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Cyclotomic Cosets: Hidden Subgroup and Quantum Sieving Algorithm for Prime-Power Moduli

arXiv cs.CRby Mathias Boucher ()1 min read3 · 3 reads
Read at arxiv.org
AI summary · grok-4.7Summaries by Vani AI

A cyclotomic quantum sieve solves CCP and related EDCP cases quickly, but not standard LWE.

The paper introduces the Cyclotomic Coset Problem, a hidden-subgroup generalization of the dihedral coset problem over prime-power cyclotomic moduli. A quantum sieve solves CCP in time and sample complexity 2 to the O_p(log n log q) with polynomial quantum space, and extends to uniform EDCP and Gaussian S|LWE in quasi-polynomial time when q is polynomial in n. The authors stress this does not by itself give a quasi-polynomial algorithm for standard LWE, because existing reductions supply too few approximate CCP states.

Key facts
  • CCP preserves an exact hidden-subgroup structure, unlike extrapolated DCP.
  • The quantum sieve runs in 2 to the O(log n log q) time and samples.
  • Uniform EDCP and Gaussian S|LWE also get quasi-polynomial time when q is polynomial.
  • Authors say known reductions do not yield a quasi-polynomial attack on standard LWE.
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Tags
post-quantumlwequantum-algorithmhidden-subgroupcryptanalysiscyclotomic

From the original article

1 min read at arxiv.org

The Learning With Errors (LWE) problem is a fundamental assumption in post-quantum cryptography. Regev established a quantum reduction from LWE to the Dihedral Coset Problem (DCP). Later, Brakerski et al.

arXiv cs.CR

By Mathias Boucher · 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.