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 scheme is based 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)
• Secure quantum channel between seller and buyer is assumed

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 for each recipient (buyer).
  • 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, buyer checks the authenticity of the signature (whether the message has come from a genuine seller).
  • The buyer performs some quantum gates on the signature quantum state, using seller's public key and message. He measures the resulting quantum state, using basis states for each qubit sent in the signature. The result thus, obtained is represnted by a classical string, in the same way as done by seller.
  • The result should reveal the random string used by seller and hence, buyer can also generate the same number of copies of the quantum digest using the publicly known QOWF.
  • Buyer, thus, compares his outputs of QOWF with the ones sent by the seller using quantum SWAP Test. If the number of matches is greater than the accepted/decided threshold value, the signature is accepted else it is rejected.

Notation

• $n$: Total number of qubits of message.
• $f$: public function to obtain public key from user's email-id
• $k_{pub}$: Seller's public key, where $k_{pub}\in \{0,1\}^{n}$.
• $k_{pri}$: Seller's private, where $k_{pri}\in \{0,1\}^{n}$.
• $k_{r}$: Random OTP number selected by PKG to denote each of Seller's signatures, where $k_{r}\in \{0,1\}^{n}$.
• $VC(x,y)$: function VC performs one time pads 'y' using quantum pad key 'x' via Quantum Vernam Cipher (1), (2).
• $k_{at}$: Shared key between the Seller and PKG where $k_{at}\in \{0,1\}^{n}$.
• $E_{k_{at}}$: Quantum Vernam cipher encrypted state which uses $k_{at}$.
• $G$: PKG's master key which is a one way function where $\{0,1\}^{n}{\xrightarrow {}}\{0,1\}^{n}$ .
• $F$: Public quantum one way function selected by Seller to generate quantum digest.
• $m$: Message sent by Seller to the Buyer, where $m\in \{0,1\}^{n}$.
• $s$: Random string of uniform distribution selected by the Seller, where $s\in \{0,1\}^{n}$.
• $t$: Random string of uniform distribution selected by the Seller, where $t\in \{0,1\}^{n}$.
• $l$: qubit address
• $|\phi \rangle _{a_{l},b_{l}}$: 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

$|\phi \rangle _{a_{l},b_{l},c_{l}}:=Y^{c_{l}}|\phi \rangle _{a_{l},b_{l}}$

• $|S\rangle _{k_{pri},m}$: Signature quantum state for message $m$ which is the quantum state

$|S\rangle _{k_{pri},m}=\bigotimes _{l=1}^{n}H^{k_{pub_{l}}\oplus k_{pri_{l}}}|\phi \rangle _{s_{l},t_{l}\oplus m_{l},m_{l}}$

• $|P\rangle$: Private key quantum state where $|P\rangle \in \{|+\rangle ,|-\rangle ,|1\rangle ,|0\rangle \}^{n}$ and it is the quantum state:

$|P\rangle :=H^{k_{pri}}|\phi \rangle _{s,t\oplus m}$

• $P$: Classical 2n-bit for $n$-qubit $|P\rangle$ where $|+\rangle$ is encoded to 10, $|-\rangle$ to 11, $|1\rangle$ to 00 and $|0\rangle$ is encoded to 01.
• $B_{l}$: This is the set of the basis of each $l^{t}h$ qubit in $|P\rangle$.

$B_{l}\in \{+,\times \}$

• $B_{l}(|P_{l}\rangle )$: Measurement of $l^{th}$ qubit in basis 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_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 b_l} : measurement result 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 l^{th}} qubit in the concerned 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 |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 Buyer.
• 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 Buyer 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.
• 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_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} : 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 g(Q)} : g is a classical function which when takes classical 2n bit string Q, gives seller's random string t as output. This function can be calculated.
• 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

• Network Stage:Prepare and Measure
• The total number of qubits used in this protocol is equal to the total number of qubits in the message.
• Secure quantum channel between seller and buyer

Properties

• 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 seller's signature private key.
• The quantum digital signature produced in this protocol is impossible to repudiate and cannot be forged in any condition.
• In the protocol the public and the private key belonging to the classical bits, only the signature cipher has quantum nature.
• No Certificate Authority is required to manage digital public-key certificate of sellers.
• 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 Distribution
Input: Seller and 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_{at}} )
Output: Seller and 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_{pri}} ); Everyone (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}} )

1. PKG 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 k_{pub}= f(} Seller's email-id)
2. PKG randomly chooses 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} , 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}
3. 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} := G(k_{pub}) \oplus k_{r} }
4. PKG 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 VC(k_{at},k_{pri})=E_{k_{at}}}
5. Seller 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 VC(k_{at}, E_{k_{at}})=k_{pri}} .

Stage 2.1 Messaging: Signature
Input: Seller (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}} , 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} , 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}} )
Output: 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} ), Buyer (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})}

1. Seller randomly chooses 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} .
2. 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 \forall l\epsilon\{1,..,n\}} , Seller operates 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 Y^{m_l}H^{s_l}U_{\frac{\pi}{4}}H^{t_l\oplus m_l}|0\rangle=|\phi\rangle_{s_l,t_l\oplus m_l, m_l}}
3. 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 \forall l\epsilon\{1,..,n\}} , Seller 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 |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}}
4. 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 \forall l\epsilon\{1,..,n\}} , Seller 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 |P\rangle_l = H^{k_{pri_l}} |\phi\rangle_{s_l, t_l\oplus m_l}}
5. 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} :
   1. Seller chooses 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_l \epsilon_R\{+,\times\}}
   2. Seller measures in basis 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_l: B_l(|P_l\rangle)=b_l}
   3. 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 B_l = +} then
      1. 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 b_l=1} then 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=00} 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 P_l=01}
   4. 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 B_l = \times} then
      1. 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 b_l=1} then 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=10} 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 P_l=11}
6. 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} :
   1. 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} :
      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 |F\rangle_l = |F(t_l||m_l||P_l|| t_l s_l)\rangle_l}
7. Seller 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 VC(k_{at},(ts, \otimes^{uw}_{l=1}|F\rangle ))=E_{k_{at}}} and sends to PKG
8. 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 VC(k_{at},E_{k_{at}})=(ts, \otimes^{uw}_{l=1}|F\rangle )}
9. PKG announces publicly that the quantum digest is ready.
10. Seller transmits 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, |S\rangle_{k_{pri}, m})} to buyer.

Stage 2.2 Messaging: Verification
Input: Buyer (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, |S\rangle_{k_{pri}, m})} , public key 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})} )
Output: Buyer accepts or rejects the signature

1. Buyer operates: 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 Y^mH^{k_{pub}}|S\rangle_{k_{pri}, m}=|V\rangle_{m, k_{pub},S}} .
2. 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} :
   1. Buyer measures 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}}} in basis 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_l: B_l(|V_l\rangle_{{(m, k_{pub},S)}_{l}})=b_l}
   2. 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 B_l = +} then
      1. 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 b_l=1} then 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=00} 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 Q_l=01}
   3. 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 B_l = \times} then
      1. 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 b_l=1} then 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=10} 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 Q_l=11}
3. Buyer obtains 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=g(Q)}
4. Buyer 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 (ts, \otimes^{w}_{l=1} |F\rangle)} from PKG.
5. For $k=1,2,...w'$:
   1. Buyer 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' = |F(t||m||Q||t s)\rangle}
   2. Buyer 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 (ts, |F\rangle)} from PKG.
   3. Buyer performs SWAP test: QSWAP(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'} )
   4. If QSWAP=true, then 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'+1}
6. 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} buyer accepts
7. Else buyer rejects

Further Information

Like most other classical digital signature schemes which provide unconditional security, this scheme also requires a trusted arbitrator who distributes the public key to the recipients. This protocol was preceded by a few other protocols which use an arbitrator to establish quantum digital signatures, most of which used entangled states.

1. Zeng and Keitel (2002)
2. Wang et al (2005)
3. Li et al (2009)
4. Zhou and Qiu (2010)
5. Guang et al (2011)
6. Zou et al (2013)
7. Wang et al (2014)
8. Li et al (2017)

References

1. BR (2000)
2. Zhou et al. (2006)

*contributed by Rhea Parekh and Shraddha Singh