# Lottery and Auction on Quantum Blockchain

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## Abstract

**:**

## 1. Introduction

- Randomness. All tickets are equally likely to win.
- Unpredictability. No player can predict the winning ticket.
- Unforgeability. Tickets cannot be forged. Especially, it is impossible to create a winning ticket after the outcome of the random process is known.
- Verifiablity. The number and the revenue of winning tickets are publicly verifiable.
- Decentralization. The random process does not rely on a single authority.

- 6.
- Unconditional security. Even an adversary with an unlimited power of computation cannot rig the lottery.

- Bid privacy. The submitted bids are not visible to other buyers during the bidding phase.
- Posterior privacy. The losing bids are not revealed to the public. In other words, only the seller knows all losing bids and their corresponding buyers.
- Bids’ binding. Buyers cannot deny or change their bids once they are committed.

- 4.
- Decentralization. The process of the auction does not rely on a single trusted third party.
- 5.
- Unconditional security. Even an adversary with an unlimited power of computation cannot manipulate the process of auction.

## 2. Background

#### 2.1. Quantum Blockchain

- Every node is a (small scale) quantum computer which can run some quantum computation on a small number of qubits. More specifically, nodes are capable of performing the quantum computation involved in at least one quantum bit commitment protocol.
- The communication between different nodes is unconditionally secure.
- There is a consensus algorithm which can be used by all miners to achieve consensus. The consensus mechanism is immune to attacks. A general definition of the consensus algorithm is given as the following.

**Definition**

**1**(consensus algorithm)

**.**

#### 2.2. Quantum Bit Commitment

**Definition**

**2**(quantum bit commitment)

**.**

- (1)
- Two finite-dimensional Hilbert spaces A and B.
- (2)
- A function $commit:\{0,1\}\mapsto A\otimes B$.
- (3)
- Two pure states $|{c}_{0}\rangle ,|{c}_{1}\rangle \in A\otimes B$, in which $|{c}_{i}\rangle =commit\left(i\right)$ is the commitment of i.
- (4)
- A quantum operation (i.e., completely positive, trace-preserving super operator) $Open$ on $A\otimes B$ such that $Open\left(\right|{c}_{0}\rangle \langle {c}_{0}\left|\right)\ne Open\left(\right|{c}_{1}\rangle \langle {c}_{1}\left|\right)$.

## 3. Lottery on Quantum Blockchain

- Ticket purchasing:
- (a)
- For every player ${p}_{i}\in \{{p}_{1},\dots ,{p}_{n}\}$, to purchase a ticket ${T}_{i}$, ${p}_{i}$ uses QBC to commit ${T}_{i}$ to all miners. At the end of this phase, every miner possesses a list of commitments $(commit\left({T}_{1}\right),\dots ,commit\left({T}_{n}\right))$.

- Ticket agreement:
- (a)
- Every player opens his commitment to every miner, so that the commitments in every miner’s possession change to $(Open\left(commit\left({T}_{1}\right)\right),\dots ,$$Open\left(commit\left({T}_{n}\right)\right))$, which essentially equals to $({T}_{1},\dots ,{T}_{n})$.
- (b)
- All the miners run a consensus algorithm to achieve a consensus on the tickets $({T}_{1},\dots ,{T}_{n})$ purchased by players. Every miner adds $({T}_{1},\dots ,{T}_{n})$ to his local copy of the blockchain.

- Winner determination:
- (a)
- The winning ticket is calculated by bit-wise XOR: $T={T}_{1}\oplus \dots \oplus {T}_{n}$.
- (b)
- A player’s revenue is determined by the Hamming distance between his ticket and the winning ticket T. The closer his ticket is to the winning ticket, the higher is his revenue (a specific rule of revenue which satisfies this principle is beyond the scope of this paper and is left for future work).

#### Analysis

**Randomness**.The winning ticket is calculated by bit-wise XOR. For every index $j\in \{1,\dots ,m\}$ in the winning ticket, $T\left[j\right]=1$ iff ${T}_{1}\left[j\right]\oplus \dots \oplus {T}_{n}\left[j\right]=1$. Therefore, the probability of $T\left[j\right]=1$ is the same as $T\left[j\right]=0$.**Unpredictability**.To predict the winning ticket a player has to know all tickets before they are opened. The concealing property of QBC ensures that even miners cannot know the players’ tickets before they are opened. Since tickets are only sent to the miners by QBC, the probability that a player knows all tickets is even lower than the probability that a miner knows them.**Unforgeability**.The binding property of QBC ensures that it is impossible to change a ticket after the ticket purchasing phase.**Verifiablity**.This is because the quantum blockchain is a transparent database. After the ticket agreement phase, the list $({T}_{1},\dots {T}_{n})$ is added to the blockchain. Every player can read all the other players’ tickets and calculate the winning ticket by himself.**Decentralization**.The random process does not rely on a single authority. Every player’s ticket essentially affects the calculation of the winning ticket.Moreover, the calculation of the winning ticket does not rely on a single miner, but on all miners.**Unconditional security**.Even an adversary with an unlimited power of computation cannot manipulate the lottery protocol. The concealing and binding property of QBC does not rely on any computational assumption. Nor does the security of the consensus algorithm. The unconditional security of the ledger is further guaranteed by the unconditional security of the digital signature schemes adopted by quantum Blockchain.

## 4. Auction on Quantum Blockchain

- The bidding phase: Every buyer ${B}_{i}$ commits his bid ${b}_{i}$ to the seller and to all miners ${M}_{j}$, where ${b}_{i}$ is a positive integer.
- The opening phase: Every buyer opens his bid to the seller.
- Decision phase: The seller calculates the winning bid, which is the highest bid (if there is a tie, then one of the maximal bids is chosen randomly), and the winning buyer, who has offered the winning bid.
- Verification phase: In this phase, the seller S and every miner ${M}_{j}$ ($1\le j\le n$) run the following procedure to convince ${M}_{j}$ that S has chosen the valid winner:
- (a)
- S sends the information about the winning buyer ${B}_{w}$ and his bid ${b}_{w}$ to the miner ${M}_{j}$.
- (b)
- S permutes losing bids to obtain a new list of $m-1$ bids $({b}_{1}^{\prime},\dots ,{b}_{m-1}^{\prime})$.
- (c)
- S sends ${b}_{1}^{\prime},\dots ,{b}_{m-1}^{\prime}$ to ${M}_{j}$.
- (d)
- ${M}_{j}$ first checks if ${b}_{w}\ge {b}_{k}^{\prime}$ for all $k\in \{1,\dots ,m-1\}$. If yes, then ${M}_{j}$ sends $({b}_{w},{b}_{1}^{\prime},\dots ,{b}_{m-1}^{\prime})$ to all buyers. Otherwise, ${M}_{j}$ sets S as a cheater by setting output to ⊥.
- (e)
- After receiving $({b}_{w},{b}_{1}^{\prime},\dots ,{b}_{m-1}^{\prime})$, every buyer ${B}_{i}$ checks if his bid is in the list, i.e., there is some ${b}_{k}^{\prime}={b}_{i}$. If yes, then ${B}_{i}$ sends the message “valid” to ${M}_{j}$. Otherwise ${B}_{i}$ opens ${b}_{i}$ to ${M}_{j}$. ${M}_{j}$ then sets S as a cheater by setting output to ⊥.
- (f)
- If ${M}_{j}$ does not output ⊥, then the seller passes the verification phase. The output of ${M}_{j}$ is now $({b}_{w},{b}_{1}^{\prime},\dots ,{b}_{m-1}^{\prime},{B}_{w})$

- Publication phase: All miners run the consensus algorithm to achieve consensus on the output of the verification phase. The consensus is then added to the blockchain.

#### Analysis

- 1.
**Bid privacy**.Every buyer only commits and opens his bids to the seller. Therefore, no buyer knows other buyers’ bids.- 2.
**Posterior privacy**.What is added to the blockchain is the winning buyer and his bid, as well as a permuted list of losing bids. Therefore, no losing buyer’s bid is revealed.- 3.
**Bids’ binding**.The binding property of quantum bit commitment ensures that buyers cannot deny or change their bids once they are committed.- 4.
**Decentralization**.There are in total n miners. The process of the auction does not rely on a single miner.- 5.
**Unconditional security**.As in the case of our lottery protocol, even an adversary with an unlimited power of computation cannot manipulate the auction protocol because the security of the quantum bit commitment and consensus algorithm does not depend on computational complexity. The unconditional security of the ledger relies on quantum Blockchain properties.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**A network of players and miners: players commit their tickets to miners. Miners use a consensus algorithm (CA) to achieve consensus about the players’ tickets.

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**MDPI and ACS Style**

Sun, X.; Kulicki, P.; Sopek, M. Lottery and Auction on Quantum Blockchain. *Entropy* **2020**, *22*, 1377.
https://doi.org/10.3390/e22121377

**AMA Style**

Sun X, Kulicki P, Sopek M. Lottery and Auction on Quantum Blockchain. *Entropy*. 2020; 22(12):1377.
https://doi.org/10.3390/e22121377

**Chicago/Turabian Style**

Sun, Xin, Piotr Kulicki, and Mirek Sopek. 2020. "Lottery and Auction on Quantum Blockchain" *Entropy* 22, no. 12: 1377.
https://doi.org/10.3390/e22121377