Secret Sharing

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Secret Sharing

• This presentation was prepared by Yufei
  Tao.http//www.cse.cuhk.edu.hk/taoyf

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Outline

• Randomized response
• Multi-party computation
• Collaborative password

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Did you ever want to commit a suicide?

• Imagine a group of people, and we want to
  estimate how many of them ever thought of
  committing a suicide.
• Any idea how to do this?

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Randomized response

• Ask each person to follow the following protocol
• Generate a random number in 0, 1.
• If the number is lt 0.4, tell the truth.
• Otherwise, lie.
• Why would people be willing to obey the protocol?
• How do we estimate from these randomized answers?

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Getting the estimate

• Our question did you ever want to kill
  yourself?
• n the number of people.
• a the number of yes received
• Assume the real number is x.
• Remember x is what we want to estimate.
• Among the yes received, in expectation
• 0.4x are truth
• 0.6(n x) are lies.
• Hence, 0.4x 0.6(n x) a.
• x 3n 5a

lies
truth
all the yes answers received
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An example

• The number of people 10000.
• The number of people that wanted to commit a
  suicide 545.
• Assume that 210 of the 545 people answer yes.
• Why not exactly 545 0.4 218?
• Also assume that 5650 of the the remaining 9455
  people answer yes.
• So totally 5860 yes received.
• Our estimate
• x 3n 5a 30000 29300 700.

lies
truth
all the yes answers received
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Exercise

• Assume an election with three candidates.
• A TV station wants to do a survey on their
  popularity.
• So it will ask people this question who will
  you vote for?
• Design a protocol to estimate of each candidates
  popularity without strong confidence in the
  preference of each voter.

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The instructor will give the protocol here
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Outline

• Randomized response
• Multi-party computation
• Collaborative password

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How much money do you have?

• Imagine a number of people, each holding a
  certain amount of money.
• We want to know totally how much money they have.
• Nobody is willing to tell others how much money
  s/he has, not even a clue.
• Any idea?

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Secure sum

• Say there are only three people.

v1 x
v1 v2 x
v1 v2 v3 x
a random number x
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Secure sum (cont.)

• What if we want v1 v2 v3?

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Secure computation

• There exist protocols to do any f(v1, v2, , vn)
  securely.

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Outline

• Randomized response
• Multi-party computation
• Collaborative password

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An old mans dying will

• A dying old man has 3 sons, and a bank account
  with a lot of money.
• He wants to make sure that all 3 sons open the
  account together.
• Say the password of the account is 353123903.
• Can you help the old man?

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Secure key

• Password 353123903
• The old man tells each son privately a number.
• Here are the numbers -1379, 473247, -39207357.
• They form an equation x3 1379x2 473247x
  39207357 0.
• This equation has three roots 353, 123, 903.
• (x 353)(x 123)(x 903) 0
• If any coefficient is missing, the equation is
  unsolvable.
• There is one final problem the son having
  39207357 can do a factorization.
• Next we give a simple solution to alleviate this
  problem.

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Secure key

• Fact Factorization of a large number is
  practically intractable.
• Password 353123903
• Make an equation (x 3521235321945)(x
  78134212391820)(x 78134290391820) 0
• Unfold it into x3 ax2 bx c 0.
• Tell a, b, c to the 3 sons respectively.
• Final note There are fast algorithms for solving
  equations with large coefficients.