{
  "id": "1502.04430",
  "version": "v1",
  "published": "2015-02-16T06:11:01.000Z",
  "updated": "2015-02-16T06:11:01.000Z",
  "title": "Distributions Attaining Secret Key at a Rate of the Conditional Mutual Information",
  "authors": [
    "Eric Chitambar",
    "Ben Fortescue",
    "Min-Hsiu Hsieh"
  ],
  "categories": [
    "quant-ph",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "In this paper we consider the problem of extracting secret key from an eavesdropped source $p_{XYZ}$ at a rate given by the conditional mutual information. We investigate this question under three different scenarios: (i) Alice ($X$) and Bob ($Y$) are unable to communicate but share common randomness with the eavesdropper Eve ($Z$), (ii) Alice and Bob are allowed one-way public communication, and (iii) Alice and Bob are allowed two-way public communication. Distributions having a key rate of the conditional mutual information are precisely those in which a \"helping\" Eve offers Alice and Bob no greater advantage for obtaining secret key than a fully adversarial one. For each of the above scenarios, strong necessary conditions are derived on the structure of distributions attaining a secret key rate of $I(X:Y|Z)$. In obtaining our results, we completely solve the problem of secret key distillation under scenario (i) and identify $H(S|Z)$ to be the optimal key rate using shared randomness, where $S$ is the G\\'acs-K\\\"orner Common Information. We thus provide an operational interpretation of the conditional G\\'acs-K\\\"orner Common Information. Additionally, we introduce simple example distributions in which the rate $I(X:Y|Z)$ is achievable if and only if two-way communication is allowed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-16T06:11:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conditional mutual information",
      "distributions attaining secret",
      "common information",
      "strong necessary conditions",
      "eve offers alice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}