{
  "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"
  }
}