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Hamming Ideals and Grobner Bases for ISD-like Syndrome Decoding

arXiv cs.CRby Roberto La Scala ()1 min read11 · 2 reads
Read at arxiv.org
AI summary · glm-5.3-flashSummaries by Vani AI

Researchers combine Grobner bases with Information Set Decoding for syndrome decoding, testing feasibility against Classic McEliece NIST Category 1 parameters.

The paper proposes GBDecode, an ISD-like decoding algorithm that fixes only a subset of an information set and solves the resulting multivariate nonlinear systems via MultiSolve, which replaces one Grobner basis computation with many computations on simpler systems. Hamming weight constraints are reformulated using elementary symmetric functions and Lucas' identity factorizations to bound equation degree. Experiments on random binary linear codes use parameters matching the NIST Security Category 1 set of the Classic McEliece cryptosystem, assessing practical feasibility rather than breaking the scheme.

Key facts
  • GBDecode generalizes ISD by fixing only a subset of an information set, reducing combinatorial search space.
  • MultiSolve replaces single Grobner basis computations with many smaller systems via exhaustive finite-field assignments.
  • Hamming variety equations derived via elementary symmetric functions; bounded degree achieved using Lucas' identity factorizations.
  • Evaluated on Classic McEliece NIST Security Category 1 parameters; feasibility assessed, cryptosystem not broken.
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Tags
classic-mceliececryptanalysisgrobner-basesinformation-set-decodingpost-quantum-cryptographysyndrome-decoding
Organizations
NIST

From the original article

1 min read at arxiv.org

We investigate an algebraic approach to the Syndrome Decoding Problem, based on a reformulation of the Hamming weight constraint and its integration with the Information Set Decoding paradigm. We begin with a systematic analysis of the Hamming variety, deriving its defining equations in terms of elementary symmetric functions.

arXiv cs.CR

By Roberto La Scala · arxiv.org

Read the full article at arXiv cs.CR

A short excerpt; the article and its images belong to the publisher. Summary, key facts and assessment above are AI-written from the article and can be wrong.