INFORMATION-THEORETIC SECURITY INFORMATION-THEORETIC SECURITY - - PowerPoint PPT Presentation

Matthieu Bloch December 5, 2019

INFORMATION-THEORETIC SECURITY INFORMATION-THEORETIC SECURITY

Lecture 4 - Elements of Information Theory

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SOURCE CODING SOURCE CODING

One-shot source coding with side information

Coding consists of encoder to encode source symbol and decoder to reconstruct source symbol Objective: transmit messages to reconstruct from and with small average probability of error Lemma (Random binning for source coding) For , let For a codebook

• f independently generated

, we have

Enc : X → [1; M] Dec : Y × [1; M] → X W X W Y (C) ≜ ( ≠ X|C) = (Dec(Enc(X), Y ) ≠ X). Pe P X ^ P γ > 0 = {(x, y) ∈ X × Y : log < γ} . Bγ 1 (x|y) PX|Y C = {Φ(x)} Φ(x) ∼ U([1; M]) [ (C)] ≤ ((X, Y ) ∉ ) + . EC Pe PPXY Bγ 2γ M

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SOURCE CODING SOURCE CODING

One-shot source coding with side information

Coding of sequences of length so that and

• Definition. (Achievable rate)

A rate is achievable if there exists a sequence of codes with length with Proposition (Achievability) The rate is achievable. Proposition (Converse) All achievable rates must satisfy .

n ∈ N∗ Enc : → [1; M] X n Dec : × [1; M] → Yn X n R n log M ≤ R ( ) = 0 lim sup

n→∞

1 n lim sup

n→∞ Pe Cn

H(X|Y ) R R ≥ H(X|Y )

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CHANNEL OUTPUT APPROXIMATION CHANNEL OUTPUT APPROXIMATION

One shot channel coding

Coding consists of encoder to approximate output statistics Lemma (Random coding for channel output approximation) For a codebook

• f independently generated codewords

, we have where .

Enc : [1; M] → X D( ∥ ) PZ QZ For γ > 0 ≜ {(x, z) ∈ X × Z) : log ≤ γ} . Cγ (z|x) WZ|X (z) QZ C ∼ Xi pX [D( ∥ )] E PZ QZ ≤ ((X, Z) ∉ ) log ( + 1) + , PPXWZ|X Cγ 1 μZ 2γ M ≜ (z) μZ minz∈suppQZ QZ

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