Short Element and Small Representative

ShortElement(I): AlgEtQIdl -> AlgEtQElt

Given an ideal \(I\) returns a non-zerodivisor in \(I\) with small coefficients (in the LLL sense). This is achieved by randomly picking an element with small coefficients in a LLL-reduced basis (wrt the T2 norm as a \({{\Bbb Z}}\)-lattice).

SmallRepresentative(I): AlgEtQIdl -> AlgEtQIdl, AlgEtQElt

Given a fractional \(R\)-ideal \(I\), it returns an isomorphic ideal \(a \cdot I\), and the element \(a\), such that \(a \cdot I\) is a subset of \(R\), and the cardinality of \(R/aI\) is small. This is achieved by computing the ShortElement \(a\) of \((R:I)\). Note that if \(I\) is invertible \(R/aI\) is isomorphic to \((R:I)/aR\).