{
  "id": "oai:papers.qnfo.org:factoring-adelic-complexity-v2",
  "slug": "factoring-adelic-complexity-v2",
  "title": "FACTORING, Adelic Complexity, and the Silent-Radix Principle",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The silent radix -- the observation that no positional numeral internally specifies its own base -- was previously explored as a cryptographic primitive through Silent Radix Encryption (SRE), a key encapsulation mechanism whose security rests on the hardness of recovering a secret base from public envelope digits. That scheme, while closing the known-plaintext attack vector via digit randomization, remains structurally linear: without noise, small numbers of ciphertexts leak algebraic constraints that lattice reduction can exploit. We present Noise-Augmented Silent Radix Encryption (N-SRE), which injects small p-adic errors into the envelope digits and formalizes the underlying hard problem as Hidden-Modulus Learning With Errors (HM-LWE). We prove that HM-LWE reduces to standard LWE when the modulus p is known to the adversary, and we argue that the unknown-modulus variant resists lattice reduction because the adversary lacks the correct geometry in which to construct a short vector. We provide a concrete KEM construction, analyze its security against known-plaintext, ciphertext-only, and lattice-based attacks, and derive parameter recommendations for classical 128-bit and post-quantum 256-bit security levels. While N-SRE does not yet match the bandwidth efficiency of NIST-standardized lattice KEMs, it demonstrates that the valuation-theoretic design space -- hiding not just a parameter but the mathematical topology of interpretation -- can support noise-augmented constructions with plausible hardness foundations.",
  "abstract_markdown": "The silent radix -- the observation that no positional numeral internally specifies its own base -- was previously explored as a cryptographic primitive through Silent Radix Encryption (SRE), a key encapsulation mechanism whose security rests on the hardness of recovering a secret base from public envelope digits. That scheme, while closing the known-plaintext attack vector via digit randomization, remains structurally linear: without noise, small numbers of ciphertexts leak algebraic constraints that lattice reduction can exploit. We present Noise-Augmented Silent Radix Encryption (N-SRE), which injects small $p$-adic errors into the envelope digits and formalizes the underlying hard problem as Hidden-Modulus Learning With Errors (HM-LWE). We prove that HM-LWE reduces to standard LWE when the modulus $p$ is known to the adversary, and we argue that the unknown-modulus variant resists lattice reduction because the adversary lacks the correct geometry in which to construct a short vector. We provide a concrete KEM construction, analyze its security against known-plaintext, ciphertext-only, and lattice-based attacks, and derive parameter recommendations for classical 128-bit and post-quantum 256-bit security levels. While N-SRE does not yet match the bandwidth efficiency of NIST-standardized lattice KEMs, it demonstrates that the valuation-theoretic design space -- hiding not just a parameter but the mathematical topology of interpretation -- can support noise-augmented constructions with plausible hardness foundations.",
  "date": "2026-07-30",
  "first_published": "2026-07-30",
  "last_revised": "2026-07-30T00:00:00Z",
  "modified": "2026-07-30T00:00:00Z",
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  "language": "en",
  "keywords": [
    "silent radix",
    "hidden radix",
    "$p$-adic LWE",
    "hidden modulus",
    "noise-augmented encryption",
    "key encapsulation mechanism",
    "post-quantum cryptography",
    "lattice reduction",
    "ultrametric"
  ],
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    "versioned research publication"
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  "url": "https://papers.qnfo.org/papers/factoring-adelic-complexity-v2",
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  "version_url": "https://papers.qnfo.org/papers/factoring-adelic-complexity-v2/v/1.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "FACTORING, Adelic Complexity, and the Silent-Radix Principle"
      }
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    "publicationYear": "2026",
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        "subject": "silent radix"
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        "subject": "hidden radix"
      },
      {
        "subject": "$p$-adic LWE"
      },
      {
        "subject": "hidden modulus"
      },
      {
        "subject": "noise-augmented encryption"
      },
      {
        "subject": "key encapsulation mechanism"
      },
      {
        "subject": "post-quantum cryptography"
      },
      {
        "subject": "lattice reduction"
      },
      {
        "subject": "ultrametric"
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        "description": "The silent radix -- the observation that no positional numeral internally specifies its own base -- was previously explored as a cryptographic primitive through Silent Radix Encryption (SRE), a key encapsulation mechanism whose security rests on the hardness of recovering a secret base from public envelope digits. That scheme, while closing the known-plaintext attack vector via digit randomization, remains structurally linear: without noise, small numbers of ciphertexts leak algebraic constraints that lattice reduction can exploit. We present Noise-Augmented Silent Radix Encryption (N-SRE), which injects small p-adic errors into the envelope digits and formalizes the underlying hard problem as Hidden-Modulus Learning With Errors (HM-LWE). We prove that HM-LWE reduces to standard LWE when the modulus p is known to the adversary, and we argue that the unknown-modulus variant resists lattice reduction because the adversary lacks the correct geometry in which to construct a short vector. We provide a concrete KEM construction, analyze its security against known-plaintext, ciphertext-only, and lattice-based attacks, and derive parameter recommendations for classical 128-bit and post-quantum 256-bit security levels. While N-SRE does not yet match the bandwidth efficiency of NIST-standardized lattice KEMs, it demonstrates that the valuation-theoretic design space -- hiding not just a parameter but the mathematical topology of interpretation -- can support noise-augmented constructions with plausible hardness foundations.",
        "descriptionType": "Abstract"
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