For context, LLL was used to prove that polynomial factorization can be solved in polynomial time. That is, given r(x) such that:

    r(x) = p(x) * q(x)

where the coefficients of each factor are integral:

    p_i, q_i in Z

LLL can be used to find p(x) and q(x) only given r(x).

There are other problems that LLL solves and there are more modern lattice reduction algorithms (PSLQ, etc.) but LLL was one of the first.