What is recorded here?
Lenstra, Lenstra and Lovász gave a polynomial-time lattice-basis reduction algorithm with short-vector approximation guarantees and used it to factor univariate polynomials over the rational numbers in polynomial time. A lattice is a regular collection of points generated by integer combinations of basis vectors. A poor basis can hide short vectors; the 1982 LLL paper gave a polynomial-time way to replace it with a better basis while preserving the same lattice. The resulting vectors are provably short relative to the shortest possibilities, although they need not be exactly shortest. The authors then represented a polynomial factor as a small vector in a lattice defined by modular divisibility. Enough modular precision separates genuine factors from other vectors, giving a polynomial-time method to factor univariate polynomials over the rational numbers. The algorithm uses earlier finite-field factorization, Hensel lifting and polynomial-size bounds, which retain their own credit. Integer factorization is a different problem: the integer content of a nonprimitive polynomial is not magically factored by this theorem. Later stronger reduction methods, cryptographic uses and faster implementations remain separate contributions. The paper also credits Cuppen for an algorithmic improvement without adding him to its three-author byline.
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Scientific scope
The algorithm reduces a lattice basis under specified rational/integer input and size-reduction/Lovász conditions; it does not generally find an exact shortest vector. Its polynomial bit bound includes input size. Polynomial factorization uses its own coefficient and precision assumptions. Later stronger reductions, floating-point variants and practical cryptographic successes are separate.
Read the original work
- Lenstra, Lenstra and Lovasz, Factoring Polynomials with Rational Coefficients, MathAnn261(4)515-534(1982), author-hosted journal original
- Publisher Crossref deposit for10.1007/BF01457454
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People and systems
A. K. Lenstra; H. W. Lenstra Jr.; L. Lovász
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English catalogue narrative
LLL: find short lattice vectors and factor rational polynomials efficiently
Lenstra, Lenstra and Lovász gave a polynomial-time lattice-basis reduction algorithm with short-vector approximation guarantees and used it to factor univariate polynomials over the rational numbers in polynomial time.
A lattice is a regular collection of points generated by integer combinations of basis vectors. A poor basis can hide short vectors; the 1982 LLL paper gave a polynomial-time way to replace it with a better basis while preserving the same lattice. The resulting vectors are provably short relative to the shortest possibilities, although they need not be exactly shortest. The authors then represented a polynomial factor as a small vector in a lattice defined by modular divisibility. Enough modular precision separates genuine factors from other vectors, giving a polynomial-time method to factor univariate polynomials over the rational numbers. The algorithm uses earlier finite-field factorization, Hensel lifting and polynomial-size bounds, which retain their own credit. Integer factorization is a different problem: the integer content of a nonprimitive polynomial is not magically factored by this theorem. Later stronger reduction methods, cryptographic uses and faster implementations remain separate contributions. The paper also credits Cuppen for an algorithmic improvement without adding him to its three-author byline.
Scientific scope
The algorithm reduces a lattice basis under specified rational/integer input and size-reduction/Lovász conditions; it does not generally find an exact shortest vector. Its polynomial bit bound includes input size. Polynomial factorization uses its own coefficient and precision assumptions. Later stronger reductions, floating-point variants and practical cryptographic successes are separate.
People and systems
A. K. Lenstra; H. W. Lenstra Jr.; L. Lovász