Arbitrated Quantum Digital Signature

This example protocol provides a quantum digital signature scheme where the public (known to all) and private (secret key preserved with the seller) keys are classical in nature, however the signature has a quantum nature. This is based directly on public-key cryptography where the seller's identity is used to generate the public-key and one-time pad generates the private key.

Tags: Quantum Digital Signature, Public key cryptography, Specific Task, Multi Party

Assumptions

The protocol assumes perfect state preparation, transmissions, and measurements.
Private-key generation (PKG) is a trusted third party, arbitrator.
In the signing process, the quantum one-way function used to create the quantum digest is assumed to take polynomial time to compute and is hard to invert.
Seller and PKG are assumed to have a pre-shared quantum key (say, using QKD)

Outline

Like other QDS protocols, it is divided into two phases: Distribution and Messaging. This scheme is presented between the seller (one who signs the message), the buyer (one whom the signed message is sent) and PKG (generates and distributes public-private key for the seller) and a buyer
. Distribution includes the generation of public and private keys as follows

Key Generation: In this step, PKG generates the public key of the seller and generates a private key which is secretly sent to Seller over the insecure classical channel.
- Seller's public key is derived from her personal information such as her email-id over a public channel. A one-way function is chosen by PKG randomly and secretly (known as the master key), which uses the classical public key as its input.
- A random OTP of the same length as the outcome of the function (random key), is used to convert it (the outcome) into seller's private key by performing bit-wise modulo 2 sum (exclusive OR gate).
- The quantum pre-shared common key (assumption) is then used to one-time pad the private key via Quantum Vernam Cipher (1), (2). The one-time padded cipher-text is then communicated to the seller (over the insecure channel).
- Seller un-pads the cipher-text to obtain the private key using the pre-shared common key. Hence, in the end, everyone knows the seller's public key and, only PKG and seller know her private key.

Messaging comprises of the following steps

Signing: In this step, the seller generates a signature quantum state using the message she wants to send, her public key and private key. The seller selects a quantum one-way function publicly to generate a quantum digest (directory) using these classical inputs. Seller repeats each step for each message bit.
- Seller selects two random strings and generates a quantum state of the message using these random strings to operate a Unitary gate and Hadamard Transform on a null/vacuum state (see Pseudo Code for operations)
- The public and the private key are used to perform Hadamard transformation on the state produced in the previous step in order to generate the signature quantum state.
- The Seller then performs some operation using her private key and measures the quantum state. It can be shown the states were one of the BB84 states and hence, can have one of the two possible bases (X basis, Z basis or + basis,x basis) and four possible states. She records the basis and classical bit representing the state obtained.
- Seller then concatenates these classical bits, the two random string bits, and a timestamp unique to the signature. The concatenated classical string is used as the input of publicly chosen QOWF, to get the output called 'quantum digest'. She produces some copies of quantum digest depending on the number of recipients.
- Seller then encrypts the timestamp and quantum output of QOWF with pre-shared common key via quantum vernam cipher. PKG unpads these and publicly announces for buyer's verification step.
- Sellers sends the signature to the buyer which includes the signature quantum state, message, timestamp and basis states.

Verification: In this method, the verifier checks the authenticity of the signature (whether the message has come from a genuine seller).
- The verifier performs some quantum gates using seller's public key and message on the signature quantum state.
- One of the randomly selected string by the Signer can be easily inferred by the Verifier from the state after the measurement. The Verifier is then able to generate their own copy of quantum digital digest using the publicly announced quantum one-way function.
- Verifier now publicly gains the timestamp and quantum digital digest from PKG and verifies that state with the produced quantum digital digest in the above step with the SWAP test. As the SWAP test has a probabilistic result, it is performed several times with the copies of quantum digital digest and then verified.
- If the test is passed the message from the Signer would be valid otherwise it is rejected.

Notation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} : Total number of qubits of message.
$k_{pub}$ : Signer's public key, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pub} \in \{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri}} : Signer's private, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri} \in \{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_r} : Random OTP number selected by PKG to denote each of Signer's signatures, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{r} \in \{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{at}} : Shared key between the Signer and PKG where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{at} \in \{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{k_{at}}} : Quantum Vernam cipher encrypted state which uses $k_{at}$ .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} : PKG's master key which is a one way function where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0,1\}^n \xrightarrow{}\{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} : Public quantum one way function selected by Signer to generate quantum digest.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} : Message sent by Signer to the Verifier, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} : Random string of uniform distribution selected by the Signer, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in \{0,1\}^n} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} : Random string of uniform distribution selected by the Signer, where $t\in \{0,1\}^{n}$ .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle_{a_l,b_l}} : Quantum state which is defined by

$|\phi \rangle _{a_{l},b_{l}}:=H^{a_{l}}U_{\frac {\pi }{4}}H^{b_{l}}|0\rangle$

$|\phi \rangle _{a_{l},b_{l},c_{l}}$ : Quantum state which is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle_{a_l,b_l,c_l} := Y^{c_l}|\phi\rangle_{a_l,b_l}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |S\rangle_{k_{pri},m}} : Signature quantum state for message $m$ which is the quantum state

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |S\rangle_{k_{pri},m} = \bigotimes^{n}_{l=1} H^{k_{pub_l}\oplus k_{pri_l}} |\phi\rangle_{s_l,t_l\oplus m_l, m_l}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P\rangle} : Private key quantum state where $|P\rangle \in \{|+\rangle ,|-\rangle ,|1\rangle ,|0\rangle \}^{n}$ and it is the quantum state:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P\rangle := H^{k_{pri}}|\phi\rangle_{s, t\oplus m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} : Classical 2n-bit for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -qubit $|P\rangle$ where $|+\rangle$ is encoded to 10, $|-\rangle$ to 11, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle} to 00 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle} is encoded to 01.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_P} : This is the set of the basis of each qubit state in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P\rangle} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_P = Basis(|P\rangle) \in \{+,\times \}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle} : Quantum digital digest received by PKG.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle'} : Quantum digital digest generated by Verifier.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} : The most number of verifiers in this scheme.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} : Safety parameter threshold for acceptance.
$w_{0}$ : Security threshold decided in advance.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w'} : Number of times SWAP test is performed.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |V\rangle_{m, k_{pub},S}} : A quantum state, where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |V\rangle_{m, k_{pub},S} := Y^m H^{k_{pub}}|S\rangle_{k_{pri}, m}} This state is also expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta|\phi\rangle_{k_{pri}\oplus s, t\oplus m}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \in \{1, -1, \iota, -\iota\}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |Q\rangle} : Result of Verifier's measurement of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |V\rangle_{m, k_{pub},S}} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} : Classical bit string denoted as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q \in \{00, 01, 10, 11\}^n} . It is proven that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=Q} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle F|F\rangle'} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta \in [0,1)} .

Hardware Requirements

The total number of qubits used in this protocol is equal to the total number of qubits in the message.
Secure quantum channel between Signer and Verifier
Quantum channel between Signer and PKG
Private database for both Signer and PKG
Measurement devices for the Verifier.

Properties

The protocol assumes the PKG to be a trusted party.
This protocol cannot be broken even if the adversary had unlimited computing power.
In this protocol, it is proven that no adversary can break the secrecy of the Signer's signature private key.
The quantum digital signature produces in this protocol is impossible to repudiate and cannot be forged in any condition.
In the protocol the public and the private key belong to the classical bits, only the signature cipher has quantum nature.
No Certificate Authority is required to manage digital public-key certificate of Signers.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle = |F\rangle'} , the measuring result Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle} occurs with probability 1, otherwise it occurs with probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1+\delta^2}{2}} . Hence, when repeated for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} times, the probability of equality is at least 1-Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\frac{1+\delta^2}{2})^w} .

Pseudocode

Stage 1: Key Generation
Output: Signer receives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri}} from the PKG.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pub}} is generated on the basis of Signer's public identity information like email or person ID-card.
PKG aquires Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pub}} through open channels.
PKG selects Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} randomly as its master key.
PKG selects Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{r}} randomly.
PKG calculates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri}} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri} := G(k_{pub}) \oplus k_{r} }

PKG uses Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{at}} to encrypt Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri}} and transmits Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{k_{at}}} to Signer.
Signer decrypts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{k_{at}}} using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{at}} and receives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{pri}} .

Stage 2: Signature
Output: PKG receives the quantum digest Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle} and the Verifier receives the Signature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ts, m, B_P,|S\rangle_{k_{pri}, m})} from the Signer.

Signer wants to sign the message Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} .
Signer selects Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l = 1, 2, ...n} :
- Signer generates the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle_{s_l,t_l\oplus m_l, m_l}} , which is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle_{s_l,t_l\oplus m_l, m_l} = Y^{m_l}H^{s_l}U_{\frac{\pi}{4}}H^{t_l\oplus m_l}|0\rangle}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l = 1, 2, ...n} :
- Signer generates the Signature quantum state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |S\rangle_{{k_{pri}}_l,m_l}} , which is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |S\rangle_{{k_{pri}}_l,m_l} = H^{k_{pub_l}\oplus k_{pri_l}}|\phi\rangle_{s_l,t_l\oplus m_l, m_l}}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l = 1, 2, ...n} :
- Signer generates the private key quantum state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P\rangle_l} , which is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P\rangle_l = H^{k_{pri_l}} |\phi\rangle_{s_l, t_l\oplus m_l}}

- The classical Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_l} is calculated based on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P\rangle_l} .
- The basis set is formed by Signer is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{P_l} = Basis(|P\rangle_l) \in \{+,\times \} }

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 1, 2, ...u w} : Different copies of the quantum digital digest state is prepared.
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l = 1, 2, ...n} :

The quantum digital digest state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle_l} is prepared by Signer, where:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle_l = |F(t_l||m_l||P_l|| t_l s_l)\rangle_l}

Signer encrypts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle} using quantum Vernam cipher and sends Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{k_{at}}(ts, \otimes^{uw}_{l=1}|F\rangle)} to PKG.
PKG decrypts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{k_{at}}(ts, \otimes^{uw}_{l=1}|F\rangle)} using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_{at}} and gets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ts, \otimes^{uw}_{l=1}|F\rangle)} .
PKG announces publicly that the quantum digest is ready.
Signer transmits Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ts, m, B_P, |S\rangle_{k_{pri}, m})} to Verifier, which is the signature.

Stage 3: Verification
Output: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is considered valid or is rejected by the Verifier.

Verifier receives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |k\rangle_{pub}} from open channels.
Verifier generates the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |V\rangle_{m, k_{pub},S}} .
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l = 1, 2, ... w} :
- Verifier measure the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |V\rangle_{{(m, k_{pub},S)}_{l}}} according to the basis (diagonal or horizontal) in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{P_l}} .
- The result of the measurement is recorded as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |Q\rangle_l} , which is converted to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_l} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} is inferred by the Verifier using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}
Verifier gains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ts, \otimes^{w}_{l=1} |F\rangle)} from PKG.
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 1, 2, ... w'} :
- Verifier generates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle'} using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} by the calculation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle' = |F(t||m||Q||t s)\rangle}

- Verifier gains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ts, |F\rangle)} from PKG.
- Verifier performs SWAP test between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle'} .
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w'>w_0} and measurement result everytime = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle'} :
- Verifier counts the message Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} as valid.
else:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is rejected by the Verifier.

Further Information

Like most other classical digital signature schemes which provide unconditional security, this scheme also requires a trusted arbitrator who distributes public key to the recipients.

References

*contributed by Rhea Parekh

Arbitrated Quantum Digital Signature

Contents

Assumptions

Outline

Notation

Hardware Requirements

Properties

Pseudocode

Further Information

References

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Arbitrated Quantum Digital Signature

Assumptions

Outline

Notation

Hardware Requirements

Properties

Pseudocode

Further Information

References

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