This thesis is solving the shortest lattice vector problem by
finding the minimum of quadratic forms. We analyse cases where
the minimum of the quadratic form alone is not sufficient to
find the shortest vector. We also analyse Semaev's exact
algorithm for dimension 3 and propose a modified extension for
an arbitrary dimension. We then apply a single iteration of
this modification to the input basis of the LLL algorithm and
compare its speed against the base LLL version. Our
modification managed to reduce the number of iterations in
dimension 20 by 18 %.