{
  "id": "acfp15-gaussian-groebner",
  "categoryId": "lwe-gaussian-algebraic-history",
  "paperId": "albrecht-cid-faugere-perret2014-algebraic",
  "year": 2014,
  "date": "2015-02-26",
  "datePrecision": "day",
  "time": {
    "coefficient": null,
    "expression": "O(2^{n[ω(1+C_A)H₂(C_A/(1+C_A)) + πlog₂(e)/(4A²)]})"
  },
  "space": {
    "coefficient": null,
    "expression": "O(2^{n[2(1+C_A)H₂(C_A/(1+C_A)) + πlog₂(e)/(4A²)]})"
  },
  "tags": [
    "conditional",
    "semi-regularity",
    "Gaussian error",
    "exponential samples"
  ],
  "review": {
    "status": "source-checked"
  },
  "certificate": {
    "status": "not-provided",
    "lean": null
  },
  "notes": "Conditional on the generated equations being semi-regular. A=1 gives the paper’s rounded time exponent (2.35ω+1.13)n and rounded memory exponent 5.85n. These are historical algebraic bounds, not best overall LWE exponents.",
  "extra": {
    "parameters": "Prime q(n) sufficiently large for degree n/A; error σ=√n/√(2π); A≥1 fixed; C_A is the degree-of-regularity constant from Lemma 6 / Proposition 3; C_1≈1.41.",
    "noise": "Discrete Gaussian with standard deviation σ=√n/√(2π), hence width αq=√n.",
    "secret": "Arbitrary secret in Z_q^n.",
    "samples": "m≈exp(πn/(4A²)); exponential oracle samples required.",
    "objective": "Recover the secret.",
    "model": "Field operations; ω is the chosen dense linear-algebra exponent; H₂ is binary entropy.",
    "assumptions": "Assumption 1: semi-regularity of the generated high-degree LWE equations.",
    "success": "At least 1−2/(π√n), conditional on the stated algebraic assumption."
  },
  "evidence": {
    "url": "https://iacr.steepath.eu/2014/1018-AlgebraicAlgorithmsforLWE.pdf",
    "locator": "§5, Lemma 6 and Theorem 5, pp. 10–11; Eq. (6)",
    "sourceVersion": "ePrint 2014/1018 revised 26 February 2015"
  }
}