[Paper Review] Controlled Alternate Quantum Walk based Block Hash Function

Dan Li, Panpan Ding+2 more|arXiv (Cornell University)|May 12, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a controlled alternate quantum walk-based block hash function (CAQWBH) that accelerates hash processing by processing messages in batches using time-position-dependent controlled quantum walks on complete graphs with self-loops. By replacing single-bit message control with batched message processing, CAQWBH achieves significantly higher efficiency while maintaining strong security through measurement irreversibility and modular post-processing, making it suitable for both classical and quantum computing platforms.

ABSTRACT

The hash function is an important branch of cryptology. Controlled quantum walk based hash function is a kind of novel hash function, which is safe, flexible, high-efficient, and compatible. All existing controlled quantum walk based hash functions are controlled by one bit message in each step. To process message in batch amounts, in this paper, controlled alternate quantum walk based block hash function is presented by using the time-position-dependent controlled quantum walks on complete graphs with self-loops. The presented hash function accelerate the hash processing dramatically, so it is more high-efficient.

Motivation & Objective

• To address the inefficiency of existing controlled quantum walk-based hash functions that process only one message bit per step.
• To enhance processing speed by enabling batch message input in each step of the quantum walk evolution.
• To maintain strong security and avalanche properties in a quantum hash function suitable for both classical and quantum hardware.
• To demonstrate practicality by enabling extensions such as message authentication codes and pseudo-random number generators.
• To ensure compatibility and flexibility in key and parameter selection for varying output lengths and security levels.

Proposed method

• Employs time-position-dependent controlled alternate quantum walks (CAQW) on complete graphs with self-loops, where N = 2^q vertices represent the state space.
• Uses a composite Hilbert space H_p ⊗ H_c, with H_p spanned by |x⟩ (x ∈ {0, ..., N−1}) and H_c by coin states, enabling superposition and entanglement.
• Applies two coin operators with adjustable angles (θ₁, θ₂) and a control mechanism that depends on both time and position, allowing dynamic message encoding.
• Processes N-bit message blocks per step by encoding message bits into the control of the coin operator, replacing single-bit control in prior works.
• Performs post-processing on the resulting pseudo-probability distribution by scaling by 10^k and taking modulo 2^k to generate a k×N-bit hash output.
• Utilizes private parameters (α_i) as a secret key (key₁) and message-dependent control as key₂, enabling secure MAC and PRNG extensions.

Experimental results

Research questions

• RQ1 Can batch message processing in controlled quantum walks significantly improve the efficiency of quantum hash functions compared to single-bit control?
• RQ2 Does the use of time-position-dependent control on complete graphs with self-loops preserve the avalanche effect and statistical randomness required for cryptographic hash functions?
• RQ3 How does the proposed CAQWBH compare in security and performance to existing CQW-based hash functions under both classical and quantum computing models?
• RQ4 Can the CAQWBH be effectively extended to build secure message authentication codes (MAC) and pseudo-random number generators (PRNG) with strong statistical properties?
• RQ5 To what extent does the infinite key space (Aleph-one) for α_i parameters ensure resistance against quantum brute-force attacks?

Key findings

• The proposed CAQWBH achieves dramatically faster hash processing by processing N-bit message blocks per step, eliminating the bottleneck of single-bit message control in prior CQW-based functions.
• Statistical analysis confirms that the hash output exhibits near-ideal performance, including strong avalanche characteristics and uniform distribution across the hash space.
• The use of private α_i parameters (key₁) creates an infinite key space (Aleph-one), making the function resistant to quantum brute-force attacks even when executed on a quantum computer.
• The CAQWBH-based message authentication code (MAC) is secure against quantum adversaries due to the combination of measurement irreversibility and the large key space.
• The CAQWBH can be adapted as a pseudo-random number generator by post-processing the probability distribution, producing long, statistically random bitstrings suitable for cryptographic applications.
• The function is fully compatible with both classical and quantum computing platforms, ensuring practical deployment under current hardware constraints despite the challenges of quantum hardware commercialization.