PaperlayerView exact revision on arXiv

Source-linked AI summary

Polar Coding for Secure Transmission and Key Agreement

O. Ozan Koyluoglu, Hesham El Gamal

arXiv:1003.1422v1cs.ITcs.CRmath.IT

TL;DR

Practical coding schemes for information-theoretic secrecy remained elusive beyond random codes. This paper uses polar coding for degraded wiretap channels and fading-channel key agreement; it achieves the secrecy capacity for symmetric main and eavesdropper channels and describes key generation using privacy amplification.

  • Problem

    Practical codes for achieving information-theoretic secrecy over given main and eavesdropper channels remained elusive beyond random-code results.

    Source: §I · p. 1

  • Method

    The paper applies polar secrecy encoding to binary-input degraded wiretap channels and fading erasure blocks, then uses privacy amplification with statistical knowledge of Eve’s channel state.

    Source: §I · p. 1, §V · p. 4, §VI · p. 6

  • Results

    Polar coding achieves perfect secrecy rate I(Wm) − I(We), and achieves secrecy capacity C(Wm) − C(We) when both channels are symmetric.

    Source: §IV · pp. 3–4

  • Takeaways & Limitations

    Alice and Bob can generate secret keys over fading channels by creating an advantage over Eve across blocks and applying privacy amplification.

    Source: §V · p. 5, §VI · p. 6

    Finding a code meeting reliability and secrecy constraints may require exhaustive search, although almost all ensemble codes do equally well as block length increases.

    Source: §IV · p. 4

Abstract

from arXiv · show

Wyner's work on wiretap channels and the recent works on information theoretic security are based on random codes. Achieving information theoretical security with practical coding schemes is of definite interest. In this note, the attempt is to overcome this elusive task by employing the polar coding technique of Ar{\i}kan. It is shown that polar codes achieve non-trivial perfect secrecy rates for binary-input degraded wiretap channels while enjoying their low encoding-decoding complexity. In the special case of symmetric main and eavesdropper channels, this coding technique achieves the secrecy capacity. Next, fading erasure wiretap channels are considered and a secret key agreement scheme is proposed, which requires only the statistical knowledge of the eavesdropper channel state information (CSI). The enabling factor is the creation of advantage over Eve, by blindly using the proposed scheme over each fading block, which is then exploited with privacy amplification techniques to generate secret keys.

I. INTRODUCTION

Practical secrecy coding remained elusive despite information-theoretic wiretap results using random codes. This work applies polar coding to degraded wiretap channels and proposes key agreement over fading channels.

I. INTRODUCTION

  • Wyner showed that a degraded wiretap channel’s lower-noise main channel can support secret bits using random codes.The work builds on information-theoretic secrecy studied first for noiseless channels.Source: §I · p. 1
  • Prior LDPC wiretap codes addressed certain binary erasure and binary symmetric channel scenarios.For other scenarios, secrecy-capacity-achieving code design was described as an open problem; secure nested codes were also considered for noiseless main channels (see also).Source: §I · p. 1
  • Polar coding achieves non-trivial secrecy rates for binary-input degraded wiretap channels and the secrecy capacity when both channels are symmetric.The proposed fading-channel key agreement scheme uses statistical knowledge of Eve’s channel state and privacy amplification.Source: §I · p. 1

II. NOTATIONS

The notation section specifies how vectors, random variables, subsets, and probability distributions are represented.

II. NOTATIONS

  • Vectors are written as x_1^N or as x when indices are omitted.For a set A, x_A denotes the subvector with indices in A.Source: §II · p. 1
  • Random variables use capital letters and their sets use calligraphic letters.For example, X is defined over a set denoted by calligraphic X.Source: §II · p. 1
  • The shorthand p(x) and p(x|y) denotes Pr(X = x) and Pr(X = x|Y = y), respectively.Source: §II · p. 1

III. POLAR CODES

Polar coding combines and splits a binary-input channel into subchannels with differing reliability, then uses the more reliable subchannels for information transmission. The resulting codes achieve symmetric capacity with low-complexity encoding and decoding.

III. POLAR CODES

  • The capacity of W is its mutual information when the input is uniformly distributed, and its Bhattacharyya parameter upper-bounds single-use maximum-likelihood decision error.Source: §III · p. 2
  • Polar codes have O(N log N) encoding and decoding complexity, with block-error probability bounded by O(2^-N^β) for fixed β < 1/2.Source: §III · p. 2
  • Channel combining and splitting transform N uses of W into N binary input subchannels, which differ in reliability.The polar encoder transmits information through subchannels whose Bhattacharyya parameters are near zero.Source: §III · p. 2
  • The information set selects indices with lower Bhattacharyya parameters, while frozen bits are supplied to the decoder and remaining indices are estimated by successive cancellation.Source: §III · p. 2
  • Polar codes achieve the symmetric capacity I(W) of a binary-input discrete memoryless channel.A polar code exists with this rate, and for symmetric channels any frozen-vector choice works.Source: §III · p. 2
  • Degradation between two binary-input channels is preserved between their corresponding polarized subchannels.Source: §III · p. 2

IV. SECURE TRANSMISSION OVER WIRETAP CHANNEL

For binary-input degraded wiretap channels, polar coding achieves a perfect secrecy rate equal to the difference between the main and eavesdropper channel mutual informations. The construction also reaches secrecy capacity when both channels are symmetric.

Secure transmission

  • The construction partitions a main-channel polar code into eavesdropper-channel code ensembles by varying frozen bits outside the main-channel frozen set.The resulting ensemble supports secrecy construction even when the eavesdropper channel is not symmetric.Source: §IV · p. 3
  • I(Wm) − I(We) is achievable as a perfect secrecy rate for binary-input degraded wiretap channels.A fixed polar code in the ensemble can satisfy reliability and secrecy for sufficiently large block length.Source: §IV · pp. 3–4
  • For a non-symmetric eavesdropper channel, finding a code satisfying both constraints may require exhaustive search, although almost all ensemble codes perform equally well as block length increases.With a symmetric eavesdropper channel, any frozen vector satisfies secrecy, leaving only the reliability constraint for code search.Source: §IV · p. 4
  • For a non-symmetric eavesdropper channel, a prefix channel can make the resulting eavesdropper channel symmetric for use with the scheme.The prefix is described by p(x|x′).Source: §IV · p. 4
  • When both channels are symmetric, polar coding achieves secrecy capacity, and every code in the ensemble satisfies reliability and secrecy.The stated capacity is C(Wm) − C(We).Source: §IV · p. 4
  • Encoding and decoding each have complexity O(N log N); for BECs, code construction can be done with complexity O(N).The BEC construction complexity is stated separately from encoding and decoding.Source: §IV · p. 4

V. SECRET KEY AGREEMENT OVER FADING WIRETAP CHANNELS

The scheme applies polar coding block by block over binary-erasure fading wiretap channels, then uses privacy amplification to distill a secret key. Its performance is expressed through accumulated main-channel capacity and the positive part of the main–eavesdropper capacity difference.

Key agreement scheme

  • Alice and Bob use polar coding at each fading block, then apply universal hashing to distill a shorter secret key from their accumulated information.The scheme targets binary-erasure main and eavesdropper channels, and the users need only statistical knowledge of Eve’s channel state.Source: §V · p. 4

Information accumulation

  • The blockwise construction splits each block’s information into secure information generated and remaining information, whose entropies are random because the channels vary across blocks.The scheme can produce no secured bits at a particular fading block.Source: §V · p. 4
  • Across fading blocks, the accumulated information is intended to be reliably decoded by Bob, while Eve’s remaining uncertainty approaches the accumulated positive capacity difference.The stated limits are H(W*) → E[C(Wm)] and H(W*|Ye*) → E[[C(Wm) − C(We)]+] as L, M, and N grow.Source: §V · p. 4

Key agreement result

  • For sufficiently large L, M, and N, Alice and Bob agree with error probability at most ε on W*, of length n over LM fading blocks.The theorem specifies W* as the concatenation of the block-level information and randomization components.Source: §V · p. 5

Privacy amplification

  • The key is formed as K = G(W*), with G chosen uniformly from a universal class of hashes mapping n bits to r bits.The passage states the hashing construction; the stated theorem’s secrecy bound is not included in these excerpts.Source: §V · p. 5

2. Hence, we obtain

The result establishes the key-agreement bounds for sufficiently large parameters and shows that the key is approximately uniform. The scheme requires only statistical Eve-CSI knowledge; extensions and secrecy-strength options are also stated.

Proof conclusion

  • For the chosen sufficiently large L, M, and N, the theorem’s desired reliability and secrecy bounds hold, and H(K) ≥ r − ε makes the key approximately uniform.The proof chooses M and N above stated thresholds and then takes L sufficiently large.Source: §2 · p. 6

Remarks

  • Unlike the discussed existing designs, the fading scheme needs only statistical knowledge of Eve’s CSI, and the main channel need not be stronger than Eve’s on average.The passage contrasts this with existing designs and the earlier section’s assumption that Eve’s channel is known.Source: §2 · p. 6
  • Setting M = 0 lets the scheme achieve strong secrecy for the wiretap channel of Section IV, rather than only a small leakage rate.The passage characterizes strong secrecy as arbitrarily small information leakage.Source: §2 · p. 6
  • The results extend to arbitrary binary-input channels, but their code-construction complexity may scale less favorably than for erasure channels.The extension uses n = LMN(E[I(Wm)] − ε*) and r = LMN(E[[I(Wm) − I(We)]+] − ε*).Source: §2 · p. 6

VI. DISCUSSION

The paper studies polar coding for binary-input discrete memoryless channels with a degraded eavesdropper, and reports non-trivial secrecy rates. It also proposes fading-channel key agreement that creates an advantage over Eve and uses privacy amplification, requiring only statistical Eve-CSI knowledge.

Discussion

  • For binary-input discrete memoryless channels with a degraded eavesdropper, polar coding achieves non-trivial secrecy rates.The discussion also notes that extension to arbitrary discrete memoryless channels may be possible using the techniques in [17].Source: §VI · p. 6
  • Over fading channels, using polar coding at each block creates an advantage over Eve that privacy amplification exploits to generate keys, with only statistical Eve-CSI knowledge required.Source: §VI · p. 6
Loading 1003.1422v1…