Information Theoretically Secure Point-to-Point Rational Secret Sharing

1. Information Theoretically Secure Point-to-Point Rational Secret Sharing Jared Saia Varsha Dani Yamel Torres

2. Game Theory and Distributed Computing • In Distributed Computing agents are considered to be either “good” or “bad”. • Game theory assumes agents are “rational” (selfish) • Goal: Find a protocol that will ensure collaboration between players.

3. Background • Selfish players • Utility functions • Nash equilibrium • Backwards induction when player’s collaboration is needed.

4. Backwards Induction • Imagine a game where Alice and Bob make a deal. Alice will invite lunch today and Bob will buy her lunch tomorrow and the same will repeat for an undefined amount of time. • Assume Alice is leaving town in 10 days, and is Bob’s turn to buy her lunch the day before she leaves.

5. Backwards Induction • Does Bob has any incentive to actually buy her lunch if he won’t get a free lunch the next day? • Furthermore now that Alice knows that he won’t buy her lunch in the last day, does she has any incentive to buy him lunch the day before?

6. Secret Sharing • (t,n) Secret Sharing • The problem defines a dealer that wants to share a secret S. The dealer must divide the secret into n pieces in a way that ensures that with less than t pieces it will be impossible to reconstruct. • The rational players prefer to learn the secret rather than not learning it at all, and prefer other players not to learn the secret. • Multiparty Computation , distributed Cryptography, and for global coin toss.

7. Rational Secret Sharing • This game will have several rounds. • The secret will be revealed in round r*(chosen randomly from a geometric distribution). • The players are informed that they received the real secret on round r* + 1. • We will use “finger prints” to ensure the correct shares have been sent between the players.

8. Rational Secret Sharing • Dealer: • Choose u.a.r polynomials G and H, s.t.: • If r = r* • Gr(0) = secret. • Hr(0) = random number from the field – {0}. • If r = r* + 1 • Gr(0) = random number from the field. • Hr(0) = 0. • Otherwise • Gr(0) = random number from the field. • Hr(0) = random number from the field – {0}.

9. Rational Secret Sharing Player 1 Player 2 Receive from dealer • Interpolate G and H with the shares received. • Evaluate Gr(0) and Hr(0) • If Hr(0) = 0, then r -1 = r* • Output Gr-1(0) • If no share is received, output Gr-1(0) Verification

10. Our Solution • Using fingerprints • Removing “online” Dealer • Send different size lists at the beginning

11. Working on… • Real t-out-of-n Secret Sharing, when t ≠ n • Assume different utility functions • Byzantine players • Scalability

12. References • S. Dov Gordon, J. Katz. (2006) Rational Secret Sharing, Revisited. • G. Kol, & M. Naor. (2008) Games For Exchanging Information, Extended Abstract STOC'08 • G. Fuchsbauer, J. Katz, & D. Naccache. (2010) Ecient Rational Secret Sharing in Standard Communication Networks. TCC'10