Zero-Knowledge Proof Protocol for Cryptarithmetic Using Dihedral Cards

Raimu Isuzugawa, Daiki Miyahara, Takaaki Mizuki

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Citations (Scopus)


Cryptarithmetic, also known as Verbal Arithmetic or Word Addition, is a popular pencil puzzle in which the aim is to deduce which letter corresponds to which numeral, given a mathematical equation in which each numeral (from 0 to 9) has been replaced with a unique letter. The most famous instance of this puzzle is probably “SEND + MORE = MONEY", whose solution is “9567 + 1085 = 10652", i.e., S = 9, E = 5, N = 6, D = 7, M = 1, O = 0, R = 8, and Y = 2. In this study, we construct a physical zero-knowledge proof protocol for a Cryptarithmetic puzzle: That is, our protocol enables a prover who knows a solution to the puzzle to convince a verifier that he/she knows the solution without revealing any information about it. The proposed protocol uses a physical deck of “dihedral cards,” which were developed by Shinagawa in 2019.

Original languageEnglish
Title of host publicationUnconventional Computation and Natural Computation - 19th International Conference, UCNC 2021, Proceedings
EditorsIrina Kostitsyna, Pekka Orponen
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages17
ISBN (Print)9783030879921
Publication statusPublished - 2021
Event19th International Conference on Unconventional Computation and Natural Computation, UCNC 2021 - Espoo, Finland
Duration: 2021 Oct 182021 Oct 22

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12984 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference19th International Conference on Unconventional Computation and Natural Computation, UCNC 2021


  • Card-based cryptography
  • Cryptarithmetic
  • Dihedral cards
  • Physical zero-knowledge proof


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