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Cryptography and Sudoku Moni Naor WEIZMANN INSTITUTE OF SCIENCE מוני נאור Joint work with: Ronen Gradwohl, Benny Pinkas, Guy Rothblum

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What is Cryptography? Traditionally: how to maintain secrecy in communication Alice and Bob talk while Eve tries to listen AliceBob Eve

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Cryptography Very ancient occupation Biblical times: Atbash in Jeremiah איך נלכדה ששך ותתפש תהלת כל הארץ איך היתה לשמה בבל בגויים Egyptian Hieroglyphs Unusual ones... Many interesting books and sources, especially about the Enigma (WW2)

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Modern Times Up to the mid 70s: classified military work Exception: Shannon, Turing* Since then - explosive growth Commercial applications Scientific work: tight relationship with Computational Complexity Theory Major works: Diffie-Hellman, Rivest, Shamir and Adleman (RSA) Recently: more involved models for more diverse tasks. How to maintain the secrecy, integrity and functionality in computer and communication system. Prevalence of the Internet: Cryptography is in the news (daily!) Cryptography is relevant to ``everyone - security and privacy issues for individuals The Study of the resources needed to solve computational problems

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Computational Complexity Theory Study the resources needed to solve computational problems Computer time Computer memory Communication Parallelism Randomness … Identify problems that are infeasible to compute by any reasonable machine Taxonomy: classify problems into classes with similar properties wrt the resource requirements Help find the most efficient algorithm for a problem A computational problem: multiplying two numbers, selecting a move in a chess position Find the shortest tour visiting all cities P=NP?

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The Crypto Arms Race: ~3000 BC - ~1980 Secure System Break Traditional crypto: 8 attack 9 defense Modern crypto (1976 -): 9 defense 8 attack Secure System+ Secure System++ Break+Break++ Secure System Secure System+

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Sudoku Fill in the empty entries in the grid so that every row, every column, and every 3 x 3 subgrid contains the digits 1 through 9.

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Sudoku Fill in the empty entries in the grid so that every row, every column, and every 3 x 3 subgrid contain the digits 1 through 9. Can be generalized to an n n grid, where n=k 2. The size of an instance is O(n 2 log(n)) bits. Nothing special about the numbers 1…9.

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The Plot I know the solution! Oh yeah? Prove it! Well, I could show you, but… …I dont want to tell you how to solve it… Veronica Paul

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Zero-Knowledge Proofs Paul wants to prove that A is true If A is true: Veronica is convinced, but doesnt learn about A! She cant prove that A is true. Blah Blah? Blah Blah? Blah! Oh!

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Authentication: prove your identity to someone using secret information, without revealing the secret Force malicious adversaries to act according to protocol Why study zero-knowledge for Sudoku? It has nice properties Its educational – everybody knows Sudoku Its FUN! Why Study Zero-Knowledge Proofs? Design protocol with benign adversaries. Then compile to withstand malicious ones

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Outline Definitions Physical model A basic protocol 2 variations

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Interactive Proof Probabilistic protocol between 2 parties: Prover and Verifier Both know instance of a problem Prover might know a witness/solution Players chat, and at the end, verifier accepts or rejects Completeness: probability that honest verifier accepts correct proof Soundness error: probability that verifier accepts incorrect proof

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Zero-Knowledge Proof Interactive Proof Zero-knowledge property: Whatever Verifier learned from Prover, could have learned by himself Exists efficient Simulator that can simulate conversation, without access to Prover zero-knowledge proof for all NP Proof of 3-colorability Proof for Hamiltonicity Set of problems that have efficient verification

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Sudoku and Complexity Sudoku is in NP Means: easy to verify solutions In fact: Sudoku is NP Complete – not all that relevant There are zero-knowledge proofs for all problems in NP Therefore there is a ZK proof for Sudoku. Direct ZK proofs for Sudoku are preferable: Efficiency: avoiding the overhead of the reduction Practicality: Implementable without the aid of computers Understandability (by non-experts!): Ensure that participants have intuitive understanding of the proof.

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Physical Objects Typical Cryptographic metaphor: Physical locked box Hard to find physical locked box that: Can never be opened Are readily available Have transparent operation Tamper-evident seal Tampering is evident Can open, but cant reseal Scratch-off card, sealed envelope

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Scratch-Off Cards Cant tell them apart (until unsealed) Can shuffle them effectively Like picking a random permutation Can triplicate them Stronger requirement Used in perfect soundness protocol

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Human Behavior Paul and Veronica are in same room Shuffling: Paul wants a fair shuffle, Veronica wants to make sure no cards were switched More benign adversary: Either protocol works, or cheating player is labeled a cheater

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Playing Cards Can use playing cards instead of scratch-off cards: Sealing = turning card face down Revealing = turning it face up Not really tamper evident Works when players in same room, watching each other

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A Simple Physical Protocol Flip coin: rows or columns?

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A Simple Physical Protocol

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Props: 81 sealed scratch-off cards, and a board with 81 cells (like Sudoku) P places a sealed card on each cell Corresponding to his solution filled-in values are unsealed V chooses one of rows/cols/subgrids P makes packet for each row, shuffles it V takes each packet, unseals cards, verifies that each contains cards 1…9 If yes -- accept, otherwise reject

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Analysis Completeness: perfect Soundness: cheating P must cheat in one of rows, columns, or subgrids P is caught with probability 1/3 Zero-knowledge: V only sees some permuted values of 1…9

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Better Soundness

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Props: 81 scratch-off cards P places 3 cards on each cell, corresponding to solution For each cell, V assigns each card to one of rows/cols/subgrids, collects to corresponding packet P shuffles each of 27 packets V takes each packet, unseals cards, verifies that each contains 1…9 If yes -- accept, otherwise reject

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Analysis of Soundness P can no longer cheat as before New way to cheat: 3 cards on a cell are not the same value Say some cell gets 3 values, not all the same. One of three cards is different from others Belongs to one of rows/cols/subgrids o/w P is always caught cheating V assigns card to correct row/col/subgrid with probability at most 1/3 Cheating P caught with probability 2/3 Actually: can show that P is caught with probability 8/9 At least 2 cells are mislabeled o/w P is always caught cheating

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Reducing Number of Shuffles Previous protocol required 27 shuffles. Too much! New protocol: same as before – 3 cards on each cell V assigns each to row/col/subgrid Make 27 packets For each packet, V assigns a random number 1…c For each i, P assembles all packets with number i P shuffles each of c piles V takes each pile, unseals cards, verifies that each contains correct number of cards 1…9. If yes -- accept, otherwise reject

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Analysis Only c shuffles required Soundness: With probability 8/9, some packet j is unbalanced However, two unbalanced packets, if shuffled together, may balance each other Suppose all packets except j are assigned to one of c piles If piles are balanced, then assigning j will cause imbalance P will be caught If 2+ piles are unbalanced P will be caught If 1 pile is unbalanced, j will balance it only if assigned to it, with probability 1/c Cheating P is caught with probability 8(c-1)/9c

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Perfect Soundness If 3 cards on each cell are guaranteed to have same value, cheating P would always get caught! Implementing triplicate: With trusted setup: 3 cards (with same value) are connected and can be torn apart Without trusted setup: Use colors instead of numbers Each card is a circle, prepared by P V cuts each card into 3 equal pieces (randomly) If card was not uniformly colored, random cut will reveal non- uniformity when card is scratched 3 3 3

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Perfect Soundness with a trusted copy machine: Prepare three copies of the solution. Puzzle should be printed on the back. One copy is cut along the rows One copy is cut along the columns One copy is cut along the subgrids Each strip is then cut into cells The cells are shuffled (or sorted by the prover) Verifier checks that all values 1…9 are there The filled-in cells have the same values on both sides To prove that the correct puzzle was solved

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Encryption Authentication Digital signatures Protocols Zero-knowledge proofs Secure computation Cryptographic Protocols ALICE BOB Cryptographic protocols: proceed by exchanging digital message Assumptions needed: existence of a one-way function

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Open problems: Implement physical protocol over the mail? Parties need not be in the same room Possible to implement commitments from scratch-off cards. However, an amplification stage requires many repetitions Not easy for humans Other puzzles?

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Cryptography Today phlegmon of the pharynx

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Cryptography Today Cryptography is a very active research area Research activities range: providing firm foundations Relationship with complexity theory providing actual constructions and analysis for specific needs. Some recent topics Obfuscation of programs Maintaining privacy of released data Voting Schemes

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Any questions?

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Based on: R.Gradwohl, M. Naor, B. Pinkas and G. Rothblum, Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles, FUN Available: PAPERS/ sudoku_abs.html

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Thank you תודה רבה

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