Computational Number Theory

Primality, factoring, lattice algorithms, and number-theoretic cryptography.

foundation tier

Computational Number Theory. Primality, factoring, lattice algorithms, and number-theoretic cryptography.

Foundations and canonical references

The standard treatments of computational number theory approach the subject from complementary angles. Cohen, A Course in Computational Algebraic Number Theory (1993) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Crandall, Prime Numbers: A Computational Perspective (2005) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text.

Open methodological questions for computational number theory include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.

Prerequisites

Sources

  • textbook · primary · 1993
    A Course in Computational Algebraic Number Theory
    cohen-henri-1993
  • textbook · primary · 2005
    Prime Numbers: A Computational Perspective
    crandall-2005, pomerance-2005

In context

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Explore

  1. 01

    Primality Testing

    AKS, Miller–Rabin, and elliptic curve primality proving.
  2. 02

    Integer Factorization

    Number field sieve, ECM, and quantum algorithms.
  3. 03

    Lattice Algorithms

    LLL, BKZ, and SVP/CVP for cryptanalysis.

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