Cryptography and Algorithmic Randomness

Algorithmic randomness and lowness

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0. This contrasts with a result of Ku cera and Terwijn 2] on sets that are low for the class of Martin-LL of random reals. The Cantor space 2 ! is the set of innnite binar...

Algorithmic Randomness and Complexity

Algorithmic Randomness and Computability

We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. Mathematics Subject Classification Primary 68Q30, 68Q15, 03D15, 03D25, 03D28, 03D30.

Kolmogorov Complexity and Algorithmic Randomness

This paper aims to provide a minimal introduction to algorithmic randomness. In particular, we cover the equivalent 1-randomness and MartinLöf randomness. After a brief review of relevant concepts in computability, we develop the basic theory of Kolmogorov complexity, including the KC theorem and more general notion of information content measures. We then build two natural definitions of rando...

Borel Normality and Algorithmic Randomness

We prove that all random sequences (in Chaitin-Martin-Lof sense) and almost all random strings (in both Kolmogorov-Chaitin and Chaitin senses) satisfy various conditions of normality ( rst introduced by Borel). All proofs are constructive.