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ITIS 6200/8200

ITIS 6200/8200. Bit commitment. Alice needs to commit a prediction which will not be revealed until later. Bob needs to make sure that Alice cannot change it. How can we do that? Example: Picking stocks: who will go first? Be careful of the forward search attack

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ITIS 6200/8200

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  1. ITIS 6200/8200

  2. Bit commitment • Alice needs to commit a prediction which will not be revealed until later. Bob needs to make sure that Alice cannot change it. How can we do that? • Example: • Picking stocks: who will go first? • Be careful of the forward search attack • Attack to such commitment: racing horses • Why this attack can be conducted: limited commitment space

  3. Bit commitment using one-way function • Can we use Hash(RA, b) to commit a bit? • If Alice does not tell Bob RA, forward search by Alice • If Alice tells Bob RA, Bob can figure out the bit • A better protocol: • Alice generates two random numbers, R1 and R2 • Alice sends (R1, Hash(R1, R2, b)) to Bob to commit the bit • Why we need R1 in plain text? • Why do we need R2 in cipher-text? • Later, Alice gives Bob R1, R2, and b to verify

  4. The advantage of this protocol: • Bob does not need to send anything • It is very difficult to find Hash(R1, R2, “0”) = Hash(R1, R2’, “1”) if R1 is long enough and the one way function has been properly designed

  5. Fair coin flip in digital world • It is different from the real world, where both parties can see the coin • The properties we need: • Alice flips the coin before Bob guesses • Alice cannot change the result after Bob guesses • Bob cannot “see” the result before taking the guess • It seems that bit commitment can solve this problem

  6. Coin flip using bit commitment • Alice commits to a bit using one of the previous protocols • Bob guesses the value of the bit • If right, Bob wins, if wrong, Alice wins • After the guess, Bob must be able to verify the result

  7. Coin flip using one-way functions • Alice generates a random number X, and sends Hash(X) to Bob • Bob guess whether X is odd or even • If Bob guesses right, Bob wins, otherwise, Alice wins • Alice reveals X so that Bob can verify. • If Alice can find two numbers (one odd, one even) having the same hash result, she can control the result every time.

  8. Coin flip using commutative encryption (where E_k1(E_k2(msg)) = E_k2(E_k1(msg)) • Alice generates two messages, (R1, Head), (R2, Tail), sends E_k1(m1) and E_k1(m2) to Bob • Bob selects one message and sends back E_k2(E_k1(m)), Alice does not know which one Bob choose • Alice decrypts the message and sends back to Bob, Bob decrypts it again and tells Alice the random number and the result • Alice and Bob reveal their keys to verify the result

  9. Coin flip using commutative encryption • Can Bob cheat? Not if he cannot guess the random string • Can Alice cheat? • Send both messages with Head. But later when they reveal the key, Alice will be caught. • Alice can lie about the value of R1 and R2: Bob can ask for their hash values before the messages are sent • An application of coin flip: • Generate session keys in a collaborative method where no party has a total control • We can flip multiple bits simultaneously

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