  Information-efficient approaches for extracting randomness from imperfect
sources have been extensively studied, but simpler and faster ones are required
in the high-speed applications of random number generation. In this paper, we
focus on linear constructions, namely, applying linear transformation for
randomness extraction. We show that linear transformations based on sparse
random matrices are asymptotically optimal to extract randomness from
independent sources and bit-fixing sources, and they are efficient (may not be
optimal) to extract randomness from hidden Markov sources. Further study
demonstrates the flexibility of such constructions on source models as well as
their excellent information-preserving capabilities. Since linear
transformations based on sparse random matrices are computationally fast and
can be easy to implement using hardware like FPGAs, they are very attractive in
the high-speed applications. In addition, we explore explicit constructions of
transformation matrices. We show that the generator matrices of primitive BCH
codes are good choices, but linear transformations based on such matrices
require more computational time due to their high densities.
