{
  "id": "aps15-lwe-primal-embedding-model",
  "categoryId": "lwe-primal-embedding-models",
  "paperId": "albrecht-player-scott2015-concrete",
  "year": 2015,
  "date": "2015-10-02",
  "datePrecision": "day",
  "time": {
    "coefficient": null,
    "expression": "T_BKZ(m+1,δ₀); log₂δ₀=[log₂(ε′τ α√(2e))]²/(4nlog₂q)"
  },
  "space": {
    "coefficient": null,
    "expression": "Depends on the SVP oracle and embedding implementation."
  },
  "tags": [
    "primal",
    "embedding",
    "uSVP",
    "historical cost model"
  ],
  "review": {
    "status": "source-checked"
  },
  "certificate": {
    "status": "not-provided",
    "lean": null
  },
  "notes": "Kannan embedding cost model stated with the empirical uSVP gap-to-reduction parameter τ. It is not a theorem that a root-Hermite factor alone solves every uSVP instance.",
  "extra": {
    "parameters": "Fix n,q, Gaussian width αq; ε′>1 controls the error tail; τ≤1 and ε_τ model uSVP success; m≈√(nlog₂q/log₂δ₀).",
    "noise": "Discrete Gaussian width αq, standard deviation αq/√(2π).",
    "secret": "General LWE secret.",
    "samples": "At least optimized m samples; cap the choice by the actual sample budget.",
    "objective": "Recover the secret by BDD-to-uSVP embedding.",
    "model": "Source uses embedding factor t=||e||; success model ε_τ for gap τδ₀^m.",
    "assumptions": "Gaussian heuristic for q-ary lattice minima; empirically calibrated uSVP success law.",
    "success": "ε_τ·[1−(ε′exp((1−ε′²)/2))^m], under the source model."
  },
  "evidence": {
    "url": "https://www.degruyterbrill.com/document/doi/10.1515/jmc-2015-0016/pdf?licenseType=open-access",
    "locator": "§5.5, Lemmas 5.17–5.18, pp. 190–191; Lemma 5.17 is attributed to Albrecht–Fitzpatrick–Göpfert Lemma 2",
    "sourceVersion": "Journal of Mathematical Cryptology 9(3), published 2 October 2015"
  }
}