5 research outputs found
On the Design of LIL Tests for (Pseudo) Random Generators and Some Experimental Results
NIST SP800-22 (2010) proposes the state of art testing suite for (pseudo)
random generators to detect deviations of a binary sequence from randomness. On
the one hand, as a counter example to NIST SP800-22 test suite, it is easy to
construct functions that are considered as GOOD pseudorandom generators by NIST
SP800-22 test suite though the output of these functions are easily
distinguishable from the uniform distribution. Thus these functions are not
pseudorandom generators by definition. On the other hand, NIST SP800-22 does
not cover some of the important laws for randomness. Two fundamental limit
theorems about random binary strings are the central limit theorem and the law
of the iterated logarithm (LIL). Several frequency related tests in NIST
SP800-22 cover the central limit theorem while no NIST SP800-22 test covers
LIL.
This paper proposes techniques to address the above challenges that NIST
SP800-22 testing suite faces. Firstly, we propose statistical distance based
testing techniques for (pseudo) random generators to reduce the above mentioned
Type II errors in NIST SP800-22 test suite. Secondly, we propose LIL based
statistical testing techniques, calculate the probabilities, and carry out
experimental tests on widely used pseudorandom generators by generating around
30TB of pseudorandom sequences. The experimental results show that for a sample
size of 1000 sequences (2TB), the statistical distance between the generated
sequences and the uniform distribution is around 0.07 (with for
statistically indistinguishable and for completely distinguishable) and the
root-mean-square deviation is around 0.005
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
Resource bounded randomness and computational complexity
AbstractThe following is a survey of resource bounded randomness concepts and their relations to each other. Further, we introduce several new resource bounded randomness concepts corresponding to the classical randomness concepts, and show that the notion of polynomial time bounded Ko randomness is independent of the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness. Lutz has conjectured that, for a given time or space bound, the corresponding resource bounded Lutz randomness is a proper refinement of resource bounded Schnorr randomness. This conjecture is answered for the case of polynomial time bound. Moreover, we will show that polynomial time bounded Schnorr randomness is a proper refinement of polynomial time bounded Kurtz randomness. In contrast to this result, we show that the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness coincide in the case of recursive sets, thus it suffices to study the notion of resource bounded Lutz randomness in the context of complexity theory. The stochastic properties of resource bounded random sequences will be discussed in detail. Schnorr has already shown that the law of large numbers holds for p-random sequences. We will show that another important law in probability theory, the law of the iterated logarithm, holds for p-random sequences too. Hence almost all sets in the exponential time complexity class are “hard” from the viewpoint of statistics
Resource Bounded Randomness and Computational Complexity
We give a survey of resource bounded randomness concepts and show their relations to each other. Moreover, we introduce several new resource bounded randomness concepts corresponding to the classical randomness concepts. We show that the notion of polynomial time bounded Ko randomness is independent of the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness. Lutz has conjectured that, for a given time or space bound, the corresponding resource bounded Lutz randomness is a proper refinement of resource bounded Schnorr randomness. We answer this conjecture for the case of polynomial time bound in this paper. Moreover, we show that polynomial time bounded Schnorr randomness is a proper refinement of polynomial time bounded Kurtz randomness too. In contrast to this result, however, we also show that the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness coincide in the case of recursive sets, whence it suffices to study the notion of resource bounded Lu..