ASYMPTOTIC ANALYSIS OF PERES’ ALGORITHM FOR RANDOM NUMBER GENERATION

2020 ◽  
pp. 1-16
Author(s):  
Zhao Ging Lim ◽  
Chen-Tuo Liao ◽  
Yi-Ching Yao
Keyword(s):  
Input Sequence ◽  
Simple Algorithm ◽  
Random Number Generation ◽  
Positive Sequence ◽  
Log P ◽  
National Bureau ◽  
Open Problems ◽  
Von Neumann ◽  
Entropy Bound ◽  
Regularly Varying

von Neumann [(1951). Various techniques used in connection with random digits. National Bureau of Standards Applied Math Series 12: 36–38] introduced a simple algorithm for generating independent unbiased random bits by tossing a (possibly) biased coin with unknown bias. While his algorithm fails to attain the entropy bound, Peres [(1992). Iterating von Neumann's procedure for extracting random bits. The Annals of Statistics 20(1): 590–597] showed that the entropy bound can be attained asymptotically by iterating von Neumann's algorithm. Let $b(n,p)$ denote the expected number of unbiased bits generated when Peres’ algorithm is applied to an input sequence consisting of the outcomes of $n$ tosses of the coin with bias $p$ . With $p=1/2$ , the coin is unbiased and the input sequence consists of $n$ unbiased bits, so that $n-b(n,1/2)$ may be referred to as the cost incurred by Peres’ algorithm when not knowing $p=1/2$ . We show that $\lim _{n\to \infty }\log [n-b(n,1/2)]/\log n =\theta =\log [(1+\sqrt {5})/2]$ (where $\log$ is the logarithm to base $2$ ), which together with limited numerical results suggests that $n-b(n,1/2)$ may be a regularly varying sequence of index $\theta$ . (A positive sequence $\{L(n)\}$ is said to be regularly varying of index $\theta$ if $\lim _{n\to \infty }L(\lfloor \lambda n\rfloor )/L(n)=\lambda ^\theta$ for all $\lambda > 0$ , where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$ .) Some open problems on the asymptotic behavior of $nh(p)-b(n,p)$ are briefly discussed where $h(p)=-p\log p- (1-p)\log (1-p)$ denotes the Shannon entropy of a random bit with bias $p$ .

scholarly journals Numerical and Non-Asymptotic Analysis of Elias’s and Peres’s Extractors with Finite Input Sequences

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 729 ◽  
Author(s):  
Amonrat Prasitsupparote ◽  
Norio Konno ◽  
Junji Shikata
Keyword(s):  
Asymptotic Analysis ◽  
Block Size ◽  
Input Sequence ◽  
Optimal Rate ◽  
Modern World ◽  
Random Numbers ◽  
Von Neumann ◽  
Information Theoretic ◽  
Running Time ◽  
Better Than

Many cryptographic systems require random numbers, and the use of weak random numbers leads to insecure systems. In the modern world, there are several techniques for generating random numbers, of which the most fundamental and important methods are deterministic extractors proposed by von Neumann, Elias, and Peres. Elias’s extractor achieves the optimal rate (i.e., information-theoretic upper bound) h ( p ) if the block size tends to infinity, where h ( · ) is the binary entropy function and p is the probability that each bit of input sequences occurs. Peres’s extractor achieves the optimal rate h ( p ) if the length of the input and the number of iterations tend to infinity. Previous research related to both extractors has made no reference to practical aspects including running time and memory size with finite input sequences. In this paper, based on some heuristics, we derive a lower bound on the maximum redundancy of Peres’s extractor, and we show that Elias’s extractor is better than Peres’s extractor in terms of the maximum redundancy (or the rates) if we do not pay attention to the time complexity or space complexity. In addition, we perform numerical and non-asymptotic analysis of both extractors with a finite input sequence with any biased probability under the same environments. To do so, we implemented both extractors on a general PC and simple environments. Our empirical results show that Peres’s extractor is much better than Elias’s extractor for given finite input sequences under a very similar running time. As a consequence, Peres’s extractor would be more suitable to generate uniformly random sequences in practice in applications such as cryptographic systems.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

scholarly journals Numerical and Non-Asymptotic Analysis of Elias’s and Peres’s Extractors with Finite Input Sequence

2018 ◽  
Author(s):  
Amonrat Prasitsupparote ◽  
Norio Konno ◽  
Junji Shikata
Keyword(s):  
Asymptotic Analysis ◽  
Block Size ◽  
Input Sequence ◽  
Optimal Rate ◽  
Modern World ◽  
Random Numbers ◽  
Von Neumann ◽  
Information Theoretic ◽  
Running Time ◽  
Better Than

Many cryptographic systems require random numbers, and weak random numbers lead to insecure systems. In the modern world, there are several techniques for generating random numbers, of which the most fundamental and important methods are deterministic extractors proposed by von Neumann, Elias, and Peres. Elias’s extractor achieves the optimal rate (i.e., information theoretic upper bound) h(p) if the block size tends to infinity, where h(·) is the binary entropy function and p is probability that each bit of input sequences occurs. Peres’s extractor achieves the optimal rate h(p) if the length of input and the number of iterations tend to infinity. The previous researches related to both extractors did not mention practical aspects including running time and memory-size with finite input sequences. In this paper, based on some heuristics, we derive a lower bound on the maximum redundancy of Peres’s extractor, and we show that Elias’s extractor is better than Peres’s one in terms of the maximum redundancy (or the rates) if we do not pay attention to time complexity or space complexity. In addition, we perform numerical and non-asymptotic analysis of both extractors with a finite input sequence with any biased probability under the same environments. For doing it, we implemented both extractors on a general PC and simple environments. Our empirical results show that Peres’s extractor is much better than Elias’s one for given finite input sequences under the almost same running time. As a consequence, Peres’s extractor would be more suitable to generate uniformly random sequences in practice in applications such as cryptographic systems.

scholarly journals ASYNCHRONOUS AUTOMATA NETWORKS CAN EMULATE ANY SYNCHRONOUS AUTOMATA NETWORK

2004 ◽  
Vol 14 (05n06) ◽  
pp. 719-739 ◽  
Author(s):  
CHRYSTOPHER L. NEHANIV
Keyword(s):  
Cellular Automaton ◽  
Local Time ◽  
Input Sequence ◽  
Clock Signal ◽  
Global State ◽  
Time Step ◽  
Open Problems ◽  
Locally Finite ◽  
Asynchronous Automata ◽  
Automata Network

We show that any locally finite automata network [Formula: see text] with global synchronous updates can be emulated by another one [Formula: see text], whose structure derives from that of [Formula: see text] by a simple construction, but whose updates are made asynchronously at its various component automata (e.g. possibly randomly or sequentially, with or without possible simultaneous updates at different nodes). By "emulation", we refer to the existence of a spatial-temporal covering 'local time', allowing one to project the behavior of [Formula: see text] continuously onto that of [Formula: see text]. We also show the existence of a spatial-temporal section of the asynchronous automata network's behavior which completely determines the synchronous global state of [Formula: see text] at every time step.We give the construction of the asynchronous automata network, establish its freedom from deadlocks, and construct local time functions and spatial-temporal sections relating any posssible behavior of [Formula: see text] to the single corresponding behavior of [Formula: see text] on a given input sequence starting from a given initial global state.This establishes that the behavior of any locally finite synchronous automata network actually can be emulated without the restriction of synchronous update, freeing us from the need of a global clock signal. Local information is sufficient to guarantee that the synchronous behavior of [Formula: see text] is completely determined by any asynchronous behavior of [Formula: see text] starting from a corresponding global state and given the same input sequence as [Formula: see text]. Moreover, the relative passage of corresponding local time at any two nodes in [Formula: see text] is bounded in a simple way by approximately one-third of the distance between them.As corollaries, any synchronous generalized cellular automaton or synchronous cellular automaton can be emulated by an asynchronous one of the same type.Implementation aspects of these asynchronous automata are also discussed, and open problems and research directions are indicated.

scholarly journals Tail Behavior of Randomly Weighted Sums

2012 ◽  
Vol 44 (03) ◽  
pp. 794-814 ◽  
Author(s):  
Rajat Subhra Hazra ◽  
Krishanu Maulik
Keyword(s):  
Mellin Transform ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Positive Sequence ◽  
Weighted Sums ◽  
Tail Behavior ◽  
Moment Conditions ◽  
Randomly Weighted Sums ◽  
Regularly Varying

Let {X t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ t , t ≥ 1} be a sequence of positive random variables independent of the sequence {X t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X (∞) = ∑ t=1 ∞Θ t X t + (where X + = max{0, X}) and max1≤k<∞∑ t=1 k Θ t X t , and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X 1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X 1, X 2,… as independent and identically distributed positive random variables. If X 1 has a regularly varying tail of index -α, where α > 0, and if {Θ t , t ≥ 1} is a positive sequence of random variables independent of {X t }, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θ t , t ≥ 1}, X (∞) = ∑ t=1 ∞Θ t X t converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X (∞) has a regularly varying tail then does X 1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ t=1 ∞Θ t along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

scholarly journals A quantum random number generator certified by value indefiniteness

2014 ◽  
Vol 24 (3) ◽  
Author(s):  
ALASTAIR A. ABBOTT ◽  
CRISTIAN S. CALUDE ◽  
KARL SVOZIL
Keyword(s):  
Random Number ◽  
Random Number Generator ◽  
Physical Principle ◽  
Mathematical Form ◽  
Number Generator ◽  
Open Problems ◽  
Efficient Procedure ◽  
Von Neumann ◽  
Quantum Bit ◽  
Temporal Correlations

In this paper we propose a quantum random number generator (QRNG) that uses an entangled photon pair in a Bell singlet state and is certified explicitly by value indefiniteness. While ‘true randomness’ is a mathematical impossibility, the certification by value indefiniteness ensures that the quantum random bits are incomputable in the strongest sense. This is the first QRNG setup in which a physical principle (Kochen–Specker value indefiniteness) guarantees that no single quantum bit that is produced can be classically computed (reproduced and validated), which is the mathematical form of bitwise physical unpredictability.We discuss the effects of various experimental imperfections in detail: in particular, those related to detector efficiencies, context alignment and temporal correlations between bits. The analysis is very relevant for the construction of any QRNG based on beam-splitters. By measuring the two entangled photons in maximally misaligned contexts and using the fact that two bitstrings, rather than just one, are obtained, more efficient and robust unbiasing techniques can be applied. We propose a robust and efficient procedure based onXORing the bitstrings together – essentially using one as a one-time-pad for the other – to extract random bits in the presence of experimental imperfections, as well as a more efficient modification of the von Neumann procedure for the same task. We also discuss some open problems.

Iso-retractable modules and rings

2019 ◽  
Vol 12 (01) ◽  
pp. 1950013 ◽  
Author(s):  
A. K. Chaturvedi
Keyword(s):  
Endomorphism Ring ◽  
Simple Module ◽  
Sufficient Conditions ◽  
Open Problems ◽  
Von Neumann ◽  
Simple Modules ◽  
Von Neumann Regular

The modules which are isomorphic to their non-zero submodules are known as iso-retractable. We characterize simple modules in terms of iso-retractable modules. We provide several sufficient conditions for iso-retractable modules to be simple. We show that if the endomorphism ring of an iso-retractable module is von-Neumann regular then [Formula: see text] is a simple module. In general, iso-retractable modules need not be projective (injective) and vice versa. We investigate some properties of iso-retractable modules with projectivity as well as injectivity. Finally, we provide some open problems.

FREE ENTROPY

2002 ◽  
Vol 34 (3) ◽  
pp. 257-278 ◽  
Author(s):  
DAN VOICULESCU
Keyword(s):  
Probability Theory ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Von Neumann Algebras ◽  
Matrix Theory ◽  
Free Probability ◽  
Open Problems ◽  
Von Neumann ◽  
Free Probability Theory ◽  
Free Entropy

Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras and its connections to random matrix theory, as well as a discussion of open problems and of a basic variational problem, connected to random multimatrix models.

scholarly journals A Non-commutative Fejér Theorem for Crossed Products, the Approximation Property, and Applications

2020 ◽  
Author(s):  
Jason Crann ◽  
Matthias Neufang
Keyword(s):  
Dynamical Systems ◽  
Compact Group ◽  
Approximation Property ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Crossed Products ◽  
Locally Compact ◽  
Open Problems ◽  
Von Neumann

Abstract We prove that a locally compact group has the approximation property (AP), introduced by Haagerup–Kraus [ 21], if and only if a non-commutative Fejér theorem holds for its associated $C^*$- or von Neumann crossed products. As applications, we answer three open problems in the literature. Specifically, we show that any locally compact group with the AP is exact. This generalizes a result by Haagerup–Kraus [ 21] and answers a problem raised by Li [ 27]. We also answer a question of Bédos–Conti [ 4] on the Fejér property of discrete $C^*$-dynamical systems, as well as a question by Anoussis–Katavolos–Todorov [ 3] for all locally compact groups with the AP. In our approach, we develop a notion of Fubini crossed product for locally compact groups and a dynamical version of the slice map property.

scholarly journals DNA synthesis for true random number generation

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Linda C. Meiser ◽  
Julian Koch ◽  
Philipp L. Antkowiak ◽  
Wendelin J. Stark ◽  
Reinhard Heckel ◽  
...  
Keyword(s):  
Random Number ◽  
Random Number Generation ◽  
Sensitive Information ◽  
Encrypted Data ◽  
Von Neumann ◽  
Number Generation ◽  
Sequencing Technologies ◽  
Encryption And Decryption ◽  
Dna Strands ◽  
Von Neumann Algorithm

AbstractThe volume of securely encrypted data transmission required by today’s network complexity of people, transactions and interactions increases continuously. To guarantee security of encryption and decryption schemes for exchanging sensitive information, large volumes of true random numbers are required. Here we present a method to exploit the stochastic nature of chemistry by synthesizing DNA strands composed of random nucleotides. We compare three commercial random DNA syntheses giving a measure for robustness and synthesis distribution of nucleotides and show that using DNA for random number generation, we can obtain 7 million GB of randomness from one synthesis run, which can be read out using state-of-the-art sequencing technologies at rates of ca. 300 kB/s. Using the von Neumann algorithm for data compression, we remove bias introduced from human or technological sources and assess randomness using NIST’s statistical test suite.

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