# A New Quantum Blind Signature Scheme with BB84-State

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Theory

#### 2.1. Properties of the Blind Signature

- Unforgeability: No one can generate an effective blind signature except the signer himself/herself. This is one of the most basic requirements.
- Non-repudiation: Once a signer has signed a message, he/she cannot deny his/her signature of the message.
- Blindness: Although a signer has signed a message, he/she cannot get the concrete content of the message.
- Unlinkability: Once the signature of the message is public, the signer cannot determine whether he/she has signed the message.
- Traceability: Once a dispute happens, the verifier can trace the signature.

#### 2.2. Quantum Fingerprint

#### 2.3. Improved QOTP Encryption

## 3. Quantum Blind Signature Scheme

#### 3.1. BB84-State Encoding

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Corollary**

**6.**

#### 3.2. The Proposed Quantum Blind Signature Scheme

#### 3.2.1. Initial Phase

#### 3.2.2. Blinding Phase

**Step B1.**The message owner A first prepares the original message m of the n-bit string. Then, A selects randomly the blind factor w of the n-bit string and blinds m to blind message $\tilde{m}$ based on the formula $\tilde{m}=m\oplus w$.

**Step B2.**According to Equations (2) and (3), A generates n-qubit blind states ${|\phi \rangle}_{\tilde{m}\oplus {k}_{AB}^{\prime},\tilde{m}}$ with the n-bit blind message $\tilde{m}$ and key ${k}_{AB}^{\prime}$, where the n-bit ${k}_{AB}^{\prime}$ is derived from the 2n-bit shared key ${k}_{AB}$ satisfying ${k}_{{AB}_{i}}^{\prime}={k}_{{AB}_{2i}}\oplus {k}_{{AB}_{(2i+1)}}$, $i\in [1,n]$. According to Equation (1), A generates quantum fingerprint $|f(\tilde{m})\rangle $ for blind message $\tilde{m}$. With the shared key ${k}_{AC}$ and ${k}_{AB}$, A applies the improved QOTP [35,36,45], which is described in Subsection 2.3, to encrypt her classical message m and blind factor w, and then obtains ${E}_{{k}_{AB}}({E}_{{k}_{AC}}(m||w))$, where the notation $\left|\right|$ denotes the concatenation of strings.

**Step B3.**A denotes $Sig{n}_{AB}\stackrel{def}{=}{\{|\phi \rangle}_{\tilde{m}\oplus {k}_{AB}^{\prime},\tilde{m}},|f(\tilde{m})\rangle ,{E}_{{k}_{AB}}({E}_{{k}_{AC}}(m||w))\}$ and transmits ${Sig{n}_{AB}}^{\otimes 2}$ to B through the quantum channel, where ${Sig{n}_{AB}}^{\otimes 2}$ represents two copies of $Sig{n}_{AB}$.

#### 3.2.3. Signing phase

**Step S1.**Analogous to the method in

**Step B1**, B obtains the n-bit key ${k}_{AB}^{\prime}$ from the shared 2n-bit key ${k}_{AB}$ between A and B. If the ${k}_{{AB}_{i}}^{\prime}$ is zero, B selects the diagonal basis $\{|+\rangle ,|-\rangle \}$, otherwise rectilinear basis $\{|0\rangle ,|1\rangle \}$. According to this basis rule, B measures all the qubits of the indistinguishable BB84-state ${|\phi \rangle}_{\tilde{m}\oplus {k}_{AB}^{\prime},\tilde{m}}$ corresponding in $Sig{n}_{AB}$ and gets the blind message ${\tilde{m}}^{\prime}$ with the key ${k}_{AB}^{\prime}$.

**Step S2.**According to Equation (1), B generates quantum fingerprint $|f({\tilde{m}}^{\prime})\rangle $. B then compares the generated $|f({\tilde{m}}^{\prime})\rangle $ with state $|f(\tilde{m})\rangle $ from $Sig{n}_{AB}$ and judges whether they are equal based on quantum fingerprint theory in [34]. If they are not equal, then B stops the scheme, otherwise draws the conclusion ${\tilde{m}}^{\prime}=\tilde{m}$ and goes on.

**Step S3.**B first selects randomly two n-bit strings u and v. According to Equation (2) and Equation (1), B then generates respectively the QBS BB84-state ${|\phi \rangle}_{\tilde{m}\oplus u,v}$ and quantum fingerprint $|f(u\left|\right|v\left|\right|\tilde{m})\rangle $ with u, v, and $\tilde{m}$. With the shared key ${k}_{BC}$, B encrypts his strings u and v and then gets ${E}_{{k}_{BC}}(u,v)$. From the receiving $Sig{n}_{AB}$, B decrypts the ${E}_{{k}_{AB}}({E}_{{k}_{AC}}(m||w))$ with his shared key ${k}_{AB}$ and gets ${E}_{{k}_{AC}}(m||w)$, then encrypts it with his shared key ${k}_{BC}$ and obtains ${E}_{{k}_{BC}}({E}_{{k}_{AC}}(m||w))$. B denotes $Sig{n}_{BC}\stackrel{def}{=}\{|f(u|\left|v\right||\tilde{m})\rangle ,{E}_{{k}_{BC}}(u,v),{E}_{{k}_{BC}}({E}_{{k}_{AC}}(m||w))\}$.

**Step S4.**B transmits ${Sig{n}_{BC}}^{\otimes 2}$ and ${{|\phi \rangle}_{\tilde{m}\oplus u,v}}^{\otimes 2}$ to A through the quantum channel.

#### 3.2.4. Unblinding Phase

**Step U1.**After receiving the blind signature ${|\phi \rangle}_{\tilde{m}\oplus u,v}$ for blind message $\tilde{m}$ signed by B, A applies ${H}^{w}$ to ${|\phi \rangle}_{\tilde{m}\oplus u,v}$ with her blind factor w and gets ${H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$, which is a quantum signature for the original message m. With the shared key ${k}_{AC}$, A generates ${E}_{{k}_{AC}}{({H}^{w}|\phi \rangle}_{\tilde{m}\oplus u,v})$ and ${E}_{{k}_{AC}}(m||w)$. A denotes $Sig{n}_{AC}\stackrel{def}{=}\{{E}_{{k}_{AC}}{({H}^{w}|\phi \rangle}_{\tilde{m}\oplus u,v}),{E}_{{k}_{AC}}(m||w)\}$.

**Step U2.**A generates ${E}_{{k}_{AT}}({Sig{n}_{AC}}^{\otimes 2},{Sig{n}_{BC}}^{\otimes 2})$ and transmits it to T through the quantum channel.

**Step U3.**T decrypts the received ${E}_{{k}_{AT}}({Sig{n}_{AC}}^{\otimes 2},{Sig{n}_{BC}}^{\otimes 2})$ and gets ${Sig{n}_{AC}}^{\otimes 2}$ and ${Sig{n}_{BC}}^{\otimes 2}$. Then, T performs the C-SWAPtest [34] to compare the two copies of $Sig{n}_{AC}$ in ${Sig{n}_{AC}}^{\otimes 2}$. The same test is also done on ${Sig{n}_{BC}}^{\otimes 2}$. Once an unequal result of the comparison occurs, T draws the conclusion that the signature is invalid and aborts the process. After T finishes the comparison tests successfully, he preserves one copy of $Sig{n}_{AC}$ and $Sig{n}_{BC}$ to be prepared to solve disputes when they arise in the future. Finally, T generates ${E}_{{k}_{CT}}(Sig{n}_{AC},Sig{n}_{BC})$ with another copy of $Sig{n}_{AC}$ and $Sig{n}_{BC}$ and transmits it to C through the quantum channel.

#### 3.2.5. Verifying Phase

**Step V1.**C first gets $Sig{n}_{AC}$ and $Sig{n}_{BC}$ from the received ${E}_{{k}_{CT}}(Sig{n}_{AC},Sig{n}_{BC})$ with his shared key ${k}_{CT}$. Then, C decrypts the ${E}_{{k}_{BC}}({E}_{{k}_{AC}}(m||w))$ in $Sig{n}_{BC}$ with his shared key ${k}_{BC}$, gets ${E}_{{k}_{AC}}^{\prime}(m||w)$, and performs the C-SWAP test [34] to compare it with ${E}_{{k}_{AC}}(m||w)$ in $Sig{n}_{AC}$. If the result of the comparison is not equal, C draws the conclusion that the signature is invalid and aborts the process. Otherwise, C then applies the key-controlled-“T” QOTP to decrypt ${E}_{{k}_{AC}}(m||w)$ with his shared key ${k}_{AC}$ and finally gets (m, w).

**Step V2.**After getting ${E}_{{k}_{BC}}(u,v)$ from the $Sig{n}_{BC}$, C decrypts ${E}_{{k}_{BC}}(u,v)$ with his shared key ${k}_{BC}$ and then gets (u, v).

**Step V3.**From the received $Sig{n}_{AC}$, C decrypts ${E}_{{k}_{AC}}{({H}^{w}|\phi \rangle}_{\tilde{m}\oplus u,v})$ with his shared key ${k}_{AC}$ and gets ${H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$. With the m obtained in

**Step V1**, C applies ${H}^{m}$ to ${H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$ and gets ${H}^{m}{H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$.

**Step V4.**With the u, v obtained in

**Step V2**, C performs single-particle measurements on the n-qubit ${H}^{m}{H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$ obtained in

**Step V3**and gets ${u}^{\prime},{v}^{\prime}$. The rules of measurement are as follows. According to Corollary 2, C uses diagonal basis $\{|+\rangle ,|-\rangle \}$ to measure the BB84-state if ${u}_{i}\oplus {v}_{i}=0$, otherwise rectilinear basis $\{|0\rangle ,|1\rangle \}$. Based on the measurement result and Equation (2), C can deduce the corresponding ${u}_{i}^{\prime}$, ${v}_{i}^{\prime}$. C aborts the process if ${u}_{i}\ne {u}_{i}^{\prime}$ or ${v}_{i}\ne {v}_{i}^{\prime}$ for some $i\in [1,n]$, otherwise goes on.

**Step V5.**C generates quantum fingerprint $|f(u\left|\right|v\left|\right|(m\oplus w))\rangle $ with the deduced (u, v) and (m, w) and then compares it with $|f(u\left|\right|v\left|\right|\tilde{m})\rangle $ in $Sig{n}_{BC}$ from B. If the result of comparison is equal, C draws the conclusion that the signature is valid, otherwise declares that the signature is not valid.

**Step V6.**According to Equation (2), C regenerates the quantum BB84-state signature ${|\phi \rangle}_{m\oplus u,v}$ with the known m, u, and v. C announces publicly the QBS correctness and declares the signature ${\{m,|\phi \rangle}_{m\oplus u,v}\}$ to the public.

## 4. Security Analyses

#### 4.1. Correctness

**Theorem**

**1.**

**Proof.**

**Step V3**, after receiving ${E}_{{k}_{AC}}{({H}^{w}|\phi \rangle}_{\tilde{m}\oplus u,v})$ from A, C decrypts it and gets ${H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$ with his shared key ${k}_{AC}$, then applies ${H}^{m}$ to ${H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$ to generate ${H}^{m}{H}^{w}{|\phi \rangle}_{\tilde{m}\oplus u,v}$, i.e.,

#### 4.2. Against External Attack

#### 4.3. Blindness

**Step U2**. In such circumstances, the two strategies of B become invalid. Thus, our proposed scheme meets the standard of blindness.

#### 4.4. Unforgeability

**Step V5**. There are two ways for A to forge. One way is that A prepares the original message pair (${m}_{1}$, ${w}_{1}$) in

**Step B1**and ${E}_{{k}_{AB}}({E}_{{k}_{AC}}({m}_{1},{w}_{1}))$ in

**Step B2**and at the same time generates ${E}_{{k}_{AC}}{({H}^{{w}_{2}}|\phi \rangle}_{\tilde{m}\oplus u,v})$ and ${E}_{{k}_{AC}}({m}_{2},{w}_{2})\}$ in

**Step U1**. In this way, C will find that (${m}_{1}$, ${w}_{1}$) is not equal (${m}_{2}$, ${w}_{2}$) and abort this signature. Thus, this strategy is unsuccessful. Another way for A’s forgery is to generate ${E}_{{k}_{AC}}{({H}^{{w}_{2}}|\phi \rangle}_{\tilde{m}\oplus u,v})$ and ${E}_{{k}_{AC}}({m}_{1},{w}_{1})\}$ in

**Step U1**. This way can pass C’s examination in

**Step V1**, but C would still find this strategy in

**Step V3**and

**Step V4**. In

**Step V3**, C applies ${H}^{{m}_{1}}$ to ${H}^{{w}_{2}}{|\phi \rangle}_{\tilde{m}\oplus u,v}$ and gets:

**Step V4**for this way. Thus, the two forgery ways for A are not effective, and the unforgeability of the proposed scheme holds.

**Step U1**. According to our scheme, ${m}^{\prime}$ and ${w}^{\prime}$ must be encrypted with the shared key ${k}_{{A}^{\prime}C}$, and then, ${E}_{{k}_{{A}^{\prime}C}}({m}^{\prime},{w}^{\prime})$ will be transmitted to C. In

**Step V1**, C decrypts ${E}_{{k}_{{A}^{\prime}C}}({m}^{\prime},{w}^{\prime})$ with the shared key ${k}_{AC}$, but C cannot get the correct ${m}^{\prime}$ and ${w}^{\prime}$ because of B’s random guess key ${k}_{{A}^{\prime}C}$. In

**Step V4**, C will find the forgery trick because the equations are not satisfied.

#### 4.5. Non-Repudiation

**Step V1**of the verifying phase, it indicates that the ${E}_{{k}_{AC}}(m||w)$ is equal in $Sig{n}_{AC}$ and $Sig{n}_{BC}$. C confirms the fact of C not aborting the signature verification procedure. If so, C can not deny this verification step. In

**Step V4**of the verifying phase, C performs single-particle measurements on the quantum signature and deduces B’s signature parameters ${u}^{\prime}$ and ${v}^{\prime}$. C would aborts the process if he finds disagreement between the derivative results $({u}^{\prime},{v}^{\prime})$ and received results $(u,v)$. Thus, C cannot deny his actions in this step. In

**Step V5**of the verifying phase, C validates the quantum fingerprint and judge A’s blind parameter w. Similarly, his announcement for A’s blindness cannot be disavowed. In the whole process, it shows that C has accepted the process of signature verification and cannot deny his validation fact if C does not abandon the verification in the verifying phase.

#### 4.6. Unlinkability

#### 4.7. Traceability

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The flow-process diagram of the proposed QBS scheme. Alice blinds the message m with blind factor w and passes $\tilde{m}$ to Bob. With the proposed encoding method of the BB84-state, Bob signs $\tilde{m}$ with u and v and then sends the blind signature ${|\phi \rangle}_{\tilde{m}\oplus u,v}$ back to Alice. Two copies of the unblinding signature are sent to Trent. After Trent’s identical verification for the two signatures, one of the signatures is transmitted to Charlie. Once Charlie verifies the validity of the signature, he publishes the signature ${\{m,|\phi \rangle}_{m\oplus u,v}\}$.

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**MDPI and ACS Style**

Chen, F.-L.; Wang, Z.-H.; Hu, Y.-M.
A New Quantum Blind Signature Scheme with BB84-State. *Entropy* **2019**, *21*, 336.
https://doi.org/10.3390/e21040336

**AMA Style**

Chen F-L, Wang Z-H, Hu Y-M.
A New Quantum Blind Signature Scheme with BB84-State. *Entropy*. 2019; 21(4):336.
https://doi.org/10.3390/e21040336

**Chicago/Turabian Style**

Chen, Feng-Lin, Zhi-Hua Wang, and Yong-Mo Hu.
2019. "A New Quantum Blind Signature Scheme with BB84-State" *Entropy* 21, no. 4: 336.
https://doi.org/10.3390/e21040336