{
  "abstract": "We study block sensitivity and certificate complexity of rotation symmetric Boolean functions, with particular emphasis on monomial rotation symmetric Boolean functions. We first show that every odd-degree monomial rotation symmetric Boolean function on $n$ variables satisfies $s=bs=C=n$. We then determine sensitivity, block sensitivity, and certificate complexity for every quadratic monomial rotation symmetric Boolean function. For the quadratic function $Q_{n,k}$ generated by $x_0x_k$, we prove, outside the short-orbit case, that \\[ C(Q_{n,k})= \\begin{cases} n,&4\\mid n/\\gcd(n,k),\\\\ n-\\gcd(n,k),&4\\nmid n/\\gcd(n,k). \\end{cases} \\] Moreover, \\(bs(Q_{n,k})=C(Q_{n,k})\\) for every quadratic MRS function, and outside the short-orbit case, strict separation \\(s<bs=C\\) occurs exactly when \\[ \\frac{n}{\\gcd(n,k)}\\equiv2\\pmod4. \\] Finally, for the consecutive quartic family \\[ R_n(x)=\\bigoplus_{i\\in\\mathbb Z_n}x_ix_{i+1}x_{i+2}x_{i+3}, \\] we determine all three measures in every residue class modulo $6$, using the sensitivity formulas and, in one residue class, finite-state results established in our companion preprint~\\cite{LiZhangRSBFSensitivity}. In particular, for $r=2,3$ and $6q+r\\ge5$ we obtain strict separation, \\[ \\boxed{s(R_{6q+r})=6q<bs(R_{6q+r})=C(R_{6q+r})=6q+1,\\qquad r=2,3.} \\] Thus this family gives an infinite collection of rotation symmetric Boolean functions for which sensitivity is strictly smaller than block sensitivity, while block sensitivity and certificate complexity coincide.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17100",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuan Li"
    },
    {
      "name": "Jing Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fk6qsf0qkdfs4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17100",
  "fulltext_url": null,
  "id": "fk6qsf0qkdfs4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17100",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221794983",
  "pdf_url": "https://arxiv.org/pdf/2610.17100",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fk6qsf0qkdfs4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17100",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17100"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Block Sensitivity and Certificate Complexity of Rotation Symmetric Boolean Functions",
  "title_text": "Block Sensitivity and Certificate Complexity of Rotation Symmetric Boolean Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/fk6qsf0qkdfs4",
  "venue": "arXiv",
  "versions": []
}