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Quantum hash function using discrete-time quantum walk on Hanoi network

Pulak Ranjan Giri

TL;DR

This work introduces a quantum hash function built from a discrete-time quantum walk on the Hanoi network HN4, where both the coin and shift operators are controlled by message bits. By leveraging long-range edges, the scheme achieves strong diffusion and low collision rates, enabling hashing of short messages without sacrificing security intuition based on the Holevo bound. The hash is obtained from the final probability distribution via a post-processing step and concatenated across network vertices to form a fixed-length output, with scalability through larger networks and precision. The approach demonstrates competitive collision resistance compared to other quantum-walk hashes and offers avenues for optimization and noise-aware refinements in the NISQ era.

Abstract

Quantum walk based hash functions have attracted a lot of attention in recent years because of its faster execution time and robust resistance against attacks compared to classical hash functions. It has been observed that the underlying graph and the way message controls the quantum walk iteration steps play a crucial role for the robustness of the hash function. We propose a quantum hash function based on the discrete-time quantum walk on a Hanoi network--a one dimensional periodic lattice with extra long-range edges of a specific form--which is highly collision resistant. The message bits of our scheme control the flow of probability amplitude through the extra long-range edges and the conditional shift operators. Our method even works for messages with small bit-lengths, contrary to most of the quantum walk based hash functions defined on a cycle, which usually work for messages with bit-lengths more than the length of the cycle.

Quantum hash function using discrete-time quantum walk on Hanoi network

TL;DR

This work introduces a quantum hash function built from a discrete-time quantum walk on the Hanoi network HN4, where both the coin and shift operators are controlled by message bits. By leveraging long-range edges, the scheme achieves strong diffusion and low collision rates, enabling hashing of short messages without sacrificing security intuition based on the Holevo bound. The hash is obtained from the final probability distribution via a post-processing step and concatenated across network vertices to form a fixed-length output, with scalability through larger networks and precision. The approach demonstrates competitive collision resistance compared to other quantum-walk hashes and offers avenues for optimization and noise-aware refinements in the NISQ era.

Abstract

Quantum walk based hash functions have attracted a lot of attention in recent years because of its faster execution time and robust resistance against attacks compared to classical hash functions. It has been observed that the underlying graph and the way message controls the quantum walk iteration steps play a crucial role for the robustness of the hash function. We propose a quantum hash function based on the discrete-time quantum walk on a Hanoi network--a one dimensional periodic lattice with extra long-range edges of a specific form--which is highly collision resistant. The message bits of our scheme control the flow of probability amplitude through the extra long-range edges and the conditional shift operators. Our method even works for messages with small bit-lengths, contrary to most of the quantum walk based hash functions defined on a cycle, which usually work for messages with bit-lengths more than the length of the cycle.

Paper Structure

This paper contains 11 sections, 24 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Quantum walk on Hanoi network
  3. Hash function with quantum walk on Hanoi network
  4. Performance of hash function
  5. Sensitivity of hash values to change in message
  6. Diffusion and confusion properties
  7. Uniform distribution
  8. Collision analysis
  9. Resistance to birthday attack
  10. Comparison with other quantum walk based hash schemes
  11. Conclusion

Figures (5)

  • Figure 1: HN4 network with $N= 2^4= 16$ vertices. Solid and dashed black curves are regular and long-range edges respectively.
  • Figure 2: Probability distribution of a pair of randomly generated $9$-bit length messages (message $1$, message $2$) as a function of the vertex location for the quantum walk on (a) a one-dimensional periodic lattice and (b) on a HN4 network for $N= 16$, $t=9$.
  • Figure 3: Plot of the hash values $h(m_1)$, $h(m_2)$, $h(m_3)$, $h(m_4)$, and $h(m_5)$. Red stars are the bit-values and blue curves are step plots of the hash values respectively.
  • Figure 4: Uniform distribution of bits on hash space
  • Figure 5: Experimental (green) and theoretical (red) collision rate ($\%$) based on $N = 10000$ pair of randomly chosen $1.5 N_v$-bit length messages as function of the number of vertices of the HN4 network for $l=7, k=16$.