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crypto-slides-16-mcommit.1x1

crypto-slides-16-mcommit.1x1 - Mutual Commitments c Eli...

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Mutual Commitments c circlecopyrt Eli Biham - August 18, 2010 440 Mutual Commitments (16)
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Example - Coin Toss Coin Toss : Alice and Bob want to toss a coin. Easy to do when they are in the same room. How can they toss a coin over the phone? Solution : Alice tosses a coin and conveys the result to Bob. Problem : Alice can choose any result. Solution : Alice and Bob each tosses a coin and then they send the results to each other. The coin toss result is the XOR of both tosses. Problem : The one who sends the result first has no influence on the result, while the second can choose any result. Note : There is a strong dependency on the simultaneous publication of the result. c circlecopyrt Eli Biham - August 18, 2010 441 Mutual Commitments (16)
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Coin Toss Alice and Bob want to toss a coin over the phone. If Alice commits to a bit, and promises not to change her choice after Bob tells her about his coin toss, we get that the protocol: Alice: Chooses a bit b A . Bob: Chooses a bit b B . Bob Alice: b B . Alice Bob: b A . Both: b = b A b B . is secure. c circlecopyrt Eli Biham - August 18, 2010 442 Mutual Commitments (16)
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Coin Toss (cont.) It is clear that Bob cannot choose the result of the coin toss. On the other hand, Alice was committed to b A before knowing b B and therefore she cannot choose the result of the coin toss, as well. Thus, we want to ensure that Alice does not change her choice after hearing b B . c circlecopyrt Eli Biham - August 18, 2010 443 Mutual Commitments (16)
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Bit Commitment This protocol is a building block for the construction of other protocols, and we will use it later. Objective : Alice chooses a bit b and uses it in some protocol, with Bob. Bob needs to make sure that Alice uses b and not b , but Alice prefers not to reveal b . The Model : Alice sends a commitment C on b to Bob, from which b cannot be reconstructed. Revealment phase: later Alice reveals b to Bob, and Bob checks the commitment. c circlecopyrt Eli Biham - August 18, 2010 444 Mutual Commitments (16)
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Bit Commitment (cont.) Example : Let f be a one-way permutation and let B be a hardcore predicate of f . Then Alice and Bob can perform the following BC protocol: Alice chooses b , and a random number r (100-bit number) such that B ( r ) = b . Alice sends the commitment C = f ( r ) to Bob. Later for the revealment phase: Alice sends b and r to Bob. Bob checks whether C = f ( r ) and b = B ( r ). If not then Alice cheated, Otherwise Bob concludes that Alice indeed committed to b . What goes wrong when B is not a hardcore predicate? What goes wrong when f is not a permutation? c circlecopyrt Eli Biham - August 18, 2010 445 Mutual Commitments (16)
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Bit Commitment (cont.) RSA based example : Alice chooses n = p · q, e, d as in RSA, and a random number 0 r < n with parity b (or another hardcore predicate B ( r ) = b ), and sends n, e and r e (mod n ) to Bob.
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