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Post-quantum hash functions using $\mathrm{SL}_n(\mathbb{F}_p)$

Corentin Le Coz, Christopher Battarbee, Ramón Flores, Thomas Koberda, Delaram Kahrobaei

TL;DR

The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions, and justify the claim that the resulting hash functions are post-quantum secure.

Abstract

We define new families of Tillich-Zémor hash functions, using higher dimensional special linear groups over finite fields as platforms. The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions. We justify the claim that the resulting hash functions are post-quantum secure.

Post-quantum hash functions using $\mathrm{SL}_n(\mathbb{F}_p)$

TL;DR

The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions, and justify the claim that the resulting hash functions are post-quantum secure.

Abstract

We define new families of Tillich-Zémor hash functions, using higher dimensional special linear groups over finite fields as platforms. The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions. We justify the claim that the resulting hash functions are post-quantum secure.
Paper Structure (18 sections, 7 theorems, 20 equations, 1 figure)

This paper contains 18 sections, 7 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper:
  3. Definition of the hash functions
  4. Background about special linear groups
  5. Expansion
  6. The general construction
  7. A concrete example
  8. Properties of the constructed hash functions
  9. Free groups and girth
  10. Preimage and collision resistance, and post-quantum heuristics
  11. The mixing property
  12. Multiplicativity and parallel computing
  13. Good and bad tails
  14. Multiplicativity
  15. Parallel computing
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $n\geq 2$ and let $p$ a prime. Write $\pi_p: \mathop{\mathrm{SL}}\nolimits_n(\mathbb{Z}) \to \mathop{\mathrm{SL}}\nolimits_n(\mathbb{F}_p)$ for the canonical projection given by reduction modulo $p$. There exist matrices $\tilde{A},\tilde{B} \in \mathop{\mathrm{SL}}\nolimits_n(\mathbb{Z})$ such

Figures (1)

  • Figure 1: Description of the maps $s_i$ in Definition \ref{['definition: concrete example']}

Theorems & Definitions (19)

  • Theorem 2.1: Arzhantseva-Biswas Arzhantseva2018Large
  • Remark 2.2
  • Proposition 2.3
  • proof : Sketch of proof
  • Definition 2.4: Special linear group based hash functions
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Proposition 3.1
  • proof
  • ...and 9 more