Whitepaper and Roadmap:
Achieving Quantum Resistance in Mining Logic(PoW) Exposed as the Most Vulnerable by Quantum Computers

Abstract

This section provides a concise explanation of the necessity to address the threats posed by advancements in quantum computing to cryptocurrency mining systems. It emphasizes the importance of achieving quantum resistance and outlines the purpose and significance of this effort.

First, the mechanism of cryptocurrency mining is implemented in a uniquely specialized way from a mathematical perspective. This very uniqueness perfectly aligns with the computational operations of quantum computers. Starting around 2024, new quantum computations became feasible, exposing this perfect alignment as a vulnerability in mining.

Quantum computations are incredibly powerful and take a fundamentally different approach to calculations compared to classical computations. As a result, we identified a counterintuitive effect on mining difficulty. Specifically, mining, which is considered difficult with traditional classical methods (using GPUs or ASICs), becomes easier for quantum computers as the difficulty level increases. This is indeed an intriguing property of numerical behavior.

Furthermore, the computational capability of quantum computers for mining far surpasses that of all ASICs operating worldwide, even if they were combined into a single unit. A small quantum computer alone would render them completely outmatched and utterly unable to compete. Allowing quantum computers to perform mining would result in the monopoly of all mining by quantum computers. Due to the reversed difficulty property, this would break the blockchain entirely!

Therefore, the SORA project prioritizes the research, development, and implementation of quantum resistance for mining above other areas, such as ECDSA or signature schemes. This is because quantum gates required to perform quantum computations for mining are easier to construct and require fewer qubits compared to breaking cryptographic systems like ECDSA or RSA. Generally, breaking ECDSA, as used in Bitcoin and similar systems, is estimated to require tens of thousands of qubits. However, for mining, only a few thousand qubits are sufficient.

As of 2024, the latest quantum computers have already achieved approximately 1,000 qubits. Therefore, the threat of quantum computers being used for mining is surprisingly imminent and much closer than one might expect.

1. Introduction

Background:
An overview of quantum computing and its impact on cryptographic systems.
Problem Statement:
Specific vulnerabilities in Proof-of-Work (PoW) mining logic.
Objective:
The realization of quantum-resistant mining logic.

2. Quantum Computing and Mining Vulnerabilities

Quantum Advantage:
Explanation of quantum algorithms (e.g., Shor’s algorithm) and their potential to compromise existing cryptographic and mining systems.

Vulnerable Aspects:
Analysis of particularly susceptible areas within PoW mining logic to quantum attacks.
Quantum computers process information using quantum bits (qubits). This allows them to perform calculations much faster than classical computers. Shor's algorithm is one of the algorithms executed on quantum computers, efficiently factoring large numbers.

\[ | \psi \rangle = \cos\left(\frac{\theta}{2}\right) | 0 \rangle + e^{i\phi} \sin\left(\frac{\theta}{2}\right) | 1 \rangle \]

Modern public key cryptography(RSA) relies on the fact that factoring large numbers is not computationally feasible in a realistic timeframe. However, quantum computers can factor large numbers in a realistic timeframe using Shor's algorithm.

Originally, the phase information φ of a quantum bit in superposition cannot be directly observed. However, by utilizing quantum Fourier transform, this phase information can be converted into the frequency domain, making it possible to transform it into an observable probability amplitude of the quantum bit's particle nature.

\[ f(x) = a^x \mod N, \;\gcd(a, N) = 1 \] \[ |x\rangle = H\; \otimes \;H\; \otimes \cdots \otimes \;H \;|00 \cdots 0\rangle = \sum_k |k\rangle \rightarrow f(x) \rightarrow |x\rangle\; \otimes \;|f(x)\rangle \] \[ |f(x)\rangle = \sum_k |k\rangle \rightarrow \text{Collapse} \rightarrow |f(x)\rangle \rightarrow f(j),\ j: \text{the chosen solution} \] \[ \therefore\;|j\rangle = \frac{1}{\sqrt{n}} ( |m_1\rangle + |m_2\rangle + \ldots + |m_n\rangle ) \] \[ \text{n: the number of superposed states on the input register} \] \[ m_1 \cdots m_n: \text{the respective states} \]

\[ \therefore\;QFT\_{2^n}\;|j\rangle \rightarrow \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n - 1} e^{2\pi i j k / 2^n} |k\rangle \]

In quantum computations, by concentrating interference effects into the output register and maintaining a structure that entangles it with the input register, a solution with enhanced probability amplitude is obtained upon measurement through collapse in the output register. Furthermore, due to the entangled nature, the strong correlation ensures that the state vector associated with the output register is also acquired, forming the basis of this mechanism.

Now, let us first illustrate the mechanism for obtaining the desired solution by enhancing interference effects, specifically the method for computing a private key from a public key in ECDSA. ECDSA is based on the closed set of points derived from the finite field properties of elliptic curves. Due to the group property formed by scalar multiplication of these coordinates, which is bijective, the coordinates are repeatedly transformed into the same values across the group, with the point at infinity acting as the boundary.

By efficiently intertwining the periodicity of this group structure with the Quantum Fourier Transform (QFT), quantum computations exploit this relationship, overcoming the limitations of classical computers, which require exponential time to compute a private key from a public key.

1. Basics of Elliptic Curve Cryptography

The public and private keys in ECDSA are defined as follows:

\begin{equation} P = kG \quad \text{where } P \text{ is the public key, } G \text{ is the generator, and } k \text{ is the private key.} \end{equation}

Here, solving the Elliptic Curve Discrete Logarithm Problem (ECDLP) to find \( k \) from \( P \) and \( G \) is computationally infeasible with classical methods.

The modulus used in the \( kG \) calculation process of ECDSA, as adopted by Bitcoin and others, is \( 2^{256} - 2^{32} - 977 \).
This becomes the prime number of the finite field.

2. Steps for Quantum Computer Attack

The private key \( k \) can be derived using the following quantum algorithm steps:

Step 1: Initialization of Superposition

A quantum state representing all possible private keys is initialized as:

\begin{equation} |\psi_{\text{initial}}\rangle = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} |k\rangle |kG\rangle, \end{equation}

where \( N \) is the order of the elliptic curve.
In this case, the \( N \) used in ECDSA, as adopted by Bitcoin and others, is \( 115792089237316195423570985008687907852837564279074904382605163141518161494337 \).

To enhance the interference effect of QFT and amplify the probability amplitude, \( M \), which can represent a number significantly exceeding \( N \), is created in superposition by increasing the number of qubits, thereby forming a larger address space.

\begin{equation} |\psi_{\text{initial}}\rangle = \frac{1}{\sqrt{M}} \sum_{k=0}^{M-1} |k\rangle |kG\rangle, \end{equation}

Step 2: Phase Oracle Application

A phase \( \pi \) is applied to the state corresponding to the target public key \( P_{\text{target}} \):

\begin{equation} |\psi_{\phi}\rangle = \frac{1}{\sqrt{M}} \sum_{k=0}^{M-1} e^{i\phi_k} |k\rangle |kG\rangle, \quad \text{where } \phi_k = \begin{cases} \pi & \text{if } kG = P_{\text{target}}, \\ 0 & \text{otherwise}. \end{cases} \end{equation}

Step 3: Quantum Fourier Transform (QFT)

The Quantum Fourier Transform is applied to exploit the periodicity of the scalar multiplication: \begin{equation} |\psi_{\text{QFT}}\rangle = \text{QFT}\Bigg(\frac{1}{\sqrt{M}} \sum_{k=0}^{M-1} e^{i\phi_k} |k\rangle\Bigg) \end{equation}

This step amplifies the frequency components corresponding to the periodicity of \( k \).

Step 4: Measurement and Collapse

Measurement of the quantum state collapses it, yielding the phase information:

\begin{equation} \phi_k = \frac{2\pi k}{r}, \end{equation} where \( r \) is the periodicity of the scalar multiplication.
In this case, the \( r \) is \( N \), therefore, \( r = N \).

Step 5: Derivation of the Private Key \( k \)

Using the extracted phase information, the private key \( k \) can be computed as:

\begin{equation} k = \frac{\phi_k \cdot N}{2\pi}, \quad \text{mod } N \end{equation}

In this way, the interference effects guide us to the desired solution. However, even more flexibility in finding solutions is achieved through quantum computer-based mining. This very flexibility is closely tied to the mechanism of cryptocurrency mining.

As a result, quantum-resistant algorithms designed for public-key cryptographic systems like ECDSA become entirely ineffective. It becomes necessary to develop new quantum-resistant solutions specifically tailored for mining.

3. Proposed Quantum-Resistant Solutions

Conceptual Framework:
Strategies for achieving quantum resistance, including modifications to PoW mechanisms.

Techniques:
Hash-based cryptography.
Post-quantum secure protocols.
Alternative consensus mechanisms with quantum resistance.

Now, these technical methods are quantum-resistant approaches widely adopted against ECDSA, which is used in Bitcoin and similar systems. Then, could these methods also be applied to strategies for achieving quantum resistance, including modifications to the PoW mechanism?

Unfortunately, these technical methods are effective as quantum-resistant solutions only for public-key cryptographic systems like ECDSA. It has been confirmed that these existing methods do not provide quantum resistance for PoW mechanisms.

4. Implementation Roadmap

Development Phases:

Research and Development:
Exploring theoretical foundations.

Prototype Testing:
Evaluating resistance through test networks.

Deployment and Scaling:
Integrating quantum-resistant mining logic into existing systems.

Technical Challenges:
Identification of obstacles and potential solutions.

5. Comparative Analysis

Performance Impact:
Evaluation of computational overhead introduced by quantum-resistant techniques.

Security Enhancements:
Comparison of vulnerability reduction pre- and post-implementation.

6. Future Directions

Scalability:
Ensuring widespread adoption across networks.

Adaptability:
Building systems capable of evolving with advancements in quantum computing.

7. Conclusion

Summary:
Recap of the importance of quantum-resistant mining logic.

Call to Action:
Encouragement for industry collaboration and continued research.

8, Appendices

Glossary:
Definitions of key terms related to quantum computing and cryptography.

Supplementary Data:
Diagrams, tables, and additional technical details.