Abstract
In this paper, we present two definitions of the orthogonality and orthogonal rate of an encryption operator, and we provide a verification process for the former. Then, four improved ternary quantum encryption schemes are constructed. Compared with Scheme 1 (see Section 2.3), these four schemes demonstrate significant improvements in term of calculation and execution efficiency. Especially, under the premise of the orthogonal rate ε as secure parameter, Scheme 3 (see Section 4.1) shows the highest level of security among them. Through custom interpolation functions, the ternary secret key source, which is composed of the digits 0, 1 and 2, is constructed. Finally, we discuss the security of both the ternary encryption operator and the secret key source, and both of them show a high level of security and high performance in execution efficiency.
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Acknowledgements
We are grateful to Qingbin Luo for useful discussions. We also thank Dr. Binju. Wang and Dina A. Sharon for their kind help in correcting and improving the language of the manuscript. We acknowledge financial support from the Education Department of Fujian Provice, China(Grant No.JA15321) and the Key Technology Support Program of Sichuan Province, China (Grant No.2015GZ0096).
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Appendix:
Appendix:
According to the QOTP scheme, the encryption operator U k is completely orthogonal and unconditionally secure. It is important that 4 guarantees the unconditional security of the QOTP scheme. Two verification processes are shown below for the orthogonality and 4, respectively.
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1)
The orthogonality
First, three tensor product properties related to the verification process are shown below, wherein A,B,C,D,M∈U k (see Remark 2).
- Property 1 :
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Suppose that the eigenvalue of the matrix A is λ(A)={λ 1,λ 2,⋅⋅⋅,λ m } and the eigenvalue of the matrix B is λ(B)={μ 1,μ 2,⋅⋅⋅,μ n }, then the eigenvalue of the matrix A⊗B can be denoted as λ(A⊗B)={λ i μ j |i=1,2,⋅⋅⋅,m;j=1,2,⋅⋅⋅,n}.
- Property 2 :
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\(\left ({A \otimes B} \right ) \cdot \left ({C \otimes D} \right ) = (A \cdot C) \otimes (B \cdot D)\).
- Property 3 :
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\({\left ({A \otimes B} \right )^{\dag } } = {A^{\dag } } \otimes {B^{\dag } }\).
Then, considering \(Tr(A) = \sum {\lambda \left (A \right )}\), the verification process will start with the calculation eigenvalues. Obviously, we can obtain \(\lambda \left (I \right ) = \left \{ {1,1} \right \}\) and \(\lambda \left (X \right ) = \lambda \left (Z \right ) = \left \{ {1, - 1} \right \}\). Meanwhile, the eigenvalue \(\lambda \left ({XZ} \right )\) is also equal to the eigenvalue \(\lambda \left ({ZX} \right ) = \left \{ {i, - i} \right \}\). Moreover, they are mutually conjugate transposes, i.e. X Z=(Z X)‡ and Z X=(X Z)‡. The verification process is described as follows.
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(1)
In the case of A=B. For n=1, M∈{I,X,Z,X Z,Z X} can be denoted as a 2×2 unitary matrix, and obviously M ‡ M=I 2 holds; for n≥2, the tensor product exists. Considering Property 2 and Property 3, we can deduce that the matrix A ‡ B has the form of the tensor product of n matrices M, which is a unitary matrix of order 2n×2n. Clearly, M ‡ M=I 2 n holds. To sum up, in the case of A=B, the equation \((A,B) = Tr({M^{\dag } }M) = \sum {\lambda ({I_{{2^{n}}}})} = {2^{n}}\) holds.
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(2)
In the case of A≠B. For n=1, note that A,B and M∈{I,X,Z,X Z,Z X}, so (A,B)=0 holds; for n≥2, combining Property 2 and Property 3, the matrix A ‡ B has the form \( \otimes _{i = 1}^{n}M\). Then considering Property 1, we can obtain that the numbers of the eigenvalues 1 and -1 are equal in matrix A ‡ B, so with the eigenvalues i and -i. As such, we can deduce \((A,B) = \sum {\lambda ({A^{\dag } }B)} = 0\). Overall, the encryption operator U k satisfies the orthogonality requirement in whole inner product space.
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2)
The totally mixed state
According to Definition 1 and the process of verifying the orthogonality, an arbitrary state ρ in inner product space can be expressed as a linear combination below,
$$ \rho = \sum\limits_{\alpha ,\beta } {{a_{\alpha ,\beta }}} {X^{\alpha} }{Z^{\beta} } $$(6)where \({a_{\alpha ,\beta }} = {{Tr(\rho {Z^{\beta } }{X^{\alpha } })} {\left / {\vphantom {{Tr(\rho {Z^{\beta } }{X^{\alpha } })} {{2^{n}}}}} \right . \kern -\nulldelimiterspace } {{2^{n}}}}\). Through (6) and X α Z β=(−1)α⋅β Z β X α (see Ref. [32]), (4) could be deduced,
$$\begin{array}{l} {\rho_{c}} = \sum\limits_{k} {{p_{k}}{U_{k}}\rho U_{k}^{\dag} } = \frac{1}{{{2^{2n}}}}\sum\limits_{\lambda ,\delta } {{X^{\lambda} }{Z^{\delta} }\rho {Z^{\delta} }{X^{\lambda} } = } \frac{1}{{{2^{2n}}}}\sum\limits_{\alpha ,\beta } {{a_{\alpha ,\beta }}} \sum\limits_{\mu ,\upsilon } {{X^{\mu} }{Z^{\upsilon} }{X^{\alpha} }{Z^{\beta} }{Z^{\upsilon} }{X^{\mu} }} \\ \quad \, = \frac{1}{{{2^{2n}}}}\sum\limits_{\alpha ,\beta } {{a_{\alpha ,\beta }}} \sum\limits_{\mu ,\upsilon } {{{(- 1)}^{\alpha \cdot \upsilon \oplus \mu \cdot \beta }}{X^{\alpha} }{Z^{\beta} }} = \sum\limits_{\alpha ,\beta } {{a_{\alpha ,\beta }}} {\delta_{\alpha ,\mathbf{{0}}}}{\delta_{\beta ,\mathbf{{0}}}}{X^{\alpha} }{Z^{\beta} }\\ \quad \, = {\alpha_{\mathbf{{0}},\mathbf{{0}}}}{X^{\mathbf{{0}}}}{Z^{\mathbf{{0}}}} = \frac{{Tr(\rho )}}{{{2^{n}}}}{I_{{2^{n}}}} = \frac{1}{{{2^{n}}}}{I_{{2^{n}}}} \end{array} $$wherein one symbol “\(\bigoplus \)” stands for modulo 2 addition, and another symbol “.” represents inner product operation. The vector 0 is defined as \(\mathbf {{0}} = {\left \{ 0 \right \}^{n}}\). Clearly, the ciphertext ρ c is a totally mixed state.
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Wang, Yq., She, K., Huang, Rf. et al. Optimal Symmetric Ternary Quantum Encryption Schemes. Int J Theor Phys 55, 4709–4722 (2016). https://doi.org/10.1007/s10773-016-3094-8
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DOI: https://doi.org/10.1007/s10773-016-3094-8