# Fast Quantum Byzantine Agreement

This example protocol is an efficient solution to the classical task of Byzantine Agreement. It allows multiple players in a network to reach an agreement in the presence of some faulty players. The protocol solves the task in the strongest possible failure model (called Byzantine failures). The quantum protocol is provably faster than any classical protocol.

Tags: Quantum Enhanced Classical Functionality, Multi Party Protocols, Specific Task, consensus task, failure-resilient distributed computing

## Assumptions

• Network: The network consists of ${\displaystyle n}$ players that are fully identified and completely connected with pairwise authenticated classical and quantum channels.
• Timing: The synchronous and asynchronous timing models are both considered.
• Message size: The size of messages (quantum and classical) are unbounded.
• Shared resources: The nodes do not share any prior entanglement or classical correlations.
• Failure: At most ${\displaystyle t (synchronous) or ${\displaystyle t (asynchronous) players show Byzantine failures. The Byzantine failed players are allowed to behave arbitrarily and collude to try and prevent the honest players from reaching agreement. The most severe model is used: Byzantine failures are adaptive, computationally unbounded and have full-information (full information of quantum states is modelled by giving a classical description of the state to the adversaries). Failures on the communication channels are not considered.

## Outline

Here we will sketch the outline of the Fast Quantum Byzantine Agreement protocol by Ben-Or (3) that solves Byzantine Agreement using quantum resources. A very nice summary of this protocol is also presented in (1). The main idea of this protocol is for each player to classically send its proposed input bit ${\displaystyle b_{i}}$ to every other player in the network and then collaborate to determine what bit is proposed by a majority of honest players. In the case where failed players make this difficult, a 'good-enough' random coin is globally flipped (using quantum resources, explained below), which is then classically post-processed to reach agreement among the honest parties. Let us make this more precise.

The protocol consists of consecutive rounds. Initially, each player sets a decision bit to its input bit. Then in each round, the players take the following steps:

• Each player transmits its current decision bit to every other player. If a player receives the same bit value from more than 2/3 of the players (including his own), then it sets his decision bit to this majority bit value. Otherwise, that player initiates a Quantum Oblivious Common Coin subroutine with all other players and sets his decision bit to the outcome of this subroutine.
• Then each player sequentially executes two classical subroutines to bias the decision value towards ${\displaystyle 0}$ or ${\displaystyle 1}$ respectively. These subroutines guarantee that if the non-faulty players are in agreement, then they will terminate and successfully output the correct agreement value.

Quantum Oblivious Common Coin subroutine: The heart of this protocol comes from the quantum enhanced Oblivious Common Coin. At the end of this subroutine, each player ${\displaystyle i}$ outputs a random bit ${\displaystyle v_{i}}$, such that with at least probability ${\displaystyle p}$ (called the fairness) ${\displaystyle v_{i}=x}$ for all players ${\displaystyle i}$ and all ${\displaystyle x\in \{0,1\}}$. Intuitively, this subroutines tosses a common coin, where all players get either ${\displaystyle 0}$ or ${\displaystyle 1}$ with probability at least ${\displaystyle p}$ each, but there may be executions (which occur with at most probability ${\displaystyle 1-2p}$) where all players do not get the same output and no common coin is actually tossed. Since the players do not know whether the outcomes are all equal or not, this type of coin tossing is referred to as oblivious common coin tossing. In particular, using quantum resources, this task can be achieved in constant rounds (in the defined model). The implementation of this subroutine makes use of a weakened version of Verifiable Quantum Secret Sharing (VQSS).

## Notation

• ${\displaystyle n:}$ number of nodes
• ${\displaystyle t:}$ number of failures
• ${\displaystyle p:}$ fairness of the Oblivious Common Coin
• ${\displaystyle k:}$ security parameter of the VQSS scheme used to implement the Oblivious Common Coin
• ${\displaystyle b_{i}:}$ input bit of player ${\displaystyle i}$
• ${\displaystyle v_{i}:}$ random bit output by player ${\displaystyle i}$ in the Oblivious Common Coin subroutine
• ${\displaystyle d:}$ the agreement value at the end of the protocol

## Requirements

• Network stage: (Fault-tolerant) Quantum computing network stage
• Relevant network stage parameters: Required number of qubits ${\displaystyle q}$.
• Benchmark values: The number of qubits ${\displaystyle q}$ required is precisely known for a finite instance of the protocol. This is calculated in (1) for ${\displaystyle n=5}$. They pick the smallest possible security parameter ${\displaystyle k=2}$ (of the VQSS scheme) and start calculating the required resources. Summarising they find that each node requires ${\displaystyle \sim 200}$ operational qubits, on which quantum circuits of depth ${\displaystyle \sim 2000}$ must be run. The consumed number of Bell pairs is 648 and the total classical communication cost is 21240 bits. It is not entirely clear if these are the expected costs or the cost per round.
• In a more asymptotic sense, it is known that the required number of qubits per node grows rapidly with the number of nodes ${\displaystyle n}$, making it, therefore, demanding on qubit requirements.

## Properties

The protocol

• solves the problem in ${\displaystyle O(1)}$ expected number of rounds, in particular independent of ${\displaystyle n}$ and ${\displaystyle t}$, whereas classically a lower bound of ${\displaystyle \Omega \left({\sqrt {n/\log(n)}}\right)}$ is known (3), (6);
• tolerates ${\displaystyle t\leq n/3}$ (synchronous) or ${\displaystyle t\leq n/4}$ (asynchronous) Byzantine failures;
• reaches agreement (each player outputs the same bit) under the validity condition (the agreement value was proposed by at least one player) and is guaranteed to terminate eventually (infinite executions occur almost never - i.e. have probability measure zero).

The Quantum Oblivious Common Coin subroutine has a single parameter ${\displaystyle k}$ (used in the Verifiable Quantum Secret Sharing scheme), but it is unclear from the works (1), (3) how the parameter ${\displaystyle k}$ influences the guarantees of the protocol.

Also note that the fairness ${\displaystyle p}$ of the Quantum Oblivious Common Coin is not a parameter, but rather a result of the specific implementation of the protocol. The global Byzantine Agreement protocol can then tolerate up to ${\displaystyle t<\left\lfloor {pn}\right\rfloor }$ failures. The Quantum Oblivious Common Coin subroutine proposed by (3) has ${\displaystyle p>1/3}$ (synchronous case, ${\displaystyle p>1/4}$ asynchronous case).

## Protocol Description

This pseudocode is based on the reference (1).

#### Fast Byzantine Agreement

Input: Each player starts with an input bit ${\displaystyle b_{i}}$ and the number of players ${\displaystyle n}$ and a security parameter ${\displaystyle k}$.
Output: Each player outputs a bit ${\displaystyle d_{i}}$. With high probability, ${\displaystyle d_{i}=d}$ for all players ${\displaystyle i}$ (agreement) and some ${\displaystyle d\in \{b_{i}\}_{i}}$ (validity).

Protocol for each player ${\displaystyle i}$:

Repeat forever (until something is returned):

1. Subroutine ${\displaystyle P_{r}(b_{i})}$: (this flips the oblivious common coin if no 2/3 majority is reached)
1. Send ${\displaystyle b_{i}}$ to all other players ${\displaystyle j\neq i}$. Receive a bit ${\displaystyle b_{j}}$ from all other players;
2. Let ${\displaystyle x=\sum _{j}b_{j}}$. If ${\displaystyle x, then set ${\displaystyle b_{i}=0}$; elseif ${\displaystyle x>2n/3}$, then set ${\displaystyle b_{i}=1}$; else set ${\displaystyle b_{i}=QOCC(n,k)}$ end;
2. Subroutine ${\displaystyle P_{0}(b_{i})}$: (classical routine - biases towards 0 and finishes if 0 is the agreement value)
1. Send ${\displaystyle b_{i}}$ to all other players ${\displaystyle j\neq i}$. Receive a bit ${\displaystyle b_{j}}$ from all other players;
2. Let ${\displaystyle x=\sum _{j}b_{j}}$. If ${\displaystyle x, then return ${\displaystyle 0}$; elseif ${\displaystyle x>2n/3}$, then set ${\displaystyle b_{i}=1}$; else set ${\displaystyle b_{i}=0}$ end;
3. Subroutine ${\displaystyle P_{1}(b_{i})}$: (classical routine - biases towards 1 and finishes if 1 is the agreement value)
1. Send ${\displaystyle b_{i}}$ to all other players ${\displaystyle j\neq i}$. Receive a bit ${\displaystyle b_{j}}$ from all other players;
2. Let ${\displaystyle x=\sum _{j}b_{j}}$. If ${\displaystyle x, then set ${\displaystyle b_{i}=0}$; elseif ${\displaystyle x>2n/3}$, then return ${\displaystyle 1}$; else set ${\displaystyle b_{i}=1}$ end;

#### Quantum Oblivious Common Coin (QOCC)

Input: Each player starts with the number of players ${\displaystyle n}$ and a security parameter ${\displaystyle k}$.
Output: Each player outputs a random bit ${\displaystyle v_{i}}$. With at least probability ${\displaystyle p}$, ${\displaystyle v_{i}=x}$ for all ${\displaystyle i}$ and all ${\displaystyle x\in \{0,1\}.}$

Protocol for each player ${\displaystyle i}$:

1. Prepare state ${\displaystyle |{\psi }\rangle =\left(\sum _{i=1}^{n}|{i}\rangle \right)^{\otimes n}}$
2. Share and verify ${\displaystyle |{\psi }\rangle }$ with a VQSS scheme (with security parameter ${\displaystyle k}$). During the verification phase, use a (classical) gradecast scheme instead of a broadcast scheme (this change is named GradedQSV in (1)). Let ${\displaystyle FP}$ denote the set of players that were caught cheating as a dealer.
3. Measure each share of player ${\displaystyle j}$ to obtain a random integer ${\displaystyle s_{i,j}}$.
4. Use gradecast to share the numbers ${\displaystyle s_{i,j},j=1,...,n}$. Add the dishonest players in the gradecast scheme to ${\displaystyle FP}$. Receive ${\displaystyle s_{l,j}}$, from each player ${\displaystyle l=1,...,n,l\neq i}$.
5. if ${\displaystyle j\in FP}$ then set ${\displaystyle S_{j}=\perp }$ else set ${\displaystyle S_{j}=\sum _{l\notin FP}s_{lj}\mod n}$
6. if ${\displaystyle S_{j}=0}$ for some ${\displaystyle j}$, then return 0; else return 1.

## Further Information

• The protocol (3) is based on the classical protocol of (7), where the classical Oblivious Common Coin is replaced by a quantum version. This Quantum Oblivious Common Coin is based on the Verifiable Quantum Secret Sharing Scheme presented in (4).
• The classical protocol of (7) also runs in constant expected time, but can only deal with limited-information adversaries. This means that the adversaries can not read communication between honest parties and read their internal state.
• The classical lower bound in the full-information Byzantine failure model of ${\displaystyle \Omega \left({\sqrt {n/\log(n)}}\right)}$ is proven in (6).
• The work (3) also provides a protocol in a weaker failure model known as fail-stop failures. Here the nodes will crash and stop working indefinitely (stop responding). Another protocol in the same model is presented in (2).
• Another weakened version of the problem, known as detectable byzantine agreement, is solved with quantum resources in (5) (and following works). In detectable byzantine agreement, the protocol is also allowed to abort (upon detecting failures) instead of reaching agreement.

## References

*contributed by Bas Dirke