Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen

TL;DR

This paper introduces physical zero-knowledge proof protocols for logic puzzles like Akari, Takuzu, Kakuro, and KenKen, enabling players to prove knowledge of solutions without revealing them, using simple office materials.

Contribution

It presents novel physical zero-knowledge proof algorithms for these puzzles, formalizes their security, and demonstrates practical implementation with everyday materials.

Findings

Protocols are simple to implement with cards and envelopes.

Security of the proofs is formally proven.

Protocols effectively verify solutions without revealing them.

Abstract

Akari, Takuzu, Kakuro and KenKen are logic games similar to Sudoku. In Akari, a labyrinth on a grid has to be lit by placing lanterns, respecting various constraints. In Takuzu a grid has to be filled with 0's and 1's, while respecting certain constraints. In Kakuro a grid has to be filled with numbers such that the sums per row and column match given values; similarly in KenKen a grid has to be filled with numbers such that in given areas the product, sum, difference or quotient equals a given value. We give physical algorithms to realize zero-knowledge proofs for these games which allow a player to show that he knows a solution without revealing it. These interactive proofs can be realized with simple office material as they only rely on cards and envelopes. Moreover, we formalize our algorithms and prove their security.