algorithmic number theory
E898460
Algorithmic number theory is a branch of mathematics and computer science that designs and analyzes efficient algorithms for solving problems involving integers, prime numbers, and related number-theoretic structures.
Statements (57)
| Predicate | Object |
|---|---|
| instanceOf |
academic discipline
ⓘ
subfield of number theory ⓘ subfield of theoretical computer science ⓘ |
| appliedIn |
coding theory
ⓘ
computational algebraic geometry ⓘ computational complexity theory ⓘ computer algebra systems ⓘ cryptanalysis ⓘ public-key cryptography ⓘ |
| fieldOfStudy |
Diophantine equations
ⓘ
algebraic number fields ⓘ algorithms ⓘ arithmetic of integers ⓘ computational complexity ⓘ elliptic curves ⓘ finite fields ⓘ lattices in number theory ⓘ modular arithmetic ⓘ number theory ⓘ prime numbers ⓘ |
| goal |
analysis of correctness of number-theoretic algorithms
ⓘ
analysis of running time of arithmetic algorithms ⓘ design of efficient number-theoretic algorithms ⓘ |
| historicalDevelopment | grew rapidly with the rise of modern cryptography in the late 20th century ⓘ |
| notableAlgorithm |
AKS primality test
GENERATED
ⓘ
LLL lattice basis reduction algorithm GENERATED ⓘ Lenstra elliptic-curve factorization method GENERATED ⓘ Miller–Rabin primality test GENERATED ⓘ Pollard rho discrete logarithm algorithm GENERATED ⓘ Pollard rho factorization algorithm GENERATED ⓘ baby-step giant-step algorithm GENERATED ⓘ number field sieve GENERATED ⓘ quadratic sieve GENERATED ⓘ |
| relatedTo |
arithmetic geometry
ⓘ
complexity theory of numerical algorithms ⓘ computational number theory ⓘ symbolic computation ⓘ |
| studies |
Diophantine approximation algorithms
ⓘ
algorithms for computing class groups ⓘ algorithms for computing unit groups ⓘ computational aspects of algebraic number theory ⓘ discrete logarithm algorithms ⓘ efficient algorithms for arithmetic problems ⓘ greatest common divisor algorithms ⓘ integer factorization algorithms ⓘ integer relation algorithms ⓘ lattice basis reduction algorithms ⓘ modular exponentiation algorithms ⓘ point counting on elliptic curves ⓘ polynomial factorization over finite fields ⓘ primality testing algorithms ⓘ |
| usesConcept |
asymptotic complexity
ⓘ
deterministic algorithms ⓘ lattice reduction ⓘ probabilistic algorithms ⓘ randomized primality tests ⓘ sieve methods ⓘ |
Referenced by (1)
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