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Introduction to Modern Cryptography, Lecture 14

Introduction to Modern Cryptography, Lecture 14. By special request: Rates of secret sharing schemes, line, cycle. What we want to prove. Legal set: {A,B}, {B,C}, {C,D} Claim: H(BC) >= 3 H(S) H(B)+H(C) >= H(BC) H(B) >= 3/2 H(S) or H(C) >= 3/2 H(S) I.e., the rate is <= 3/2. Definitions.

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Introduction to Modern Cryptography, Lecture 14

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  1. Introduction to Modern Cryptography, Lecture 14 By special request: Rates of secret sharing schemes, line, cycle

  2. What we want to prove • Legal set: {A,B}, {B,C}, {C,D} • Claim: H(BC) >= 3 H(S) • H(B)+H(C) >= H(BC) • H(B) >= 3/2 H(S) or H(C) >= 3/2 H(S) • I.e., the rate is <= 3/2

  3. Definitions Entropy: Conditional Entropy: Entropy of Joint variable: Mutual Information:

  4. Definitions Mutual Information: Conditional Mutual Information:

  5. Secret Sharing • S – random variable (secret) from distribution • A- random variable of shares of a legal subset of users • H(S|A) = 0 – knowing the shares determines the secret • A – random variable of shares of a non-legal subset of users • H(S|A) = H(S)

  6. Lemma Y – random variable of shares for a non-legal subset of users X U Y – rand. Var. for legal subset H(X|Y) = H(S) + H(X|YS)

  7. Main lemma H(BC)>= 3 H(S). H(S) ≤ H(C|AD) ≤ H(C|A) = H(C|AS) ≤ H(CB|AS) = H(B|AS)+H(C|ABS) ≤ H(B|AS) + H(C|BS) = H(B|A)-H(S)+H(C|B)-H(S) ≤ H(BC)-2H(S)

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