A Pseudorandom Sequence Generated over a Finite Field Using The Möbius Function

The aim of this paper is to generate and examine a pseudorandom sequence over a finite field using the Möbius function. In the main part of the paper, after generating a number of sequences using the Möbius function, we examine the sequences’ pseudorandomness using autocorrelation and prove that the second half of any sequence in $\mathbb{F}_{3^n}$ is the same as the first, but for the sign of the terms. I reach the conclusion, that it is preferable to generate sequences in fields of the form $\mathbb{F}_{3^n}$, thereby obtaining a sequence of the numbers $-1$,$0$,$1$, each of which appear in the same amounts. There is a variety of applications of the discussed pseudorandom generator and other generators such as cryptography or randomized algorithms.

Quantum generators of random numbers

Generation of random numbers is a central problem for many applications in the field of information processing, including, e.g., cryptography, in classical and quantum regime, but also mathematical modeling, Monte Carlo methods, gambling and many others. Both, the quality of the randomness and efficiency of the random numbers generation process are crucial for the most of these applications. Software produced pseudorandom bit sequences, though sufficiently quick, do not fulfill required randomness quality demands. Hence, the physical hardware methods are intensively developed to generate truly random number sequences for information processing and electronic security application. In the present paper we discuss the idea of the quantum random number generators. We also present a variety of tests utilized to assess the quality of randomness of generated bit sequences. In the experimental part we apply such tests to assess and compare two quantum random number generators, PQ4000KSI (of company ComScire US) and JUR01 (constructed in Wroclaw University of Science and Technology upon the project of The National Center for Research and Development) as well as a pseudorandom generator from the Mathematica Wolfram package. Finally, we present our new prototype of fully operative miniaturized quantum random generator JUR02 producing a random bit sequence with velocity of 1 Mb/s, which successfully passed standard tests of randomness quality (like NIST and Dieharder tests). We also shortly discuss our former concept of an entanglement-based quantum random number generator protocol with unconditionally secure public randomness verification.

A General-Purpose Uniform Random Number Package

As computer capacities and simulation technologies advance, simulation has become the method of choice for modeling and analysis. The fundamental advantage of simulation is that it can tolerate far less restrictive modeling assumptions, leading to an underlying model that is more reflective of reality and thus more valid, leading to better decisions. Simulation studies are typically preceded by transforming in a more or less complicated way of a sequence of numbers between 0 and 1 produced by a pseudorandom generator into an observation of the measure of interest. Random numbers are a fundamental resource in science and technology. A facility for generating sequences of pseudorandom numbers is a fundamental part of computer simulation systems. Furthermore, random number generators also play an important role in cryptography and in the blockchain ecosystem. All samples of the sequence are generated independently of each other, and the value of the next sample in the sequence cannot be predicted, regardless of how many samples have already been produced.

Bootstrapping variables in algebraic circuits

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One needs only to consider size-s degree-s circuits that depend on the firstlog○c svariables (where c is a constant and composes a logarithm with itself c times). Thus, the hitting-set generator (hsg) manifests a bootstrapping behavior—a partial hsg against very few variables can be efficiently grown to a complete hsg. A Boolean analog, or a pseudorandom generator property of this type, is unheard of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w.r.t. variables. This is repeated c times in an efficient way. Pushing the envelope further we show that (i) a quadratic-time blackbox PIT for 6,913-variate degree-s size-s polynomials will lead to a “near”-complete derandomization of PIT and (ii) a blackbox PIT for n-variate degree-s size-s circuits insnδtime, forδ<1/2, will lead to a near-complete derandomization of PIT (in contrast,sntime is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable,O(s1.49)-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of Kabanets and Impagliazzo [Kabanets V, Impagliazzo R (2003)Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing STOC ’03].