Quantum Computational Complexity of SHA-256 Hash Function: Optimized Attack Time via Grover's Algorithm and Quantum Parallelism

4월 13, 2025

Title: Quantum Computational Complexity of SHA-256 Hash Function: Optimized Attack Time via Grover's Algorithm and Quantum Parallelism

Abstract:

This paper presents a theoretical framework for solving the SHA-256 hash function in significantly reduced time using quantum computing. By leveraging Grover's algorithm for efficient searching and quantum parallelism for concurrent computation, we derive an optimized mathematical model that predicts a dramatic reduction in the time complexity required to break SHA-256 from thousands of years to mere seconds. We discuss the impact of quantum hardware on computational efficiency and propose a formal expression for the attack time based on quantum computational parameters.

1. Introduction

The SHA-256 (Secure Hash Algorithm 256-bit) is one of the most widely used cryptographic hash functions. It is primarily used in blockchain technologies, including Bitcoin, to guarantee data integrity and security. The computational difficulty of reversing SHA-256 relies on the sheer number of possible hash values, $2^{256}$ , making traditional brute-force attacks infeasible within a human timescale.

However, with the advent of quantum computing, classical computational limits may be significantly surpassed. Quantum algorithms, specifically Grover's algorithm, offer quadratic speedup for unstructured search problems, such as reversing hash functions. This paper investigates the potential for solving SHA-256 using quantum computing, including an in-depth exploration of Grover’s algorithm and quantum parallelism, and derives a formal mathematical expression for the time required to break the hash function.

2. SHA-256 Hash Function and Classical Attack Complexity

SHA-256 operates by generating a 256-bit hash value from an arbitrary-length input. Given the bit-length of the hash, the total number of possible outputs is $2^{256}$ . The task of reversing a hash, i.e., finding the input that produces a specific hash, is a brute-force search problem, where all $2^{256}$ possible inputs must be checked.

In classical computing, this requires $O(2^{256})$ time, which is computationally infeasible. However, quantum computing promises significant speedups, especially through algorithms like Grover’s algorithm, which solves unstructured search problems in $O(2^{n/2})$ time. For a 256-bit hash function, Grover’s algorithm would reduce the search space from $2^{256}$ to $2^{128}$ .

3. Grover’s Algorithm and Quantum Parallelism

Grover’s algorithm provides a quadratic speedup over classical brute-force search methods. Specifically, for a function $f(x)$ , where $f(x) = 0$ for a desired input $x_0$ , Grover’s algorithm requires approximately $O(2^{n/2})$ steps to find the correct solution. In the case of SHA-256, the attack time using Grover’s algorithm can be expressed as:

T_{\text{grover}} = k_g \cdot 2^{n/2} \cdot t_q

Where:

$n = 256$ (the bit-length of the SHA-256 hash),
$k_g$ is a constant representing the efficiency of the algorithm (e.g., gate complexity),
$t_q = 10^{-9}$ seconds is the time for one quantum gate operation.

Substituting $n = 256$ into the equation:

T_{\text{grover}} = k_g \cdot 2^{128} \cdot 10^{-9}

Assuming $k_g = 1000$ as a conservative estimate for the complexity of the algorithm:

T_{\text{grover}} = 1000 \cdot 2^{128} \cdot 10^{-9}

The result is:

T_{\text{grover}} \approx 3.4 \times 10^{29} \, \text{nanoseconds} \approx 10^{12} \, \text{years}

Thus, the time required to break SHA-256 using Grover's algorithm in isolation is still impractically long—on the order of a trillion years.

4. Quantum Parallelism for Attack Optimization

The core advantage of quantum computing lies in quantum parallelism, the ability to simultaneously evaluate multiple candidate inputs. A quantum computer can process an exponentially large number of inputs in parallel using a superposition of states. This reduces the effective computational time further, allowing multiple calculations to be performed simultaneously.

Let $N$ represent the number of parallel quantum operations available. With this parallelism, the search time complexity of finding a solution can be expressed as:

T_{\text{quantum-parallel}} = \frac{O(2^{n/2})}{N} \cdot t_q

Where:

$N$ is the number of parallel operations that can be performed concurrently,
$O(2^{n/2})$ is the search complexity from Grover’s algorithm.

If a quantum computer with $N = 10^6$ parallel operations were available, the attack time would reduce to:

T_{\text{quantum-parallel}} = \frac{1000 \cdot 2^{128} \cdot 10^{-9}}{10^6} = 3.4 \times 10^{23} \, \text{nanoseconds} \approx 0.34 \, \text{seconds}

This calculation shows that by utilizing quantum parallelism, the attack time on SHA-256 can be reduced to less than a second, assuming sufficient quantum resources.

5. Optimized Attack Model: Reducing Time from Millennia to Seconds

The optimized attack time for SHA-256, leveraging Grover’s algorithm and quantum parallelism, can be expressed as:

T_{\text{optimized}} = \frac{k_{\text{opt}} \cdot 2^{128} \cdot t_q}{N}

Where:

$k_{\text{opt}}$ is the optimized constant representing improved efficiency from algorithmic advances and quantum gate optimization,
$N$ is the number of parallel quantum operations available.

In practical terms, a sufficiently powerful quantum computer could perform on the order of $10^6$ to $10^9$ parallel quantum operations, potentially reducing the search time to a few seconds. If $N = 10^6$ operations are possible, the time required is:

T_{\text{optimized}} = \frac{1000 \cdot 2^{128} \cdot 10^{-9}}{10^6} = 0.34 \, \text{seconds}

This implies that, with a high degree of quantum parallelism, SHA-256 could be broken in a fraction of a second, reducing the attack time from millennia to seconds.

6. Conclusion

The theoretical analysis demonstrates that with quantum computing and Grover's algorithm, the time required to reverse SHA-256 can be drastically reduced. Quantum parallelism further optimizes the attack, potentially allowing SHA-256 to be broken in seconds, compared to the classical brute-force time of $O(2^{256})$ , which would require millennia.

This highlights the urgency of transitioning to post-quantum cryptographic algorithms that are resistant to quantum attacks. Future developments in quantum hardware and algorithms will play a pivotal role in determining the feasibility of such attacks, with the timeline for such breakthroughs still uncertain. Nevertheless, this analysis provides a critical insight into the potential risks posed to blockchain technologies and cryptographic systems relying on SHA-256.

References:

Shor, P. W. (1997). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer.
Grover, L. K. (1996). A Fast Quantum Mechanical Algorithm for Database Search.
Chen, L. et al. (2016). Report on Post-Quantum Cryptography. NIST.
Aggarwal, D. et al. (2017). Quantum attacks on Bitcoin and how to protect against them.
Gidney, C., & Ekerå, M. (2019). How to factor 2048-bit RSA integers in 8 hours using 20 million noisy qubits.

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