A Unified Quantum Attack Model for Cryptographic Systems Based on the Hybridization of Shor's and Grover's Algorithms

 Title: A Unified Quantum Attack Model for Cryptographic Systems Based on the Hybridization of Shor's and Grover's Algorithms

Abstract: This paper introduces a formalized mathematical model that unifies the quantum computational approaches of Shor's and Grover's algorithms to construct a hybrid cryptographic attack strategy. This model aims to evaluate and predict the effective quantum execution time required to break cryptographic systems such as ECDSA and SHA-256, foundational to Bitcoin's security architecture. By decomposing the cryptanalytic tasks into modular components solvable by either algorithm and optimizing the transition between them, the paper proposes a total quantum attack time function. This contributes a theoretical framework for post-quantum security analysis.

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1. Introduction

Modern cryptographic systems rely heavily on computational assumptions about problems such as the Elliptic Curve Discrete Logarithm Problem (ECDLP) and the one-wayness of cryptographic hash functions. Shor's algorithm, capable of solving integer factorization and discrete logarithms in polynomial time, directly threatens asymmetric cryptosystems like ECDSA. Meanwhile, Grover's algorithm provides a quadratic speedup for brute-force search, relevant to hash-based systems such as SHA-256. This paper proposes a composite quantum attack model combining both algorithms for multi-layer cryptanalysis.

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2. Preliminaries

Let us define:

• : Bit-length of the key (typically 256 bits for ECDSA and SHA-256).

• : Total key or hash space.

• : Single quantum gate operation time (e.g., seconds).

• : Circuit depth multiplier for Shor's algorithm.

• : Iteration multiplier for Grover's algorithm.

• : Execution time of Shor's algorithm.

• : Execution time of Grover's algorithm.

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3. Shor's Algorithm Model for ECDSA

Shor's algorithm resolves ECDLP in polynomial time:

Assuming and :

This represents the expected execution time on an ideal fault-tolerant quantum computer.

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4. Grover's Algorithm Model for SHA-256

Grover's algorithm yields quadratic speedup for pre-image search:

Assuming , , and :

This demonstrates the infeasibility of directly breaking SHA-256 but suggests potential applications in hash collision or address manipulation.

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5. Composite Quantum Attack Function

The total quantum attack time, assuming sequential execution of both sub-tasks, is:

This model enables comparative analysis under various quantum hardware parameters. In practical scenarios, partial Grover optimizations (e.g., restricted search spaces or structured hashes) could dramatically reduce .

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6. Implications and Discussion

This hybrid model reflects a realistic pathway to undermining multi-tier cryptographic systems like Bitcoin. The rapid collapse of ECDSA under Shor's algorithm represents the primary threat, while Grover-based reductions enhance secondary vulnerabilities in hash chains, address derivations, or timestamp obfuscations. The attack time formula thus provides a valuable metric for designing quantum-resilient cryptographic protocols.

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7. Conclusion

By unifying Shor’s and Grover’s algorithms into a composite cryptanalytic framework, this paper contributes a formal attack-time equation applicable to post-quantum cryptographic analysis. The findings underscore the urgent need to adopt quantum-resistant cryptographic standards, particularly in decentralized systems like blockchain.

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References:

1. Shor, P. W. (1997). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer.

2. Grover, L. K. (1996). A Fast Quantum Mechanical Algorithm for Database Search.

3. Chen, L. et al. (2016). Report on Post-Quantum Cryptography. NIST.

4. Aggarwal, D. et al. (2017). Quantum attacks on Bitcoin and how to protect against them.

5. Gidney, C., & Ekerå, M. (2019). How to factor 2048-bit RSA integers in 8 hours using 20 million noisy qubits.