Physical ZKP for Makaro Using a Standard Deck of Cards

TL;DR

This paper introduces a practical physical zero-knowledge proof protocol for Makaro using a standard deck of all different cards, improving practicality and efficiency over previous methods by reducing the number of cards needed.

Contribution

It presents a novel ZKP protocol for Makaro that employs a standard deck and introduces a general encoding method for numbers with all different cards.

Findings

Reduces the number of cards needed compared to previous protocols

Enables secure computation of numerical functions like comparison and maximum

Provides a practical approach for physical ZKPs with standard decks

Abstract

Makaro is a logic puzzle with an objective to fill numbers into a rectangular grid to satisfy certain conditions. In 2018, Bultel et al. developed a physical zero-knowledge proof (ZKP) protocol for Makaro using a deck of cards, which allows a prover to physically convince a verifier that he/she knows a solution of the puzzle without revealing it. However, their protocol requires several identical copies of some cards, making it impractical as a deck of playing cards found in everyday life typically consists of all different cards. In this paper, we propose a new ZKP protocol for Makaro that can be implemented using a standard deck (a deck consisting of all different cards). Our protocol also uses asymptotically less cards than the protocol of Bultel et al. Most importantly, we develop a general method to encode a number with a sequence of all different cards. This allows us to securely compute several numerical functions using a standard deck, such as verifying that two given numbers are different and verifying that a number is the largest one among the given numbers.