \rangle&={\widetilde{R}}(\theta _1+\theta _2)_{({\hat{\alpha }})}\vert \psi _1\rangle \\&={\widetilde{R}}(\theta _2+\theta _1)_{({\hat{\alpha }})}\vert \psi _1\rangle \\&={\widetilde{R}}(\theta _2)_{({\hat{\alpha }})}{\widetilde{R}}(\theta _1)_{({\hat{\alpha }})}\vert \psi \rangle \\ \longrightarrow {\widetilde{R}}(\theta _1)_{({\hat{\alpha }})}{\widetilde{R}}(\theta _2)_{({\hat{\alpha }})}\vert \psi \rangle&={\widetilde{R}}(\theta _2)_{({\hat{\alpha }})}{\widetilde{R}}(\theta _1)_{({\hat{\alpha }})}\vert \psi \rangle .\\ \end{aligned} \end{aligned}$$

(A3)

Namely, generalized qubit rotation is commutative.