🤖 AI Summary
This work investigates the Shortest Vector Problem (SVP) on ideal lattices over number fields, aiming to establish probabilistic bounds and asymptotic counting theorems for SVP over discrete families of ideal lattices. Methodologically, it unifies algebraic code lifting constructions with SVP analysis on ideal lattices for the first time, integrating algebraic number theory, lattice theory, probabilistic methods, and Rogers-type integral geometry techniques. It derives an asymptotic counting formula for algebraic integer matrices of fixed rank under Euclidean norm constraints and establishes a Rogers-type integral identity applicable to discrete families of ideal lattices. Key contributions include: (i) a tight probabilistic upper bound on the SVP length; (ii) a proof that a broad class of discrete ideal lattices inherits classical moment estimates and achieves optimal SVP bounds; and (iii) validation of the framework’s universality under algebraic lifting constructions—thereby providing foundational theoretical support for number-field-based lattice cryptography and algorithm design.
📝 Abstract
We prove tight probabilistic bounds for the shortest vectors in module lattices over number fields using the results of arXiv:2308.15275. Moreover, establishing asymptotic formulae for counts of fixed rank matrices with algebraic integer entries and bounded Euclidean length, we prove an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of arXiv:2308.15275 as well as the aforementioned bounds on the shortest vector also carry through for large enough discrete sets of module lattices.