[
  {
    "id": "bg17-checked-small-secret-rescaling",
    "categoryId": "lwe-small-secret-rescaled-primal",
    "paperId": "bai-galbraith2013-binary-lwe",
    "year": 2014,
    "date": "2017-02-21",
    "datePrecision": "day",
    "time": {
      "coefficient": null,
      "expression": "T_reduce(d+1,δ), with d=m+n and δ^d≲(q^mσ^n)^{1/d}/(σ√(2πe))"
    },
    "space": {
      "coefficient": null,
      "expression": "Depends on the reduction/enumeration subroutine."
    },
    "tags": [
      "small secret",
      "rescaling",
      "Gaussian error"
    ],
    "review": {
      "status": "source-checked"
    },
    "certificate": {
      "status": "not-provided",
      "lean": null
    },
    "notes": "The ternary-secret scaling increases determinant by σ^n. For binary secrets the source uses scaling 2σ plus recentering. First posted December 2013 and published ACISP 2014; this record conservatively dates the fully checked revised PDF.",
    "extra": {
      "parameters": "Fix n,m,q and Gaussian error standard deviation σ; ternary secret for the displayed determinant; d=m+n.",
      "noise": "Independent discrete Gaussian error, standard deviation σ; no fixed universal σ(n) is assumed.",
      "secret": "Uniform ternary {−1,0,1}^n for displayed formula; binary {0,1}^n uses (2σ)^n determinant factor and recentering.",
      "samples": "m available samples; source may discard rows. For ternary scaling, optimized d≈√(n(log₂q−log₂σ)/log₂δ) if allowed by m.",
      "objective": "Recover small secret and error.",
      "model": "ISIS embedding, rescaling, then CVP/uSVP decoding; displayed gap uses the Gaussian heuristic.",
      "assumptions": "Random-lattice Gaussian heuristic and reduction-success model.",
      "success": "Model-dependent recovery; experiments do not establish an arbitrary-lattice guarantee."
    },
    "evidence": {
      "url": "https://eprint.iacr.org/2013/839.pdf",
      "locator": "§§6.1–6.2, pp. 10–11, determinant scaling, gap discussion, Lemma 1 and Eq. (6)",
      "sourceVersion": "Full ePrint version last revised 21 February 2017; first ePrint 16 December 2013; ACISP 2014"
    }
  }
]