DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

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Finite discrete memoryless wiretap channel and code model

Definition

Let $X, Y, Z$ be finite nonempty alphabets and let $W (y, z ∣ x)$ be a stochastic kernel from $X$ to $Y \times Z$ (Finite stochastic kernels). Its Bob and Eve marginals are $W_{Y} (y ∣ x) : = \sum_{z} W (y, z ∣ x)$ and $W_{Z} (z ∣ x) : = \sum_{y} W (y, z ∣ x)$ ; each is a finite DMC in the sense of Finite discrete memoryless channels.

For each integer $n \geq 1$ , use one-based coordinate labels for blocks: for $a^{n} \in A^{n}$ , a function on ${0, \dots, n - 1}$ , write $a_{i} : = a^{n} (i - 1)$ for $1 \leq i \leq n$ . With these labels, the joint channel law is

$W^{n} (y^{n}, z^{n} ∣ x^{n}) : = \prod_{i = 1}^{n} W (y_{i}, z_{i} ∣ x_{i}) .$

The product formula is a stochastic kernel by The finite wiretap product law is stochastic. Finite distributivity gives $\sum_{z^{n}} W^{n} (y^{n}, z^{n} ∣ x^{n}) = \prod_{i} \sum_{z_{i}} W (y_{i}, z_{i} ∣ x_{i}) = \prod_{i} W_{Y} (y_{i} ∣ x_{i}) = W_{Y}^{n} (y^{n} ∣ x^{n})$ , and the analogous finite sum over $y^{n}$ gives marginal $W_{Z}^{n}$ . The pair $(Y^{n}, Z^{n})$ may be correlated conditional on $X^{n}$ ; neither marginal changes if that coupling changes while the two marginal channels stay fixed.

An $(n, M_{n})$ wiretap code has a finite message set $M_{n}$ , a uniform message $M_{n}$ , a stochastic encoder $P_{X^{n} ∣ M_{n}}$ using private randomness independent of channel noise, and a decoder ${\hat{M}}_{n} : Y^{n} \to M_{n} \cup {e}$ , where $e \notin M_{n}$ denotes erasure and is counted as error. The encoder uses no feedback and the parties have no preshared secret key or forward public aid. The public code description is known to Eve, who observes $Z^{n}$ . Average error is $P_{e, a v}^{(n)} : = \Pr [{\hat{M}}_{n} (Y^{n}) \neq M_{n}]$ ; maximal error is $P_{e, m a x}^{(n)} : = \max_{m \in M_{n}} \Pr [{\hat{M}}_{n} (Y^{n}) \neq m ∣ M_{n} = m]$ . The rate is $n^{- 1} \ln ∣ M_{n} ∣$ nats per channel use. A secrecy metric is specified separately.

Depends on

Used by

Finite degraded wiretap channel Definition
Operational weak and strong secrecy capacities Definition
Weak and strong secrecy for wiretap code sequences Definition
Weak secrecy does not imply strong secrecy Example
The finite wiretap auxiliary has cardinality at most |X| Lemma
Strong-secrecy capacity of a finite DMC wiretap channel Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Abbas El Gamal and Young-Han Kim, Network Information Theory, arXiv:1001.3404v3 (standard reference, not scraped)
Paul Cuff, Princeton ELE 538: Information Theoretic Security, Lecture 12 (standard reference, not scraped)