Tag
Computational Number Theory
58 resources tagged “Computational Number Theory” across the knowledge library.
58 items · page 1 of 3
Guide6 Aug 2026Algebraic Numbers and Number FieldsAlgebraic numbers and integers, minimal polynomials, number fields as finite extensions of Q, real and complex embeddings, the signature, and the primitive element theorem.5 min readRead MoreGuide6 Aug 2026Algorithms for Elliptic Curves over ℚComputing minimal Weierstrass models, Tate's algorithm for reduction type and conductor, torsion subgroup determination, heights, and descent for rank computation.5 min readRead MoreGuide6 Aug 2026Applications of the LLL AlgorithmPractical applications of lattice reduction: integer kernel and image computation, integer relation detection, recovering minimal polynomials from numerical approximations, and simultaneous Diophantine approximation.5 min readRead MoreGuide6 Aug 2026Baby-Step Giant-Step and Class Group StructureThe baby-step giant-step method for discrete logarithms and group order, its application to class groups when an approximation to the order is available, and determination of group structure.4 min readRead MoreGuide6 Aug 2026Chinese Remainder Theorem AlgorithmsThe Chinese remainder theorem in constructive form, Garner's incremental algorithm, and the multi-modular strategy that controls coefficient growth in exact computation.4 min readRead MoreGuide6 Aug 2026Class Group and Unit Computation in General Number FieldsThe general sub-exponential algorithm for class groups and units: factor base selection, ideal reduction, relation collection, the relation matrix and its kernel, and verification against the analytic class number formula.5 min readRead MoreGuide6 Aug 2026Class Groups, Units and the RegulatorThe ideal class group, Dirichlet's unit theorem, the regulator, Minkowski's bound, and the analytic class number formula used to verify computed values.4 min readRead MoreGuide6 Aug 2026Class Numbers of Imaginary Quadratic FieldsComputing class numbers of imaginary quadratic fields by enumeration of reduced forms, by analytic class number formulas, and by modular form methods, with the Gauss class number problem.4 min readRead MoreGuide6 Aug 2026Classical Factoring: Trial Division, Fermat and LehmanTrial division and wheel factorisation, Fermat's difference of squares method, Lehman's improvement, and the role of these methods as a preprocessing stage.5 min readRead MoreGuide6 Aug 2026Classical Primality Proofs: Pocklington and LehmerThe Pocklington-Lehmer N−1 test, partial factorisation requirements, the N+1 test with Lucas sequences, and combined methods.4 min readRead MoreGuide6 Aug 2026Complex Multiplication and Class FieldsComplex multiplication, isogenies, the relation between CM curves and imaginary quadratic orders, Hilbert and Weber class polynomials, and the CM method for curve construction.5 min readRead MoreGuide6 Aug 2026Compositeness Tests: Fermat and Miller–RabinThe Fermat test and its failure on Carmichael numbers, the strong probable prime test of Miller-Rabin, error bounds, deterministic base sets for bounded ranges, and Baillie-PSW.4 min readRead MoreGuide6 Aug 2026Computational Algebraic Number Theory: Discipline OverviewStructural overview of computational algebraic number theory: the algorithm layers, the four central computational tasks of a number field, and how lattice reduction underpins the whole subject.6 min readRead MoreGuide6 Aug 2026Computing Galois Groups of Number FieldsThe resolvent method for determining Galois groups, the Frobenius cycle-type approach via Dedekind's theorem, transitive group classification by degree, and test polynomials.5 min readRead MoreGuide6 Aug 2026Computing the Maximal Order: the Round 2 AlgorithmThe Pohst-Zassenhaus theorem, the Dedekind criterion, the radical and the ring of multipliers, and the Round 2 algorithm for computing the ring of integers.5 min readRead MoreGuide6 Aug 2026Continued Fraction ExpansionsSimple continued fractions, convergent recurrences and best-approximation properties, Lagrange's periodicity theorem, and the expansion of a square root used for Pell's equation and factoring.4 min readRead MoreGuide6 Aug 2026Decomposition of Prime Numbers in Number FieldsRamification indices and residue degrees, the fundamental identity, Dedekind's theorem relating prime decomposition to polynomial factorisation modulo p, and computing valuations.5 min readRead MoreGuide6 Aug 2026Discriminants and Integral BasesThe field discriminant, integral bases, the index of an equation order, and why computing the maximal order reduces to factoring the polynomial discriminant.5 min readRead MoreGuide6 Aug 2026Elliptic Curve L-functions and the Birch–Swinnerton-Dyer ConjectureThe zeta function of a curve, the L-function as an Euler product, modularity and analytic continuation, and the Birch-Swinnerton-Dyer conjecture with its computational uses.4 min readRead MoreGuide6 Aug 2026Elliptic Curve Primality ProvingGoldwasser-Kilian and Atkin-Morain elliptic curve primality proving: the group order downstep, the CM method for avoiding point counting, certificate structure and verification.5 min readRead MoreGuide6 Aug 2026Elliptic Curves: Definitions and the Group LawWeierstrass forms, the discriminant and j-invariant, the group law with explicit formulas, torsion, and the structure of the group of points over finite fields.5 min readRead MoreGuide6 Aug 2026Factorisation of Polynomials Modulo a PrimeSquarefree decomposition, distinct-degree factorisation by gcd with x^(q^d) − x, equal-degree splitting by Cantor-Zassenhaus, and Berlekamp's linear-algebra approach.5 min readRead MoreGuide6 Aug 2026Gram–Schmidt OrthogonalisationThe Gram-Schmidt procedure, the mu coefficients and their role in reduction conditions, the numerical instability of the classical algorithm, and exact integral alternatives.4 min readRead MoreGuide6 Aug 2026Hensel Lifting and Factorisation over the IntegersHensel's lemma and quadratic lifting, Mignotte's coefficient bounds, the exponential recombination problem, and the LLL-based polynomial-time solution.5 min readRead More