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A Complexity Hierarchy of Shuffles in Card-Based Protocols

Tomoki Ono, Suthee Ruangwises

Abstract

Card-based cryptography uses physical playing cards to construct protocols for secure multi-party computation. Existing card-based protocols employ various types of shuffles, some of which are easy to implement in practice while others are considerably more complex. In this paper, we classify shuffle operations into several levels according to their implementation complexity. We motivate this hierarchy from both practical and theoretical perspectives, and prove separation results between several levels by showing that certain shuffles cannot be realized using only operations from lower levels. Finally, we propose a new complexity measure for evaluating card-based protocols based on this hierarchy.

A Complexity Hierarchy of Shuffles in Card-Based Protocols

Abstract

Card-based cryptography uses physical playing cards to construct protocols for secure multi-party computation. Existing card-based protocols employ various types of shuffles, some of which are easy to implement in practice while others are considerably more complex. In this paper, we classify shuffle operations into several levels according to their implementation complexity. We motivate this hierarchy from both practical and theoretical perspectives, and prove separation results between several levels by showing that certain shuffles cannot be realized using only operations from lower levels. Finally, we propose a new complexity measure for evaluating card-based protocols based on this hierarchy.
Paper Structure (19 sections, 3 theorems, 5 equations, 2 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 5 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Preliminaries
  5. Scramble Shuffle
  6. Random Cut
  7. Pile-Scramble Shuffle
  8. Random Pile Cut
  9. Practical Viewpoint
  10. 1. The resulting permutation is unknown to everyone.
  11. 2. All players can verify that no cheating occurs.
  12. Conclusion.
  13. Theoretical Viewpoint
  14. Levels of Shuffles
  15. Shuffles Beyond the Four Fundamental Operations
  16. ...and 4 more sections

Key Result

Theorem 1

For $n=3$, the shuffle $\{\text{id},(1\,2\,3),(1\,2\,3)^2\}$ cannot be realized using only ss operations and deterministic permutations.

Figures (2)

  • Figure 1: Boxes with sliding covers
  • Figure 2: Small envelopes inside a large envelope

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof