# Solving Confirmation Time in Sharded Blockchain with PFQN

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## Abstract

**:**

## 1. Introduction

#### 1.1. Research Background

#### 1.2. Related Works

#### 1.3. Motivation and Challenge

- We decouple the input of the sharded blockchain through the product-form queue network (PFQN) and solve the transactions at different stages to obtain the average expected value of the transaction confirmation time applicable to the sharded blockchain;
- We additionally consider the transaction confirmation process on the main chain, and add a new confirmation queue F after applying the PFQN model, making the model more in line with the actual transaction confirmation situation in the blockchain;
- We utilize the PFQN model to assess the impact of quantum-resistant technologies on sharded blockchain transaction times, enhancing security against quantum threats.

## 2. Materials and Methods

#### 2.1. Why PFQN?

#### 2.2. Blockchain Setting

#### 2.3. Model Assumption

#### 2.4. Model and Derivation

#### 2.4.1. PFQN Model

_{Nc}. N distributes c to the nodes in the shard at a service rate μ

_{Nc}. Miners who have received c will add c to their transaction memory pool, representing c entering the shard’s consensus queue P. The service rate μ

_{Nc}represents the service rate of the transaction in the network. Since μ

_{Nc}is large in reality, the service time can be negligible. Therefore, we simply see c entering queue P at rate λ

_{Pc}.

_{d}is

_{d}.

#### 2.4.2. Derivation of Transaction Confirmation Delay

## 3. Results

_{all}, the expected service time E[S] is less than the arrival rate λ. Figure 6 illustrates the transaction latency in a simulated sharded blockchain environment under real transaction conditions, with 100,000 Ethereum transactions injected at a constant rate. The simulations were performed with different numbers of shards, specifically 2, 4, 50, and 100, while maintaining the number of nodes within each shard at four. The figure compares the transaction delays within the sharded blockchain with the expected delays across different numbers of shards.

## 4. Discussion

#### 4.1. PFQN and Sharded Blockchain Simulation

#### 4.2. Security Analysis

#### 4.2.1. Prior Research on Quantum-Safe Blockchain

#### 4.2.2. Attack Models and Assumptions

^{12}true random numbers per second using Grover and Shor algorithms, surpassing the current classical methods.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Parameter | Description |

M | Set of queues |

λ | Customer input rate per shard |

d | Total number of destination fields in a TX |

D[d] | The probability distribution for ‘d’ |

b | Maximum number of TXs allowed in a block |

c | Regular customer |

${\mathrm{c}}_{\mathrm{k}}^{+}$ | K stages cross-shard signal |

s | Block components |

${\mathrm{\alpha}}_{\mathrm{J}\mathrm{e}}$ | The arrival rate of customer type e to queue J |

${\mathrm{\alpha}}_{\mathrm{J}{\mathrm{s}}_{\mathrm{i}}}^{+}$ | The arrival rate of positive signal ${\mathrm{s}}_{\mathrm{i}}^{+}$ to queue J |

${\mathrm{\alpha}}_{\mathrm{J}{\mathrm{c}}_{\mathrm{i}}}^{+}$ | The arrival rate of cross-shard positive signal ${\mathrm{c}}_{\mathrm{i}}^{+}$ to queue J |

U | Maximum stage achievable for signal ${\mathrm{c}}_{\mathrm{i}}^{+}$ |

${\mathrm{R}}_{{\mathrm{J}}^{\mathrm{\u2019}}\mathrm{J}}^{\mathrm{k}}$ | The service completion rate for a receipt in network queue J′ leading to a stage k signal ${\mathrm{c}}_{\mathrm{k}}^{+}$ for network queue J |

${\mathrm{\mu}}_{\mathrm{J}\mathrm{e}}$ | Service rate for customer type e in a standard queue J |

${\mathrm{\rho}}_{\mathrm{J}\mathrm{e}}$ | The utilization factor incurred by customer type e on a typical queue J |

## Appendix A

**Definition**

**A1**

**(A Secure Sharding Blockchain).**

**Definition**

**A2**

**(Consistency).**

**Definition**

**A3**

**(Liveness).**

**Definition**

**A4**

**(Persistence).**

**Assumption**

**A1.**

**Assumption**

**A2.**

**Lemma**

**A1.**

**Proof**

**of**

**Lemma**

**A1.**

**Theorem**

**A1.**

**Proof**

**of**

**Theorem**

**A1.**

**Theorem**

**A2.**

**Proof**

**of**

**Theorem**

**A2.**

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**Figure 3.**(

**a**) Impact of ρp on arrival rate λ; (

**b**) impact of transaction degree dmax on arrival rate λ.

**Figure 4.**(

**a**) impact of number of shards M on arrival rate λ; (

**b**) impact of block size b on arrival rate λ.

Reference | Methodology | Focus Area | Key Findings | Contributions to the Field |
---|---|---|---|---|

[8] | GI/M/1 queue with batch-service | Transaction confirmation in Single-chain systems | Developed a queueing theory model for blockchain systems, identifying average transaction numbers and confirmation times | Introduced an analytical approach for blockchain queueing systems |

[9] | M/G/1 for delay characterization, machine learning for transaction classification | Transaction confirmation in Single-chain systems | Proposed a machine learning framework for transaction classification and queueing theory for delays | Enhanced understanding of blockchain delays and transaction confirmation dynamics |

[10] | PFQN | Sharded blockchain efficiency | Established a model for sharded blockchain and derived maximum throughput | Introduced a new model for analyzing sharded blockchain performance |

[11,12] | M/G/1 queue with batch service | Transaction confirmation time in Bitcoin | Analyzed transaction confirmation time in Bitcoin using queue theory | Applied queue theory to understand Bitcoin’s transaction dynamics |

Attack Type | Affected Blockchain Component | Attack Purpose | Means of Attack |
---|---|---|---|

Block Replacement Attack | Blockchain Historical Records | To replace the existing blockchain rewrite historical records. | Using Grover’s algorithm to calculate nonces |

Signature Forgery Attack | Transaction and Message Signatures | To tamper with or forge transactions | Using Shor’s algorithm to break public key encryption systems |

NIST Level | Encryption Standard |
---|---|

1 | AES 128 |

2 | SHA3-256 |

3 | AES192 |

4 | SHA3-384 |

5 | AES256 |

Algorithm Category | Cryptographic Algorithm | Private Key Length (bytes) | Public Key Length (bytes) | NIST Level | Approximate Probability of Compromise |
---|---|---|---|---|---|

Post-quantum encryption algorithm | CYSTAL-Dilithium3 | 1952 | 4000 | 3 | ${2}^{-192}$ |

FALCON | 1793 | 2305 | 5 | ${2}^{-256}$ | |

Classic | RSA | 3072 | 3072 | 1 | ${2}^{-128}$ |

ECDSA | 256 | 512 | 1 | ${2}^{-128}$ |

Signatures/s | Verifications/s | Max Safe Transactions | Expected Encryption Time per Transaction | |
---|---|---|---|---|

CYSTAL-Dilithium3 | 6506.33 | 17,561.33 | $\approx \frac{{2}^{21}}{\mathrm{U}}$ | 0.000154 |

FALCON | 1446.52 | 9782.67 | $\approx \frac{{2}^{85}}{\mathrm{U}}$ | 0.000691 |

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**MDPI and ACS Style**

Wu, J.; Du, H.; Chen, J.; Ren, W.
Solving Confirmation Time in Sharded Blockchain with PFQN. *Electronics* **2024**, *13*, 1220.
https://doi.org/10.3390/electronics13071220

**AMA Style**

Wu J, Du H, Chen J, Ren W.
Solving Confirmation Time in Sharded Blockchain with PFQN. *Electronics*. 2024; 13(7):1220.
https://doi.org/10.3390/electronics13071220

**Chicago/Turabian Style**

Wu, Junting, Haotian Du, Jin Chen, and Wei Ren.
2024. "Solving Confirmation Time in Sharded Blockchain with PFQN" *Electronics* 13, no. 7: 1220.
https://doi.org/10.3390/electronics13071220