Introduction to Modern Cryptography, Lecture 14

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Introduction to Modern Cryptography, Lecture 14

• By special request
• Rates of secret sharing schemes, line, cycle

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What we want to prove

• Legal set A,B, B,C, C,D
• Claim H(BC) gt 3 H(S)
• H(B)H(C) gt H(BC)
• H(B) gt 3/2 H(S) or H(C) gt 3/2 H(S)
• I.e., the rate is lt 3/2

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Definitions
Entropy
Conditional Entropy
Entropy of Joint variable
Mutual Information
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Definitions
Mutual Information
Conditional Mutual Information
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Secret Sharing

• S random variable (secret) from distribution
• A- random variable of shares of a legal subset of
  users
• H(SA) 0 knowing the shares determines the
  secret
• A random variable of shares of a non-legal
  subset of users
• H(SA) H(S)

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Lemma

• Y random variable of shares for a non-legal
  subset of users
• X U Y rand. Var. for legal subset
• H(XY) H(S) H(XYS)

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Main lemma

• H(BC)gt 3 H(S).
• H(S) H(CAD) H(CA) H(CAS) H(CBAS)
  H(BAS)H(CABS) H(BAS) H(CBS)
  H(BA)-H(S)H(CB)-H(S) H(BC)-2H(S)