Abstract-Linear transformations have many applications in information theory, like data compression and error-correcting codes design. In this paper, we study the power of linear transformations in randomness extraction, namely linear extractors, as another important application. Comparing to most existing methods for randomness extraction, linear extractors (especially those constructed with sparse matrices) are computationally fast and can be simply implemented with hardware like FPGAs, which makes them very attractive in practical use. We mainly focus on simple, efficient and sparse constructions of linear extractors. Specifically, we demonstrate that random matrices can generate random bits very efficiently from a variety of noisy sources, including noisy coin sources, bit-fixing sources, noisy (hidden) Markov sources, as well as their mixtures. It shows that lowdensity random matrices have almost the same efficiency as highdensity random matrices when the input sequence is long, which provides a way to simplify hardware/software implementation. Note that although we constructed matrices with randomness, they are deterministic (seedless) extractors -once we constructed them, the same construction can be used for any number of times without using any seeds. Another way to construct linear extractors is based on generator matrices of primitive BCH codes. This method is more explicit, but less practical due to its computational complexity and dimensional constraints.
Sparse extractors for restricted sources Motivated by certain applications, Zhou and Bruck REF show that low density random matrices can efficiently extract random bits from some restricted noisy sources, such as bit fixing sources and Markov sources.
Linear extractors for extracting randomness from noisy sources
2011 IEEE International Symposium on Information Theory Proceedings
2011 IEEE International Symposium on Information Theory Proceedings
Computer Science