Download
1 / 7
70 likes | 83 Vues
Lecture 14 delves into the rates of secret sharing schemes, exploring concepts such as entropy, conditional entropy, mutual information, and definitions of secret sharing. The goal is to prove specific claims regarding the information rates in a legal set scenario.
E N D
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 Entropy: Conditional Entropy: Entropy of Joint variable: Mutual Information:
Definitions Mutual Information: Conditional Mutual Information:
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)
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)
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)
