[
  {
    "id": "albrecht17-scaled-sparse-dual",
    "categoryId": "lwe-sparse-secret-dual-models",
    "paperId": "albrecht2017-dual-small-secret",
    "year": 2017,
    "date": "2017-05-18",
    "datePrecision": "day",
    "time": {
      "coefficient": null,
      "expression": "Scaled-dual log₂ cost ≈ [2L/(2L+1)] · (unscaled-dual log₂ cost), in the stated sparse-FHE regime"
    },
    "space": {
      "coefficient": null,
      "expression": "Depends on the chosen dual-lattice solver."
    },
    "tags": [
      "dual",
      "sparse secret",
      "relative improvement",
      "FHE regime"
    ],
    "review": {
      "status": "source-checked"
    },
    "certificate": {
      "status": "not-provided",
      "lean": null
    },
    "notes": "A relative improvement in the source’s fixed-width sparse-FHE regime q=n^L·2^{O(L)}. It is not a universal exponent for LWE and need not beat exhaustive sparse-secret search when h is constant.",
    "extra": {
      "parameters": "q(n) with log₂q=L log₂n+O(L), fixed L>0, constant Gaussian width αq=8 and fixed sparse weight h (source example h=64); source scaled dimension m′=2n.",
      "noise": "Discrete Gaussian width αq=8, i.e. standard deviation 8/√(2π), independent of n.",
      "secret": "Uniform sparse ternary secret of fixed Hamming weight h, as in the source’s HElib asymptotic model.",
      "samples": "Enough samples for the selected dual-lattice dimension; actual m must be enforced when instantiating.",
      "objective": "Decision-LWE under a scaled dual-lattice attack.",
      "model": "Relative exponent scaling 2L/(2L+1) against the corresponding unscaled dual model.",
      "assumptions": "Source lattice-reduction and small-secret scaling model; fixed h and L are essential.",
      "success": "Use the same target distinguishing advantage in both compared dual cost models."
    },
    "evidence": {
      "url": "https://github.com/malb/publications/blob/2fc91ddbe45e21d7c337bfc658e4fd8926dd887c/on-dual-attacks-against-small-secret-lwe/small-sparse-secret.tex",
      "locator": "Abstract and Section Modulus switching / asymptotic behaviour",
      "sourceVersion": "Author TeX commit 2fc91ddbe45e21d7c337bfc658e4fd8926dd887c, 18 May 2017; associated ePrint last revised 6 May 2017"
    }
  },
  {
    "id": "albrecht17-sparse-dimension-halving",
    "categoryId": "lwe-sparse-secret-dual-models",
    "paperId": "albrecht2017-dual-small-secret",
    "year": 2017,
    "date": "2017-05-18",
    "datePrecision": "day",
    "time": {
      "coefficient": null,
      "expression": "O(2^h · C(n/2,α,q)); more generally O(p_{h,d}^{−1}·C(n−n/d,α,q))"
    },
    "space": {
      "coefficient": null,
      "expression": "Space of the reduced-dimension solver C."
    },
    "tags": [
      "sparse secret",
      "dimension reduction",
      "relative improvement"
    ],
    "review": {
      "status": "source-checked"
    },
    "certificate": {
      "status": "not-provided",
      "lean": null
    },
    "notes": "Exploits the event that a discarded coordinate subset misses the sparse support. C denotes a specified residual LWE solver, so this is a reduction/tradeoff rather than a standalone universal LWE exponent.",
    "extra": {
      "parameters": "Fix h≥0 and d>1 independent of n; p_{h,d}=lim_{n→∞}binom(n−h,n/d)/binom(n,n/d)=(1−1/d)^h.",
      "noise": "Original error distribution/width αq is retained by the zero-support branch.",
      "secret": "Uniform support of Hamming weight h with small ternary nonzero values; basic support argument does not depend on their magnitudes.",
      "samples": "Same original sample budget passed to residual instances; residual solver requirements remain explicit.",
      "objective": "Solve LWE using independent support-avoidance trials and a residual solver C.",
      "model": "C(N,α,q) is cost at the fixed required high success probability; d=2 gives p=2^{−h}.",
      "assumptions": "Uniform/random sparse support; high-success residual solver; fixed h,d.",
      "success": "Constant support-avoidance probability per trial, amplified by O(1/p_{h,d}) trials."
    },
    "evidence": {
      "url": "https://github.com/malb/publications/blob/2fc91ddbe45e21d7c337bfc658e4fd8926dd887c/on-dual-attacks-against-small-secret-lwe/small-sparse-secret.tex",
      "locator": "Section Sparse secrets, Asymptotic behaviour lemma and following d=2 remark",
      "sourceVersion": "Author TeX commit 2fc91ddbe45e21d7c337bfc658e4fd8926dd887c, 18 May 2017; associated ePrint last revised 6 May 2017"
    }
  }
]