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What is a Random Sequence
 The Mathematical Association of America, Monthly
, 2002
"... there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a ..."
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there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a
Entropy Tests for Random Number Generators
, 1997
"... : Uniformity tests based on a discrete form of entropy are introduced and studied in the context of empirical testing of uniform random number generators. Comparisons are made with tests based on the Pearson chisquare statistic. Numerical results are provided and several currently used generators f ..."
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: Uniformity tests based on a discrete form of entropy are introduced and studied in the context of empirical testing of uniform random number generators. Comparisons are made with tests based on the Pearson chisquare statistic. Numerical results are provided and several currently used generators fail the tests. The linear congruential and nonlinear inversive generators with poweroftwo modulus perform especially badly. CR Categories and Subject Descriptors: G.3 [Probability and Statistics]: Random Number Generation General Terms: Algorithms, statistics Additional Key Words and Phrases: Random number generators; statistical tests; goodnessoffit; entropy. Authors' Addresses: P. L'Ecuyer and J.F. Cordeau, D'epartement d'Informatique et de Recherche Op'erationnelle (IRO), Universit'e de Montr'eal, C.P. 6128, Succ. CentreVille, Montr'eal, H3C 3J7, Canada; email: lecuyer@iro.umontreal.ca A. Compagner, Faculty of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600...
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"... Wstęp. We współczesnej kryptografii wszelkie informacje podlegające ochronie to ciągi binarne, czyli ciągi zer i jedynek. Już to pierwsze zdanie pracy można uznać za mało precyzyjne, istnieje bowiem na przykład kryptografia wizualna, w której obiektami zainteresowania sa obrazy, my jednak w tej prac ..."
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Wstęp. We współczesnej kryptografii wszelkie informacje podlegające ochronie to ciągi binarne, czyli ciągi zer i jedynek. Już to pierwsze zdanie pracy można uznać za mało precyzyjne, istnieje bowiem na przykład kryptografia wizualna, w której obiektami zainteresowania sa obrazy, my jednak w tej pracy będziemy się poruszać w obrębie pewnego modelu, pomijając całe bogactwo obiektów i metod występujących w kryptografii. Tak więc wiadomościami będą ciągi binarne (o ustalonej lub nieograniczonej długości), a ich ochrona będzie polegała na utajnieniu zawartości, czyli sprawieniu, że nikt niepowołany nie będzie mógł poznać treści tych wiadomości (to również uproszczenie, ponieważ poufność ciągu to tylko jedna z wielu jego cech, które można chronić, por. [6], [16]). Mamy zatem ciąg binarny postaci (1) {01100111010001001...}, będący tekstem jawnym (odkrytym). Po wykonaniu odpowiedniej operacji szyfrowania nasz ciąg binarny staje się szyfrogramem, czyli innym ciągiem bitów mającym tę własność, że na jego podstawie nie można rozpoznać znaczenia tekstu odkrytego. Zatem nasza wiadomość zaszyfrowana (szyfrogram) jest to także ciąg binarny, tym razem postaci (zakładamy tu, że tekst odkryty i szyfrogram są tej samej długości) (2)
and
, 1992
"... The generalized feedback shift register (GFSR) algorithm suggested by Lewis and Payne is a widely used pseudorandom number generator, but has the following serious drawbacks: 1. An initialization scheme to assure higher order equidistribution is involved and is timeconsuming. 2. Each bit of the gen ..."
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The generalized feedback shift register (GFSR) algorithm suggested by Lewis and Payne is a widely used pseudorandom number generator, but has the following serious drawbacks: 1. An initialization scheme to assure higher order equidistribution is involved and is timeconsuming. 2. Each bit of the generated words constitutes an msequence based on a primitive trinomial, which shows poor randomness with respect to weight distribution. 3. Large working area is necessary. 4. The period of sequence is far shorter than the theoretical upper bound. This paper presents the twisted GFSR (TGFSR) algorithm, a slightly but essentially modified version of the GFSR, which solves all the above problems without loss of merit. Some practical TGFSR generators were implemented and they passed strict empirical tests. These new generators are most suitable for simulation of a large distributive system, which requires a number of mutually independent pseudorandom number generators with compact size. 1
The Algorithmic Theory of Randomness
, 2001
"... this paper we won't discuss this very important topic. We will focus instead on the admittedly less ambitious but more manageable question of whether it is possible at least to obtain a mathematically rigourous (and reasonable) denition of randomness. That is, in the hope of clarifying the conc ..."
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this paper we won't discuss this very important topic. We will focus instead on the admittedly less ambitious but more manageable question of whether it is possible at least to obtain a mathematically rigourous (and reasonable) denition of randomness. That is, in the hope of clarifying the concept of chance, one tries to examine a mathematical model or idealization, that might (or might not) capture some of the intuitive properties associated with randomness. In the process of rening our intuition and circunscribing our concepts, we might be able to arrive at some fundamental notions. With luck (no pun intended) , these might in turn bring some insight into the deeper problems mentioned before. At least it could help one to discard some of our previous intuitions or to decide for the need of yet another mathematical model.
Computability and Algorithmic Complexity in Economics
, 2012
"... Rabin's effectivization of the GaleStewart Game [42] remains the model methodological contribution to the field for which Velupillai coined the name Computable Economics more than 20 years ago. Alain Lewis was the first to link Rabin's work with Simon's fertile concept of bounded rat ..."
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Rabin's effectivization of the GaleStewart Game [42] remains the model methodological contribution to the field for which Velupillai coined the name Computable Economics more than 20 years ago. Alain Lewis was the first to link Rabin's work with Simon's fertile concept of bounded rationality and interpret them in terms of Alan Turing's work. Solomonoff (1964), one of the three the other two being Kolmogorov and Chaitinacknowledged pioneers of algorithmic complexity theory, had his starting point in one aspect of what Velupillai [72] came to call the Modern Theory of Induction, an aspect which had its origins in Keynes [23]. Kolmogorov's resurrection of von Mises [80] and the genesis of Kolmogorov complexity via computability theoretic foundations for a frequency theory of probability has given a new lease of life to finance theory [49]. Rabin's classic of computable economics stands in the long and distinguished tradition of game theory that goes back to Zermelo [84], Banach & Mazur [5], Steinhaus [62] and Euwe [14]. This is an outline of the origins and development of the way computability theory and algorithmic complexity theory were incorporated into economic and nance theories. We try to place, in the context of the development of computable economics, some of the classics of the subject as well as those that have,