How to Implement a Random Bisection Cut

Transcription

1 How to Implement a Random Bisection Cut Itaru UEDA 1 Akihiro NISHIMURA 1 Yu ichi HAYASHI 2 Takaaki MIZUKI 1 Hideaki SONE 1 1 Tohoku University 2 Tohoku Gakuin University TPNC 2016

2 Introduction What is secure computation a a {0,1} b {0,1} b a b By using a deck of cards We can realize a secure computations 2

3 One example of AND secure computations 1. Add one black and red card each a 0 b 3. Apply a Random bisection cut [ ] 2. Rearrange the order 4. Rearrange again 5. Turn over the two left-most cards 0 = 1 = a b or 3 a b

4 One example of AND secure computations 4

5 One example of AND secure computations 1. Add one black and red card each a 0 b 3. Apply a Random bisection cut [ ] 2. Rearrange the order 4. Rearrange again 5. Turn over the two left-most cards 0 = 1 = a b or 5 a b

6 Definition of bit values We use a black card and a red card Definition of bit values 0 = 1 = Represent the value of a bit as above input bits a and b follow this encoding 6

7 Definition of a commitment Bit values are given as backside is up Commitment Commitment a b Left cards constitute a commitment to a Right cards constitute a commitment to b 7

8 The property of cards The cards which we use are Face up The property of cards Face down same color cards and all cards of backside are identical 8

9 The Sone-Mizuki AND computation 1. Add one black and red card each a 0 b 3. Apply a Random bisection cut [ ] 2. Rearrange the order 4. Rearrange again 5. Turn over the two left-most cards 0 = 1 = a b or 9 a b

10 A Random Bisection Cut (RBC) A Random bisection cut [ ] A probability of 1 A probability of or [ ] 10

11 A Random Bisection Cut (RBC) 11

12 AND Secure Computation Protocols Random Cut had been used before the RBC was designed AND protocols Crépeau and Kilian [CRYPTO 93] Niemi and Renvall [TCS, 1998 (journal )] Stiglic [TCS, 2001 (journal)] Mizuki and Sone [FAW 09] # of colors # of cards Type of shuffle Avg. # of trials 4 10 Random cut Random cut Random cut Random bisection cut 1 12

13 Random Cut (RC) operation Random Cut A cyclic shuffling operation

14 Random Cut (RC) operation 14

15 Benefit of Random Bisection Cut Card-based protocols have become more efficient thanks to RBC AND protocols Crépeau and Kilian [CRYPTO 93] Niemi and Renvall [TCS, 1998 (journal )] Stiglic [TCS, 2001 (journal)] Mizuki and Sone [FAW 09] # of colors # of cards Type of shuffle Avg. # of trials 4 10 Random cut Random cut Random cut Random bisection cut 1 15

16 Security of Random Bisection Cut [ ] There exist a concern that The two halves of the cards are swapped or not. The result of the shuffle may leak visually 16

17 Objective of Our Study We provides some methods for executing a Random Bisection Cut securely Our proposal methods A method using auxiliary tools ( 2) A method to reduce a RBC to the RC using dummy cards ( 3) 17

18 Contents 1. Introduction 2. Executing a Random Bisection Cut Using Auxiliary Tools 3. Executing a Random Bisection Cut Using Dummy Cards 4. Secrecy of Implementations of the Random Cut 5. Conclusion 18

19 Contents 1. Introduction 2. Executing a Random Bisection Cut Using Auxiliary Tools 3. Executing a Random Bisection Cut Using Dummy Cards 4. Secrecy of Implementations of the Random Cut 5. Conclusion 19

20 Random Bisection Cut (RBC) 20

21 A Method Using Envelopes or Boxes One solution Shuffle behind his/her back or under a table envelopes It is desirable for all actions to be performed in front of other players and/or third parties than to be hidden boxes 21

22 A Method Using a Separator Card and Rubber Band 1. Use a separator card 3. Flip the bundle The separator prevents information regarding the color of cards from being leaked. 2. Place one half on it 4. Place the other half on and fix them together 22

23 A Method Using a Separator Card and Rubber Band 5. Toss the pile with a spin Execute a RBC securely even in front of other players and/or third parties 23

24 Contents 1. Introduction 2. Executing a Random Bisection Cut Using Auxiliary Tools 3. Executing a Random Bisection Cut Using Dummy Cards 4. Secrecy of Implementations of the Random Cut 5. Conclusion 24

25 Another idea IDEA a Random Cut is able to be implemented easily we reduce a RBC to Random Cuts by using dummy cards 1. A method which utilizes cards of other colors 2. A method which uses vertical asymmetricity of the backs of cards 25

26 Another idea IDEA a Random Cut is able to implement easily we reduce a RBC to Random Cuts by using dummy cards 1. A method which utilizes cards of other colors 2. A method which uses vertical asymmetricity of the backs of cards 26

27 1. A Method Which Utilizes Cards of Other Colors Remember, we want to apply a RBC to two sets of cards [ ] n cards n cards Assume every card is or 27

28 1. A Method Which Utilizes Cards of Other Colors We use 2s and 2t as dummy cards that differ from and We realize a RBC only by Random Cuts (All of the backs have to be identical ) The procedure is as follows 28

29 1. A Method Which Utilizes Cards of Other Colors 1. Place dummy cards with their faces down s + t dummies n cards s + t dummies n cards s cards t cards 29

30 1. A Method Which Utilizes Cards of Other Colors 2. Apply a Random Cut 2(n + s + t) cards 3. Turn over the left-most card 2(n + s + t) cards 3-a) If the face-up card is Remove the dummy cards one by one as below t cards (s + t) cards n cards n cards DONE 30

31 1. A Method Which Utilizes Cards of Other Colors 3-b) If the face-up card is Remove the dummy cards (s + t) cards n cards n cards s cards 3-c) If the face-up card is or Turn it over again and return to Step 2 31

32 Average Number of Trials In Step 3, The probability that either (a) or (b) occurs is + = The # of dummy cards The # of all of cards in the deck 2(s+t) 2(n+s+t) A RBC is implemented by 2(s + t) dummies after an average of n+s+t Random Cut execution s+t 32

33 Other Proposal Methods IDEA a Random Cut is able to implement easily we reduce a RBC to Random Cuts by using dummy cards 1. A method which utilizes cards of other colors 2. A method which uses vertical asymmetricity of the backs of cards 33

34 s + t dummies n cards s + t dummies n cards s cards t cards 34

35 2. A Method Which Uses Vertical Asymmetricity of the backs of cards 1. Arrange the 2m dummies m dummies n cards m dummies n cards 2. Apply a Random Cut 2(n + m) cards 35

36 2. A Method Which Uses Vertical Asymmetricity of the backs of cards 4. Remove all dummy cards m dummies n cards m dummies n cards n cards n cards By 2m additional cards and one Random Cut We can implement a Random Bisection Cut. 36

37 The Proposal Methods By using dummies, We reduce a RBC to Random Cuts Do we implement a Random Cut securely anyways Need to discuss secure implementations of Random Cuts 37

38 Contents 1. Introduction 2. Executing a Random Bisection Cut Using Auxiliary Tools 3. Executing a Random Bisection Cut Using Dummy Cards 4. Secrecy of Implementations of the Random Cut 5. Conclusion 38

39 Secrecy of Implementations Do we implement a Random Cut securely anyways We conducted a basic experiment how we can implement Random Cuts securely 39

40 Discussion Regarding Implementations We asked 11 students to track a specific card 40

41 Discussion Regarding Implementations RESULT 3 students could follow and track with a high probability. They could visually observe how many cards were moved at every cut. The key is to make it impossible to count moved cards 41

42 Discussion Regarding Implementations How to increase the security Move cards from the bottom to the top Ensure that cards are not out of alignment We propose the Hindu cut Those 3 students all gave up following the Hindu cut shuffle 42

43 Confirming the Security of the Hindu Cut Manner of experiment Participants Use: 8 cards including 2 dummies Request: Watch a movie of a Hindu cut and track a specific card 72 participants who came to our lab s booth in the Open Campus of Tohoku Univ. in

44 Confirming the Security of the Hindu Cut 44

45 The Result of The Basic Experiment 72 participants answered as (1) I have no idea. :64 persons (89%) (2) Definitely, it must be A. : 5 persons (7%) (3)Definitely, it must be B. : 3 persons (4%) To rule out wild guesses, we requested the participants who answered (2) or (3) to watch additional movies No one can answer correctly for all of movies We recognize that a more careful and wide ranging investigation is still required 45

46 Contents 1. Introduction 2. Executing a Random Bisection Cut Using Auxiliary Tools 3. Executing a Random Bisection Cut Using Dummy Cards 4. Secrecy of Implementations of the Random Cut 5. Conclusion 46

47 Conclusion Card-based protocols have become more efficient thanks to the RBC But, implementation issues has not previously been discussed We have proposed some novel methods for implementing the RBC We demonstrated that humans are able to implement it in practice securely 47