Done on 20121119:
  The linear system as we want to solve it in crypto applications. Give
  figures showing how it is infeasible to use dense algorithms.
  Definition of Krylov subspace.
  The Lanczos algorithm.  The Wiedemann algorithm. Blackboard proofs of a
  few basic facts about these algorithm. Notably, show that a generator
  can be regarded as a denominator for a rational fraction description of
  the power series.
  Block algorithms (sketch): block Lanczos, block Wiedemann. Show how
  block Wiedemann can be useful over distinct clusters.

  A proof of Lanczos' algorithm in a randomized setting can be obtained
  following Eberly and Kaltofen's 1997 paper ``on randomized Lanczos
  algorithm''.