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On the computational power of random strings

Adam R. Day
2009 Annals of Pure and Applied Logic  
There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of non-random strings and the overgraph.  ...  This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and Allender and co-authors.  ...  The author also acknowledges the support of the New Zealand Tertiary Education Commission. The author would like to thank the anonymous referees for their comments.  ... 
doi:10.1016/j.apal.2009.03.001 fatcat:5y4ipkmxcbbihgp5nymxgf7vim

Limits on the Computational Power of Random Strings [chapter]

Eric Allender, Luke Friedman, William Gasarch
2011 Lecture Notes in Computer Science  
(All of these inclusions hold, no matter which "universal" Turing machine one uses in the definitions of C(x) and K(x).)  ...  Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let R C and R K denote the sets of strings that are "random" according to these measures; both R K and R C are undecidable  ...  Introduction The motivation for the research reported here comes from a body of work that seems to indicate that the set of "random strings" is quite powerful, where "randomness" is defined in terms of  ... 
doi:10.1007/978-3-642-22006-7_25 fatcat:i35itblntfhufkftijxjbpqb34

Limits on the computational power of random strings

Eric Allender, Luke Friedman, William Gasarch
2013 Information and Computation  
How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings?  ...  We present the first upper bound on the class of computable sets in P R and NP R .  ...  The first two authors were supported in part by NSF Grants CCF-0830133, CCF-0832787, and CCF-1064785.  ... 
doi:10.1016/j.ic.2011.09.008 fatcat:qaroprmu4beb5iex42cn7qgzfi

Power from random strings

E. Allender, H. Buhrman, M. Koucky, D. van Melkebeek, D. Ronneburger
The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.  
When no resource bounds are present, the set of Kolmogorov random strings R K is easily seen to be co-r.e. and not decidable, but the complement of the halting problem is not reducible to R K via a many-one  ...  The sets R µ of Kolmogorov random strings are good examples of sets with a lot of information content that is difficult to access.  ...  We acknowledge illuminating discussions with Lance Fortnow, as well as helpful conversations of the first author with Paul Beame, Henry Cohn, Chris Umans, and Ramarathnam Venkatesan during a visit to Microsoft  ... 
doi:10.1109/sfcs.2002.1181992 dblp:conf/focs/AllenderBKMR02 fatcat:jhsdtmr3krbrvo6r62flc6wc24

Patterns hidden from simple algorithms

Madhu Sudan
2011 Communications of the ACM  
The key, this theory argues, is that randomness is really in the "eyes of the beholder", or rather in the computing power of the "tester" of randomness.  ...  Knowing how an algorithm, or a computational process, works in the presence of somewhat random strings becomes the essence of a recurring question.  ...  Given such major obstacles to analyzing the seeming randomness in strings, it is no surprise that the study of pseudo-randomness remains in its infancy.  ... 
doi:10.1145/1924421.1924445 fatcat:j4e276ibyjco7d4muo2ekyzwai

Fast Spectral Tests for Measuring Nonrandomness and the DES [chapter]

Frank A. Feldman
1988 Lecture Notes in Computer Science  
One test, developed by J. Gait [l], considered properties of power spectra obtained from the discrete Fourier transform of finite binary strings.  ...  By varying the number of DES rounds from 1 to 16, it was thought possible to gradually vary the degree of randomness of the resulting strings.  ...  Nevertheless, on the basis of twenty-seven sample points taken from the power spectrum of one such sequence, the DES output was judged to be random.  ... 
doi:10.1007/3-540-48184-2_22 fatcat:bgcixlkue5gg3coya7al4qq5oa

Tunable, Hardware-based Quantum Random Number Generation using Coupled Quantum Dots [article]

Heath McCabe, Scott M. Koziol, Gregory L. Snider, Enrique P. Blair
2019 arXiv   pre-print
A random bit string may be generated using a sequence of repeated measurements on the same double quantum dot (DQD) system.  ...  Basic quantum mechanical principles are used to study power dissipation and timing considerations for the generation of random bit strings.  ...  ACKNOWLEDGMENT The authors thank Craig S. Lent of the University of Notre Dame for discussion on the topics of quantum systems and measurement as applied to this concept.  ... 
arXiv:1907.00795v2 fatcat:rd6ygrkenvegbd36ptuwv2gmym

An Encryption Technique for Provably Secure Transmission from a High Performance Computing Entity to a Tiny One

Miodrag J. Mihaljević, Aleksandar Kavčić, Kanta Matsuura
2016 Mathematical Problems in Engineering  
An encryption/decryption approach is proposed dedicated to one-way communication between a transmitter which is a computationally powerful party and a receiver with limited computational capabilities.  ...  It is shown that deliberate and secret key controlled insertion of random bits into the basic ciphertext provides security enhancement of the resulting encryption scheme.  ...  Acknowledgments The Ministry of Education, Science and Technological Development, Serbia, has partially funded this work.  ... 
doi:10.1155/2016/7920495 fatcat:26jbk65vmvgyvcpdavttjuibqq

Crossover phenomenon in the performance of an Internet search engine [article]

Lucas Lacasa, Jacopo Tagliabue, Andrew Berdahl
2012 arXiv   pre-print
In this work we explore the ability of the Google search engine to find results for random N-letter strings.  ...  Interestingly, we find that the probability of finding such strings sharply drops from one to zero at Nc = 6.  ...  Acknowledgments The authors would like to acknowledge generous support from the National Science Foundation, the Santa Fe Institute and grants FIS2009-13690 and S2009ESP-1691.  ... 
arXiv:1205.1505v1 fatcat:b4k26jecwbfnhit77jaepzg7ae

Page 2093 of Mathematical Reviews Vol. , Issue 88d [page]

1988 Mathematical Reviews  
In the Kolmogorov theory of randomness, there is no difference between the notions of random strings and random functions, since the cost of coding from one form into the other is ignored.  ...  The notion of one-way functions is conceptually simpler than the notion of pseudo-random strings, and is obviously important for cryptography.  ... 

Randomness and Nondeterminism [chapter]

Leonid A. Levin
1995 Proceedings of the International Congress of Mathematicians  
As long as the seed is not disclosed, no computational power that may fit in the whole universe can distinguish this array from truly random ones.  ...  The next step came from cryptography of the 1980s. Many applications require only few properties of random strings.  ... 
doi:10.1007/978-3-0348-9078-6_137 fatcat:ylg6q5gvkfbcte5cltjzyq7i6q

Non-Randomness of Google's Quantum Supremacy Benchmark [article]

Sangchul Oh, Sabre Kais
2021 arXiv   pre-print
Our results imply that random matrices and the Wasserstein distance could be new tools for analyzing the performance of quantum computers.  ...  The calculation of the Wasserstein distances shows that Google's random bit-strings are farther away from the Haar-measure random bit-strings than the classical random bit-strings.  ...  Department of Energy, Office of Science, National Quantum Information Science Research Centers. We also acknowledge the National Science Foundation under award number 1955907.  ... 
arXiv:2110.06046v2 fatcat:si4ukasb3vdd7fpkok63u72dku

Public-Randomness in Public-Key Cryptography [chapter]

Alfredo De Santis, Giuseppe Persiano
1991 Lecture Notes in Computer Science  
In this work WC investigate the power of Public Randomness in the context of Public-key cryptosystcms.  ...  We consider the Diffie-Bellman Public-key model in which an additional short random string is shared by all users.  ...  All is needed is the availability of a. common random reference string, whose length does not depend on the number of users, but only on the desired level of security.  ... 
doi:10.1007/3-540-46877-3_5 fatcat:kksw4dij4vgxdaadfjpnjm424e

Unexpected Power of Random Strings

Shuichi Hirahara, Michael Wagner
2020 Innovations in Theoretical Computer Science  
In addition, we provide an exact characterization of S exp 2 in terms of exponential-time-computable nonadaptive reductions to arbitrary dense subsets of random strings.  ...  strings: Buhrman, Fortnow, Koucký, and Loff (CCC 2010 [10]) showed that every language in BPP is reducible to the set of random strings via a polynomial-time nonadaptive reduction (irrespective of the  ...  The Unexpected Power of Kolmogorov-Random Strings In this section, we show that every language in the exponential-time hierarchy is reducible to the set of Kolmogorov-random strings under PH reductions  ... 
doi:10.4230/lipics.itcs.2020.41 dblp:conf/innovations/Hirahara20 fatcat:piztb2rwvbcvxh3mhpsljgubsy

The Tsallis entropy and the Shannon entropy of a universal probability [article]

Kohtaro Tadaki
2008 arXiv   pre-print
We study the properties of Tsallis entropy and Shannon entropy from the point of view of algorithmic randomness.  ...  In this paper, we investigate the properties of the Shannon entropy, the power sum, and the Tsallis entropy of a universal probability by means of the notion of program-size complexity.  ...  We first consider the notion of the weak Chaitin D-randomness of the power sum of a universal probability.  ... 
arXiv:0805.0154v1 fatcat:c5l7myptzfbr5jsd5qqu4vel24
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