Tag
Foundational Algorithms
11 resources tagged “Foundational Algorithms” across the knowledge library.
11 items
Guide6 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 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 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 2026Integer Square Roots and Perfect Power DetectionComputing exact integer square roots by Newton's method, fast rejection of non-squares by modular filters, and detection of perfect powers and prime powers.5 min readRead MoreGuide6 Aug 2026Legendre, Jacobi and Kronecker SymbolsQuadratic residues, Euler's criterion, the Legendre symbol, its Jacobi and Kronecker extensions, and the reciprocity-based algorithm that evaluates them in logarithmic time.4 min readRead MoreGuide6 Aug 2026Modular Exponentiation and Powering AlgorithmsLeft-to-right and right-to-left binary powering, sliding window exponentiation, addition chains, and the generic monoid formulation that makes one routine serve many algebraic structures.6 min readRead MoreGuide6 Aug 2026Multiprecision Integer ArithmeticRepresentation of multiprecision integers, schoolbook and fast multiplication, division, modular reduction strategies, and how to choose the right base ring for a computation.6 min readRead MoreGuide6 Aug 2026Solving Polynomial Equations Modulo pRoot-finding modulo a prime: the gcd with x^p − x, equal-degree splitting by random shifts, and the special low-degree cases that admit closed forms.4 min readRead MoreGuide6 Aug 2026Square Roots Modulo a PrimeComputing modular square roots: the p ≡ 3 (mod 4) shortcut, the Tonelli–Shanks algorithm and its 2-adic structure, and Cornacchia's algorithm for x² + dy² = p.4 min readRead MoreGuide6 Aug 2026The Euclidean Algorithm and GCD ComputationEuclid's algorithm and its worst case, the binary GCD, Lehmer's method for multiprecision operands, and how to choose between them by operand size.6 min readRead MoreGuide6 Aug 2026The Extended Euclidean Algorithm and Modular InversesThe extended Euclidean algorithm, its loop invariants, half-extended variants, modular inversion, and rational reconstruction from a partial remainder sequence.5 min readRead More