{"paper":{"title":"Module Lattice Security (Part III): Structured CVP Distance on the Log-Unit Lattice","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The L² CVP distance from a random short ring element to the log-unit lattice converges to π/(2√6) √n as the dimension n tends to infinity.","cross_cats":["cs.CR","math.NT","math.ST","quant-ph","stat.TH"],"primary_cat":"cs.DS","authors_text":"Ming-Xing Luo","submitted_at":"2026-05-17T12:00:59Z","abstract_excerpt":"We prove that the $L^2$ CVP distance from a random short ring element to the log-unit lattice of $\\Q(\\zeta_{2^k})$ converges to $\\frac{\\pi}{2\\sqrt{6}}\\sqrt{n}$ as $n=2^{k-1}\\to\\infty$. We then show that this target lies inside the Voronoi cell of the origin for $k\\ge 4$. For the $L^\\infty$ norm, the maximum over $n$ sub-Gaussian coordinates yields $O(\\sqrt{\\log n})$ which translates into a sub-polynomial approximation factor for the Short Generator Problem. We show a Coarse Lattice Theorem that Babai's algorithm returns zero for all structured targets, yet exactly recovers unit perturbations o"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the L² CVP distance from a random short ring element to the log-unit lattice of Q(ζ_{2^k}) converges to π/(2√6) √n as n=2^{k-1}→∞. ... combined with Parts I and II, we reduce the CDPR factor for ML-KEM from exp(O(√n)) to a sub-polynomial value.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The model of a 'random short ring element' in the ring of integers of Q(ζ_{2^k}) is sufficiently representative that its embedding statistics match the sub-Gaussian coordinates used to derive the limit and the Voronoi cell membership (abstract, first sentence and L^∞ 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