Linear extractors for extracting randomness from noisy sources

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 low-density random matrices have almost the same efficiency as high-density 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

Code generator matrices as RNG conditioners

We quantify precisely the distribution of the output of a binary random number generator (RNG) after conditioning with a binary linear code generator matrix by showing the connection between the Walsh spectrum of the resulting random variable and the weight distribution of the code. Previously known bounds on the performance of linear binary codes as entropy extractors can be derived by considering generator matrices as a selector of a subset of that spectrum. We also extend this framework to the case of non-binary codes

Physical key-protected one-time pad

We describe an encrypted communication principle that forms a secure link between two parties without electronically saving either of their keys. Instead, random cryptographic bits are kept safe within the unique mesoscopic randomness of two volumetric scattering materials. We demonstrate how a shared set of patterned optical probes can generate 10 gigabits of statistically verified randomness between a pair of unique 2 mm^3 scattering objects. This shared randomness is used to facilitate information-theoretically secure communication following a modified one-time pad protocol. Benefits of volumetric physical storage over electronic memory include the inability to probe, duplicate or selectively reset any bits without fundamentally altering the entire key space. Our ability to securely couple the randomness contained within two unique physical objects can extend to strengthen hardware required by a variety of cryptographic protocols, which is currently a critically weak link in the security pipeline of our increasingly mobile communication culture