# Unconditionally Secure Quantum Signatures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Properties of Signature Schemes

- (1)
- Unforgeability: A dishonest party should not be able to successfully send a message pretending to be someone else.
- (2)
- Non-repudiation: A signer should be unable to successfully deny that he sent a message signed with his signature.
- (3)
- Transferability: If a verifier accepts a signature, he should be confident that any other verifier (e.g., a judge) would also accept the signature.

## 3. “Classical” Signature Schemes

#### 3.1. Public-Key Digital Signatures

#### 3.2. Cryptographic and Universal Hash Functions

- (1)
- Pre-image resistance: Given $h\left(x\right)$, it should be difficult to find x, that is, these hash functions are one-way functions.
- (2)
- Second pre-image resistance: Given ${x}_{1}$, it should be difficult to find an ${x}_{2}$, such that $h\left({x}_{1}\right)=h\left({x}_{2}\right)$.
- (3)
- Collision resistance: It should be difficult to find any distinct pair ${x}_{1},{x}_{2}$, such that $h\left({x}_{1}\right)=h\left({x}_{2}\right)$.

#### 3.3. Lamport–Diffie One-Time Signatures

#### 3.4. Unconditionally Secure “Classical” Signature Schemes

- (1)
- A signing key: ${s}_{i}=F({U}_{i},{y}_{1},\dots ,{y}_{\omega},z)$;
- (2)
- A pair of verification keys: ${v}_{i}$ and ${\tilde{v}}_{i}=F(x,{v}_{i},z)$.

## 4. Quantum Signature Schemes

#### 4.1. Quantum One-Way Functions and Quantum Hash Functions

**Definition 1.**Quantum one-way function: Let $\psi :{\{0,1\}}^{n}\to \mathcal{H}$ be the mapping $k\to |{\psi}_{k}\rangle $. Then, ψ is called a quantum one-way function if it is easy to compute, i.e., $|{\psi}_{k}\rangle $ for a particular k can be determined using a polynomial-time algorithm, but impossible to invert, in the sense that if given a quantum state prepared in one of the states $|{\psi}_{k}\rangle $, one cannot, using any procedure allowed by quantum mechanics, with certainty determine which state one has been given.

**Definition 2.**(n,s,δ)-quantum hash function: Let ψ be a quantum one-way function whose domain has size ${2}^{n}$ and whose range is a Hilbert space with dimension ${2}^{s}$. Suppose further that any distinct $w,{w}^{\prime}$ give δ-orthogonal outputs ($\delta <1$), i.e., $|\langle \psi \left(w\right)|\psi \left({w}^{\prime}\right)\rangle |<\delta .$ Then, ψ is an (n,s,δ)-quantum hash function.

#### 4.2. Quantum Digital Signatures

- (1)
- Alice chooses M pairs of L-bit classical strings, $\{{k}_{0}^{i},{k}_{1}^{i}\}$, $1\le i\le M$. The ${k}_{0}$’s will be used if the future message is chosen to be the bit zero and the ${k}_{1}$’s will be used if the future message is chosen to be the bit one. Increasing the value of M will increase the security level of the protocol (security is exponential in M).
- (2)
- Alice assigns each of the L-bit strings to a different element in the set of fingerprinting states, or whatever the chosen set of output quantum states is, according to a mapping known to all participants. That is, all participants know the one-way function, but not the L-bit strings used as input. She then distributes the quantum states to the t participants, so that each participant has a suitable number of copies of each of the the $2M$ quantum states $\left\{\right|{\psi}_{{k}_{0}^{i}}\rangle ,|{\psi}_{{k}_{1}^{i}}\rangle \}$, $1\le i\le M$.
- (3)
- Unless the distribution is managed by a trusted third party, the participants should perform some sort of test to ensure that they all received the same public keys. In the three-party setting, Gottesman and Chuang suggest that Alice sends two copies of each public key to each participant. Bob and Charlie would then both perform a SWAP test on their two keys to check that they are the same. One participant, say Bob, would then pass one of his keys to Charlie, who would perform a SWAP test on this key and one of his own to determine if they are equal. Bob performs a similar test on a key received directly from Alice and one forwarded by Charlie. The keys used in these last SWAP tests would then be discarded, and Bob and Charlie are left with one copy of the public key each. If any of the SWAP tests fail, the protocol is aborted. If none of the tests fail, the participants have evidence that Alice did in fact send out the same public key states. By sending each participant more copies of the public key, the probability of discovering a cheating Alice can be made arbitrarily close to one.In the more general t-party setting, the authors suggest a distributed symmetry test performed by all participants. In this case, Alice would distribute $t+1$ copies of the public key to each participant. They would perform a test to check for complete symmetry of their $t+1$ copies. If the test is passed, they would send one copy of the public key to each of the other $t-1$ participants, keeping one copy to use for signature verification and the remaining copy for a further symmetry test using all of the public keys received from other participants.
- (4)
- To later send the message b, Alice would send $(b,{k}_{b}^{1},{k}_{b}^{2},\dots ,{k}_{b}^{M})$. From this, a recipient can easily compute $|{\psi}_{{k}_{b}^{i}}\rangle $ for each i and compare the state to the public keys they previously received from Alice (again using a SWAP test). The recipient counts the number of mismatches he gets.
- (5)
- If the mismatch rate is less than some rate ${s}_{a}$, the recipient will accept the message. If the message is forwarded on from another recipient, the mismatch rate must be less than ${s}_{v}$ to be accepted, where ${s}_{v}>{s}_{a}$.
- (6)
- All used and unused keys are discarded.

#### 4.3. Quantum Signatures Using a Multi-Port without Quantum Memory

**Figure 1.**Taken from [39]. The multi-port used by Bob and Charlie to symmetrise the quantum states they receive from Alice. Whatever overall quantum state Alice sends to Bob ($|\alpha \rangle $) and Charlie ($|\beta \rangle $), the outputs of the signal ports will be symmetric with respect to Bob and Charlie. Bob and Charlie can either store the signal port states in quantum memory or, alternatively, if one wishes to also remove the need for quantum memory, they can perform some type of quantum measurement on the signal ports directly when the states are received from Alice. Detecting photons in the null ports helps to detect cheating.

#### 4.4. Quantum Signatures with Quantum Key Distribution Components

#### 4.5. Security against Tampering with the Quantum Channels

#### 4.6. Coherent State Mappings

## 5. Conclusions

## Acknowledgements

## Conflicts of Interest

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Amiri, R.; Andersson, E.
Unconditionally Secure Quantum Signatures. *Entropy* **2015**, *17*, 5635-5659.
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Amiri R, Andersson E.
Unconditionally Secure Quantum Signatures. *Entropy*. 2015; 17(8):5635-5659.
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Amiri, Ryan, and Erika Andersson.
2015. "Unconditionally Secure Quantum Signatures" *Entropy* 17, no. 8: 5635-5659.
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