[
  {
    "id": "acfp15-binary-error-linear-samples",
    "categoryId": "lwe-binary-error-algebraic-models",
    "paperId": "albrecht-cid-faugere-perret2014-algebraic",
    "year": 2014,
    "date": "2015-02-26",
    "datePrecision": "day",
    "time": {
      "coefficient": null,
      "expression": "O(n² · 2^{ωn(1+β)H₂(β/(1+β))+o(n)}), β=C−1/2−√(C(C−1))"
    },
    "space": {
      "coefficient": null,
      "expression": "O(n² · 2^{2n(1+β)H₂(β/(1+β))+o(n)})"
    },
    "tags": [
      "binary error",
      "linear samples",
      "conditional",
      "semi-regularity"
    ],
    "review": {
      "status": "source-checked"
    },
    "certificate": {
      "status": "not-provided",
      "lean": null
    },
    "notes": "Conditional linear-sample tradeoff. For m=2n the source rounds the time exponent to 0.43ωn; for m≥6.6n it reports approximately 0.344n time and 0.289n memory. Rounded decimals are not ingested as exact coefficients.",
    "extra": {
      "parameters": "Odd prime q=q(n), polynomially bounded; n variables; m=Cn with fixed C>1; β=C−1/2−√(C(C−1)); 2≤ω<3.",
      "noise": "Independent uniform errors in {0,1}.",
      "secret": "Arbitrary s∈Z_q^n; this is binary ERROR, not an assumption of binary secret.",
      "samples": "Exactly m=Cn; changing C changes the exponent.",
      "objective": "Recover s by solving the error-free quadratic system.",
      "model": "Arithmetic in Z_q; H₂(x)=−xlog₂x−(1−x)log₂(1−x). Formula follows Theorem 7 plus Proposition 1’s binomial coefficient.",
      "assumptions": "Assumption 2: the specific quadratic equations are semi-regular.",
      "success": "Conditional on Assumption 2 and the intended unique-solution regime."
    },
    "evidence": {
      "url": "https://iacr.steepath.eu/2014/1018-AlgebraicAlgorithmsforLWE.pdf",
      "locator": "§6, Theorem 7(ii)–(iii), Eqs. (10)–(11), pp. 12–13; proof Eq. (14) and Proposition 1",
      "sourceVersion": "ePrint 2014/1018 revised 26 February 2015"
    }
  },
  {
    "id": "acfp15-binary-error-quasilinear-samples",
    "categoryId": "lwe-binary-error-algebraic-models",
    "paperId": "albrecht-cid-faugere-perret2014-algebraic",
    "year": 2014,
    "date": "2015-02-26",
    "datePrecision": "day",
    "time": {
      "coefficient": null,
      "expression": "O(m² · 2^{ωn log₂log₂log₂(n)/(8log₂log₂(n))})"
    },
    "space": {
      "coefficient": null,
      "expression": "O(m² · 2^{2n log₂log₂log₂(n)/(8log₂log₂(n))})"
    },
    "tags": [
      "binary error",
      "quasilinear samples",
      "conditional",
      "subexponential"
    ],
    "review": {
      "status": "source-checked"
    },
    "certificate": {
      "status": "not-provided",
      "lean": null
    },
    "notes": "Conditional subexponential bound in a quasilinear-sample regime. The hidden sample-count constant must be retained when instantiating the result.",
    "extra": {
      "parameters": "Odd prime q=q(n), polynomially bounded; source regime m=O(n log log n); 2≤ω<3.",
      "noise": "Independent uniform errors in {0,1}.",
      "secret": "Arbitrary s∈Z_q^n.",
      "samples": "Quasilinear samples m=O(n log log n), as specified in Theorem 7(iv); not a claim for every smaller m.",
      "objective": "Recover the secret via the generated quadratic equations.",
      "model": "Field operations; dense linear-algebra exponent ω.",
      "assumptions": "Assumption 2: semi-regularity.",
      "success": "Conditional on semi-regularity and the intended unique-solution regime."
    },
    "evidence": {
      "url": "https://iacr.steepath.eu/2014/1018-AlgebraicAlgorithmsforLWE.pdf",
      "locator": "§6, Theorem 7(iv), Eq. (12), p. 13",
      "sourceVersion": "ePrint 2014/1018 revised 26 February 2015"
    }
  }
]