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Exponential decreasing rate of leaked information in universal random privacy amplification

Masahito Hayashi

arXiv:0904.0308v5cs.ITcs.CRmath.AC

TL;DR

The paper addresses how to bound Eve’s information when extracting a secret key from shared randomness. It extends universal2 privacy amplification to Rényi order 1+s, applies the bound to wire-tap channels, and derives an exponential bound that improves on Hayashi’s. The construction also supports linear or additive-channel settings and secret key agreement with public communication.

  • Problem

    Existing exponential bounds for Eve’s information and universal2 hashing were limited by prior approaches based on Hayashi’s exponent or Rényi entropy of order 2.

  • Method

    The paper extends universal2 privacy amplification using Rényi entropy of order 1+s, applies it to wire-tap-channel random coding, and combines linear codes with Toeplitz-plus-identity hashing.

  • Results

    The resulting exponential upper bound for Eve’s information improves on Hayashi’s bound and is applied to secret key agreement with public communication.

  • Takeaways & Limitations

    The framework provides a privacy-amplification-based route to exponentially decreasing Eve information across wire-tap channels, additive settings, and public-discussion key agreement.

Abstract

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We derive a new upper bound for Eve's information in secret key generation from a common random number without communication. This bound improves on Bennett et al(1995)'s bound based on the Rényi entropy of order 2 because the bound obtained here uses the Rényi entropy of order $1+s$ for $s \in [0,1]$. This bound is applied to a wire-tap channel. Then, we derive an exponential upper bound for Eve's information. Our exponent is compared with Hayashi(2006)'s exponent. For the additive case, the bound obtained here is better. The result is applied to secret key agreement by public discussion.

I. INTRODUCTION

The paper develops stronger exponential bounds on Eve’s information by extending universal2 privacy amplification from Rényi order 2 to order 1+s, then applies the result to wire-tap channels and secret key agreement.

  • Motivation: The paper seeks to improve the exponential decreasing-rate bound for Eve’s information in wire-tap channels.This rate estimates how quickly Eve’s information vanishes for finite-length codes.
  • Contribution: The main theorem generalizes Bennett et al.’s universal2 hashing bound from Rényi order 2 to order 1+s for s ∈ [0,1].The resulting order-1+s bound is shown to outperform the smooth Rényi entropy approach.
  • Contribution: For wire-tap channels, the paper derives a random-coding upper bound for Eve’s information using a universal2 hash function.This approach contrasts with Hayashi’s channel-resolvability method.
  • Contribution: The paper addresses the lack of existing linear-code constructions with exponentially vanishing Eve information below the Bob–Eve mutual-information gap.It combines arbitrary linear codes with privacy amplification using a Toeplitz matrix concatenated with the identity.
  • Applications: The result is applied to secret key agreement with public discussion, yielding a lower bound on Eve-information’s exponential decreasing rate below H(A|E) − H(A|B).The protocol combines a wire-tap-channel code with Toeplitz-plus-identity privacy amplification.
  • Proof and construction: The main theorem is proved in Appendix A, while Appendix B establishes that Toeplitz matrix concatenated with the identity is a universal2 hashing function.

A. Method based on R´enyi entropy of order 1 + s

The method extracts a key with a universal2 hash function and bounds Eve’s information using conditional Rényi entropy of order 1+s. In the i.i.d. setting, this yields exponential decay whenever the key rate is below H(A|E).

  • Problem setup: The task is to map a shared random variable A to a key f(A) that is nearly independent of Eve’s correlated variable E.Alice and Bob apply the same function without communication.
  • Rényi-entropy method: The analysis defines Rényi entropy and conditional Rényi entropy of order 1+s for the privacy-amplification bound.The paper mainly uses a modified conditional quantity because the standard form requires more complicated calculations.
  • Universal2 hashing: A universal2 hash ensemble limits the collision probability of outputs for every distinct input pair to at most 1/M.The ensemble is indexed by a random variable describing the stochastic choice of function.
  • Main bound: Theorem 1 extends Bennett et al.’s inequality to every 0 < s ≤ 1, whereas their result covered only s = 1.Theorem 2 then guarantees the existence of a deterministic function attaining the corresponding bound.
  • i.i.d. exponent: In the i.i.d. setting, Eve’s information decreases exponentially when the generation rate satisfies R < H(A|E).The exponent quantifies the speed of convergence as the block length grows.

B. Method based on smooth min-entropy

The paper reviews smooth min-entropy approaches to bounding leaked information and derives an alternative bound based on variational distance and continuity of Shannon entropy.

  • Renner and Wolf introduced smooth min-entropy, while Renner developed a different definition for evaluating leaked information after hashing.
  • The method modifies the distribution using variational distance and bounds the resulting entropy difference through continuity of Shannon entropy.
  • The resulting alternative bound minimizes over ǫ and combines a smooth min-entropy term with a continuity correction 2η(2ǫ, log |A| · M).
  • Applying Cramér’s theorem and optimizing the auxiliary rate yields an exponent whose maximizing parameter satisfies an equality between entropy and rate terms.
  • The optimized value is smaller than the earlier bound given by equation (5), while a related Renner–Wolf formula is not applied because it does not simplify the derivation.

III. THE WIRE-TAP CHANNEL IN A GENERAL FRAMEWORK

The paper constructs wire-tap channel codes by combining random coding with universal2 hashing, obtaining codes with controlled decoding error and Eve’s information.

  • A wire-tap channel has separate channels from Alice to Bob and Eve, with Alice choosing an encoding to limit Eve’s information.
  • A code is specified by its message size, input distributions, and disjoint decoding sets, and its performance includes size, Bob’s average error probability, and Eve’s information.
  • The random-coding protocol generates LM independent codewords according to p and applies a function from an ensemble satisfying universal hashing conditions.
  • The wire-tap code samples uniformly from hash preimages, lets Bob recover and hash the transmitted random variable, and then decodes the message.
  • Theorem 3 guarantees the existence of a code for arbitrary integers L and M and any input distribution p, with explicit performance bounds.
  • For n-fold discrete memoryless channels, additivity of φ gives corresponding codes for L^n and M^n, and the rate max_p I(p : W B) − I(p : W E) can be asymptotically attained.
  • When the sacrifice information rate is R, the paper’s decreasing-rate bound is eψ(R|W E, p), compared with Hayashi’s eφ(R|W E, p).

IV. COMPARISON WITH EXISTING BOUND

The comparison shows that the paper’s bound is at least as favorable as Hayashi’s in the analyzed wire-tap settings, with a stronger advantage characterized for additive channels.

  • Near the mutual-information rate: Near R = I(p : W), Taylor expansions show that the two quantities agree through second order and differ in their third-order terms.
  • Hölder’s inequality establishes nonnegativity of ˜I3(p : W), implying eψ(s|W E,p) is smaller than eφ(s|W E,p).
  • The paper notes that this inequality gives an advantage only in the third order when s is small.
  • Additive channels: For additive channels, the channel has the form W_x(z) = W_0(z − x) = P(z − x), and the corresponding exponents are compared with Bennett et al.’s order-2 bound.
  • Additive channels: Figure 1 compares the present paper’s normal-line exponent with Hayashi’s thick-line exponent and Bennett et al.’s dashed-line exponent.
  • General additive channels: For the general additive channel, eψ(R|W E, pmix) ≥ eφ(R|W E, pmix), and the paper states that its upper bound is better than Hayashi’s.

V. WIRE-TAP CHANNEL WITH LINEAR CODING

The paper constructs wire-tap codes by combining linear error-correcting codes with random linear privacy amplification. For additive or general additive channels, this yields bounds on Eve’s information and can achieve the corresponding rate under stated coding conditions.

  • Linear-code construction: The wire-tap code uses a submodule C2 of a linear code C1, with uniform coset encoding and decoding over C1/C2.The construction treats C1 as the usual message encoding and C2 as the privacy-amplification subcode.
  • Information bound: Theorem 4 bounds the expected Eve information for codes whose random subcode satisfies Condition 3.The theorem evaluates the uniform coset encoder associated with C1/C2(X).
  • Additive channels: For additive channels, ψ(s|W E, Pmix,C1) ≤ φ(s|W E, C1) ≤ φ(s|W E, X) for 0 < s ≤ 1.The bound follows from translation invariance and concavity of the relevant exponent function.
  • Linear-code construction: Random linear privacy amplification constructs C2(X) as the kernel of a concatenated Toeplitz matrix and identity matrix.For C1 equivalent to Fq^m and |C2(X)| = q^k, the construction uses (X, I).
  • Achievable coding: If W E is additive or general additive and C1 has effective encoding and decoding, the resulting wire-tap code satisfies the paper’s information inequalities.The construction uses random linear privacy amplification and the concatenation of a Toeplitz matrix with the identity.
  • Achievable coding: For n-fold discrete memoryless channels with general additive W E, the construction can achieve I(Pmix,X : W B) − I(Pmix,X : W E) when an error-correcting code reaches I(Pmix,X : W B).The result combines such an error-correcting code with random linear privacy amplification.

VI. SECRET KEY AGREEMENT

The paper applies its linear-code and privacy-amplification framework to secret key agreement with public discussion. It recovers the conditional-entropy difference as the achievable key-generation rate and states that the bound can be attained with the proposed construction.

  • Protocol setting: Secret key agreement aims to produce a shared random variable nearly independent of Eve using public communication.Performance is evaluated by key size, agreement probability, and mutual information with Eve.
  • Protocol setting: The protocol assumes that Alice’s variable space A has a module structure, allowing the linear-code construction to be applied.The paper notes that any finite set can be regarded as a cyclic group.
  • Rényi-entropy analysis: In the i.i.d. finite-field setting, the conditional Rényi entropy satisfies ˜H1+s(An|En|(P A,E)n) = n ˜H1+s(A|E|P A,E).The paper uses this additivity relation in constructing codes for the secret key agreement setting.
  • Achievable rate: The achievable key-generation rate is H(A|E) − H(A|B), equal to I(A : B) − I(A : E).This is the rate previously obtained by Maurer and Ahlswede–Csiszár.
  • Achievable construction: For the induced general additive channels, the bound can be attained by combining a linear code with random privacy amplification.This applies the construction developed for the wire-tap channel to public-discussion secret key agreement.

VII. DISCUSSION

The paper extends the universal-hashing bound from Rényi order 2 to order 1+s and applies it to wire-tap channels. It obtains an exponential upper bound, improves the existing wire-tap bound, and applies the result to public-discussion secret key agreement.

  • Discussion: The universal2-hashing bound for Eve’s information is extended from Rényi entropy of order 2 to order 1+s for s ∈ [0,1].This generalizes Bennett et al.’s result for secret-key generation from a common random number without communication.
  • Discussion: Applying the generalized bound to wire-tap channels yields an exponential upper bound for Eve’s information that improves the existing bound.The paper also compares the resulting exponent with the prior wire-tap analysis.
  • Discussion: The proposed privacy amplification uses a concatenated Toeplitz matrix and identity when the error-correcting code is linear and the channel is additive or general additive.The result is also applied to secret key agreement with public communication.

APPENDIX A PROOF OF THEOREM 1

The appendix develops the universal-hashing and Toeplitz constructions and contrasts variational-distance and mutual-information leakage criteria. It shows that small variational distance need not imply small mutual information.

  • Proof and construction: Condition 1 is used with subadditivity of x^s and concavity of the logarithm to derive the universal-hashing information bound.The proof also uses log(1 + x) ≤ x to obtain the final inequality.
  • Proof and construction: The Toeplitz construction uses m − 1 random finite-field variables to define the matrix X in the concatenation (X, I).The identity component has size (m − k) × (m − k).
  • Leakage criteria: The appendix constructs an example where variational-distance leakage is small while mutual-information leakage is large.The example is used to compare the two security criteria.
  • Leakage criteria: Small d1(P A,E, P A,mix × P E) does not guarantee small I(A : E), whereas small mutual-information leakage guarantees small variational distance.The paper therefore treats mutual information as the more restrictive leaked-information criterion.

APPENDIX D PROOF OF (12)

The appendix proves equation (12) through a sequence of inequalities using concavity of log x, stated conditions, and bounds valid for 0 < s ≤ 1.

  • Proof of (12): The proof uses concavity of log x to derive several intermediate inequalities.The passage identifies inequalities (32), (34), (38), and (40) as consequences of concavity.
  • Proof of (12): The bounds (1 + x)^s ≤ 1 + xs and log(1 + x) ≤ x hold for positive x and 0 < s ≤ 1.
  • Proof of (12): The derivation simplifies the random variable f_Φ(Y)(k) to Y_k and uses conditional expectations involving X when condition C holds.
  • Proof of (12): Applying the derived inequality yields the stated result, and using (36) produces equation (21).
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