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Arjen K. Lenstra works in computational number theory and cryptography. With Hendrik W. Lenstra and László Lovász, he authored the 1982 paper introducing the LLL algorithm and applying lattice reduction to factoring polynomials with rational number coefficients in polynomial time.^[ref]

A rational factorization can be converted to an integer-coefficient problem, then studied through modular information and short lattice vectors. Reduction helps recover an exact factor from that indirect arithmetic data. The achievement concerns polynomial factorization; it does not provide a polynomial-time algorithm for general integer factorization. His EPFL interview describes his introduction to randomized algorithms and his approach to teaching mathematical computation.^[ref]

Hendrik W. Lenstra is a mathematician working in computational number theory. He introduced the elliptic-curve factorization method, replacing the multiplicative groups in an earlier factorization approach by groups of points on variable elliptic curves.^[ref]

The method computes modulo a composite integer $N$ without first knowing its prime divisors. If an elliptic-curve operation requires inverting a denominator $d$ with $1<\gcd (d,N)<N$, the failed inversion reveals a proper divisor. Varying the elliptic curve varies the group orders modulo the unknown primes, offering repeated chances for a favourable order. He also coauthored the LLL algorithm with Arjen K. Lenstra and László Lovász.^[ref]

László Lovász is a mathematician whose work connects combinatorics, graph theory and efficient algorithms. With Arjen K. Lenstra and Hendrik W. Lenstra, he developed the LLL algorithm, which replaces a basis of a Euclidean lattice by a more geometrically controlled basis of the same lattice. Their 1982 paper uses lattice reduction to factor polynomials over the rational numbers in polynomial time.^[ref]

The point of reduction is not to change the lattice: integer basis changes preserve it. Instead, shorter, better organized vectors expose relations hidden by a poor basis. This is a striking bridge from geometric reasoning to exact algebraic computation. Lovász shared the 2021 Abel Prize with Avi Wigderson for contributions linking discrete mathematics and theoretical computer science.^[ref]

In the geometry of numbers, a lattice algorithm works with a Euclidean lattice, usually specified by linearly independent vectors $b_1,\dots ,b_m$ whose integer linear combinations formThe basis of a Euclidean lattice is not unique. For example, $(1,0)$ and $(100,1)$ generate the same Euclidean lattice as $(1,0)$ and $(0,1)$: replace the second vector by itself minus 100 times the first. This integer linear combination is reversible, so it changes the representation without changing the set of Euclidean lattice points.

Lattice reduction seeks a more useful basis of a Euclidean lattice, with shorter and less nearly dependent vectors. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm combines integer basis changes with Gram-Schmidt orthogonalization and comparisons controlling the quality of the result. It is a polynomial-time example, but it does not generally return an exactly shortest vector.^[ref] Better bases make geometric searches and lattice-based cryptography analysis more practical. The geometry and the sizes of the integer coordinates both matter: a small number of arithmetic steps need not imply a small bit complexity.