Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles
We consider cryptographic and physical zero-knowledge proof schemes for Sudoku, a popular combinatorial puzzle. We discuss methods that allow one party, the prover, to convince another party, the verifier, that the prover has solved a Sudoku puzzle, without revealing the solution to the verifier. The question of interest is how a prover can show: (i) that there is a solution to the given puzzle, and (ii) that he knows the solution, while not giving away any information about the solution to the verifier.
In this paper we consider several protocols that achieve these goals. Broadly speaking, the protocols are either cryptographic or physical. By a cryptographic protocol we mean one in the usual model found in the foundations of cryptography literature. In this model, two machines exchange messages, and the security of the protocol relies on computational hardness. By a physical protocol we mean one that is implementable by humans using common objects, and preferably without the aid of computers. In particular, our physical protocols utilize scratch-off cards, similar to those used in lotteries, or even just simple playing cards.
The cryptographic protocols are direct and efficient, and do not involve a reduction to other problems. The physical protocols are meant to be understood by ”lay-people” and implementable without the use of computers.
KeywordsCryptographic Protocol Playing Card Random Coin Computational Hardness Commitment Protocol
Unable to display preview. Download preview PDF.
- 2.Blum, M.: How to Prove a Theorem So No One Else Can Claim It. In: Proc. of the International Congress of Mathematicians, Berkeley, California, USA, pp.1444–1451 (1986)Google Scholar
- 5.Oded Goldreich, Modern Cryptography, Probabilistic Proofs and Pseudorandomness, Springer, Algorithms and Combinatorics, vol. 17 (1998)Google Scholar
- 9.Gradwohl, R., Naor, M., Pinkas, B., Rothblum, G.N.: Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles. http://www.wisdom.weizmann.ac.il/~naor/PAPERS/sudoku_abs.html
- 10.Gradwohl, R., Naor, E., Naor, M., Pinkas, B., Rothblum, G.N.: Proving Sudoku in Zero-Knowledge with a Deck of Cards (January 2007), http://www.wisdom.weizmann.ac.il/~naor/PAPERS/SUDOKU_DEMO/
- 11.Hayes, B.: Unwed Numbers. American Scientist Vol. 94(1), http://www.americanscientist.org/template/AssetDetail/assetid/48550 (January- February 2006)
- 15.Naor, M., Naor, Y., Reingold, O.: Applied kid cryptography or how to convince your children you are not cheating (March 1999), http://www.wisdom.weizmann.ac.il/~naor/PAPERS/waldo.ps
- 16.Jean-Jacques, Q., Myriam, Q., Muriel, Q., Michaël, Q., Louis, G., Marie Annick, G., Gaïd, G., Anna, G., Gwenolé, G., Soazig, G., Berson, T.: How to explain zero-knowledge protocols to your children. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 628–631. Springer, Heidelberg (1990)Google Scholar
- 17.Schneier, B.: The solitaire encryption algorithm (1999), http://www.schneier.com/solitaire.html
- 18.Vadhan, S.P.: Interactive Proofs & Zero-Knowledge Proofs, lectures for the IAS/Park City Math Institute Graduate Summer School on Computational Complexity, http://www.eecs.harvard.edu/~salil/papers/pcmi-abs.html
- 19.Sudoku, Wikipedia, the free encyclopedia (based on Oct 19th 2005 version), http://en.wikipedia.org/wiki/Sudoku
- 20.Yato, T.: Complexity and Completeness of Finding Another Solution and its Application to Puzzles, Masters thesis, Univ. of Tokyo, Dept. of Information Science. (January 2003) Available: http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/MasterThesis.ps