as computer modeling e.g., Markov chain s , statistics, experimental design, etc. Pseudorandomness ... A Conceptual Perspective. Cambridge University Press. 2008. ref This notion of pseudorandomness ... Press 2008. ISBN 978 0 521 88473 0. Font color gray Limited preview at Google Books Category Pseudorandomness ... more details
Let math U U n n in mathbb N math be a uniform ensemble and math X X n n in mathbb N math be an distribution ensemble ensemble . The ensemble math X math is called pseudorandom if math X math and math U math are Computationally indistinguishable indistinguishable in polynomial time . References Goldreich, Oded 2001 . Foundations of Cryptography Volume 1, Basic Tools . Cambridge University Press. ISBN 0 521 79172 3. Fragments available at the http www.wisdom.weizmann.ac.il oded frag.html author s web site . Category Algorithmic information theory Category Pseudorandomness Category Cryptography crypto stub ... more details
A binary sequence BS is a sequence of math N math bits, math a j math for math j 0,1,...,N 1 math , i.e. math m math ones and math N m math zeros. A BS is pseudorandomness pseudo random PRBS if its autocorrelation function math C v sum j 0 N 1 a ja j v math has only two values math C v begin cases m, mbox if v equiv 0 mbox mod N mc, mbox otherwise end cases math where math c frac m 1 N 1 math is called the duty cycle of the PRBS. A PRBS is random in a sense that the value of an math a j math element is independent of the values of any of the other elements, similar to real random sequences. It is pseudo because it is deterministic and after math N math elements it starts to repeat itself, unlike real random sequences, such as sequences generated by radioactive decay or by white noise. The PRBS is more general than the n sequence , which is a special pseudo random binary sequence of n bits generated as the output of a linear shift register. An n sequence always has a 1 2 duty cycle and its number of elements math N 2 k 1 math . PRBS s are used in telecommunication , encryption , simulation , correlation technique and time of flight spectroscopy . Practical implementation Pseudorandom binary sequences can be generated using linear feedback shift register s. ref Paul H. Bardell, William H. McAnney, and Jacob Savir, Built In Test for VLSI Pseudorandom Techniques , John Wiley & Sons, New York, 1987. ref See also Pseudorandom number generator Gold code Complementary sequences Bit Error Rate Test References reflist External links http www.scriptwell.net correlation.htm Refimprove date January 2008 Category Pseudorandomness Category Binary sequences de Pseudo random bit stream pl PRBS ru uk ... more details
In cryptography , a pseudorandom function family , abbreviated PRF , is a collection of efficiently computable Function computer science functions which emulate a random oracle in the following way no efficient algorithm can distinguish with significant Advantage cryptography advantage between a function chosen randomly from the PRF family and a random oracle a function whose outputs are fixed completely at random . Pseudorandom functions are vital tools in the construction of cryptographic primitive s, especially secure encryption encryption schemes . Pseudorandom functions are not to be confused with Cryptographically secure pseudorandom number generator pseudorandom generators PRGs . The guarantee of a PRG is that a single output appears random if the input was chosen at random. On the other hand, the guarantee of a PRF is that all its outputs appear random, regardless of how the corresponding inputs were chosen, as long as the function was drawn at random from the PRF family. A pseudorandom function family can be constructed from any pseudorandom generator, using, for example, the construction given by Goldreich, Shafi Goldwasser Goldwasser , and Micali. ref Oded Goldreich , Shafi Goldwasser , Silvio Micali 1986 How to Construct Random Functions , Journal of the ACM , vol.33, no.4, p.792 807. doi 10.1145 6490.6503 http theory.lcs.mit.edu cis pubs shafi 1986 jacm.pdf preprint http www.math.weizmann.ac.il oded ggm.html web page and preprint ref See also Pseudorandom permutation References references Category Theory of cryptography Category Cryptographic primitives Category Pseudorandomness de Pseudozuf llige Funktion he ... more details
In computer science math epsilon math biased sample spaces, also known as math epsilon math biased generators or small bias probability spaces, refer to small probability spaces that approximate larger spaces as defined below. Efficient constructions of math epsilon math biased sample spaces have found many applications in computer science, some of which are derandomization of algorithms, construction of error correcting codes , and construction of PCP theorem PCP s . Definition Let math X 1 , X 2 , ldots, X n math be binary random variables and math D math be their joint probability distribution . The random variables math X 1 , X 2 , ldots, X n math are said to be math epsilon math biased if for all subsets math S subseteq 1,2, ldots,n math , math Pr D sum i in S X i 0 Pr D sum i in S X i 1 epsilon . math References http www.wisdom.weizmann.ac.il naor PAPERS bias abs.html Gives efficient construction of math epsilon math biased spaces http www.wisdom.weizmann.ac.il oded PS RND l07.ps Introduction from a course by Oded Goldreich http www.cs.utexas.edu diz 395T 01 lec7.ps Brief description of a construction and an application to constructing almost k wise independent spaces Category Pseudorandomness Category Theoretical computer science ... more details
Context date January 2011 The pseudorandom function advantage PRF advantage of an algorithm on a pseudorandom function family is a measure of how effectively the algorithm can distinguish between a member of the family and a random oracle . Consequently, the maximum pseudorandom advantage attainable by any algorithm with a fixed amount of computational resources is a measure of how well such a function family emulates a random oracle. Say that an adversary algorithm has access to an oracle that will apply a function to inputs that are sent to it. The algorithm sends the oracle a number of queries before deciding whether the oracle is a random oracle or simply an instance of the pseudorandom function family. Say also that there is a 50 chance that the oracle is a random oracle and a 50 chance that it is a member of the function family. The pseudorandom advantage of the algorithm is defined as two times the probability that the algorithm guesses correctly minus one. ref Shafi Goldwasser Goldwasser, S. and Mihir Bellare Bellare, M. http cseweb.ucsd.edu mihir papers gb.html Lecture Notes on Cryptography . Summer course on cryptography, MIT, 1996 2001 ref ref Li, Ninghui Fall 2004 , Security of Symmetric Ciphers, retrieved December 6, 2010 from http www.cs.purdue.edu homes ninghui courses Fall04 lectures lect07.pdf ref References See Wikipedia Footnotes on how to create references using ref ref tags which will then appear here automatically Reflist External links http cseweb.ucsd.edu mihir papers gb.html Categories Category Theory of cryptography Category Pseudorandomness ... more details
An math N,M,D,K, epsilon math extractor is a bipartite graph with math N math nodes on the left and math M math nodes on the right such that each node on the left has math D math neighbors on the right , which has the added property that for any subset math A math of the left vertices of size at least math K math , the distribution on right vertices obtained by choosing a random node in math A math and then following a random graph theory edge to get a node x on the right side is math epsilon math close to the Uniform distribution continuous uniform distribution in terms of total variation distance . A disperser is a related graph. An equivalent way to view an extractor is as a bivariate function math E N times D rightarrow M math in the natural way. With this view it turns out that the extractor property is equivalent to for any source of randomness math X math that gives math n math bit s with min entropy math log K math , the distribution math E X,U D math is math epsilon math close to math U M math , where math U T math denotes the uniform distribution on math T math . Extractors are interesting when they can be constructed with small math K,D, epsilon math relative to math N math and math M math is as close to math KD math the total randomness in the input sources as possible. Extractor functions were originally researched as a way to extract randomness from weakly random sources. See randomness extractor . Using the probabilistic method it is easy to show that extractor graphs with really good parameters exist. The challenge is to find explicit or polynomial time computable examples of such graphs with good parameters. Algorithms that compute extractor and disperser graphs have found many applications in computer science . References Ronen Shaltiel, http www.cs.haifa.ac.il ronen online papers survey.ps Recent developments in extractors a survey Category Graph families Category Pseudorandomness Category Theoretical computer science ... more details
orphan date February 2010 In theoretical computer science a pseudorandom generator for low degree polynomials is an efficiently computable function whose output is indistinguishable from the uniform distribution by evaluation of low degree polynomials in the following sense. Definition A pseudorandom generator math G mathbb F s rightarrow mathbb F n math for polynomials of degree math d math over a Finite field F is an efficient procedure that stretches math s n math field elements into math n math field elements and fools any polynomial of degree d in n variables over F For every such polynomial p, the Total variation Total variation distance in probability theory statistical distance between the distributions math p U n math , for uniform math U n math in math mathbb F n math , and math p G U s math , for uniform math U s math in math mathbb F s math , is at most a small math epsilon math . Construction The case of linear polynomials is solved by Epsilon Biased Sample Spaces small bias spaces which give constructions with seed length math s O log n log 1 epsilon math this is optimal up to constant factors . Following the sequence of papers http www.ccs.neu.edu home viola papers gen.pdf , http shachar.lovett.googlepages.com prg poly.pdf it was established in http www.ccs.neu.edu home viola papers d.pdf that a sum of math d math small bias spaces fools degree math d math polynomials. This gives a construction with seed length math s O log n 2 d log 1 epsilon math . References http www.ccs.neu.edu home viola papers gen.pdf The paper proposed taking a sum of independent small bias spaces for fooling low degree polynomials . http shachar.lovett.googlepages.com prg poly.pdf The paper gave the first unconditional result showing that sum of math 2 d math small bias spaces fools low degree polynomials . http www.ccs.neu.edu home viola papers d.pdf The paper shows that sum of math d math small bias spaces fools low degree polynomials . Category Pseudorandomness ... more details
also Maximum length sequence Pseudorandom number generator Pseudorandomness White noise References Reflist Cdma Category Noise Category Pseudorandomness ar de Pseudozufallsrauschen ... more details
refimprove date March 2011 Stochastic screening or FM screening is a halftone process based on Pseudorandomness pseudo random distribution of halftone dots, using frequency modulation FM to change the density of dots according to the gray level desired. Traditional amplitude modulation halftone screening is based on a geometric and fixed spacing of dots, which vary in size depending on the tone color represented for example, from 10 to 200 micrometre s . The stochastic screening or FM screening instead uses a fixed size of dots for example, about 25 micrometres and a distribution density that varies depending on the color s tone. The technique of stochastic screening, which has existed since the seventies, Citation needed date March 2011 has had a revival in recent times thanks to increased use of Computer to plate computer to plate CTP techniques. In previous techniques, computer to film , during the exposure there could be a drastic variation in the quality of the plate. It was a very delicate and difficult procedure that was not much used. Today, with CTP during the creation of the plate you just need to check a few parameters on the density and tonal correction curve. When you make a plate with stochastic screening you must use a tone correction curve, this curve allows one to align the tone reproduction of an FM screen to that of an industry standard. Given the same final presswork tone value, an FM screen utilizes more halftone dots than an AM XM screen. The result is that more light is filtered by the ink and less light simply reflects off the surface of the substrate. The result is that FM screens exhibit a greater color gamut than conventional AM XM halftone screen frequencies. The creation of a plate with stochastic screening is done the same way as is done with an AM XM screen. A tone reproduction compensation curve is typically applied to align the stochastic ... References Category Printing processes Category Printing terminology Category Pseudorandomness ... more details
dablink For other uses of the word, see Nonce . Image Nonce cnonce uml.svg thumb right 345px Typical client server communication during a nonce based authentication process including both a server nonce and a client nonce. In security engineering , nonce is an arbitrary number used only once to sign a cryptographic communication. It is similar in spirit to a nonce word , hence the name. It is often a randomness random or pseudo random number issued in an authentication protocol to ensure that old communications cannot be reused in replay attack s . For instance, nonces are used in HTTP digest access authentication to calculate an MD5 digest of the password . The nonces are different each time the 401 authentication challenge list of HTTP status codes response code is presented, thus making replay attacks virtually impossible. A nonce may be used to ensure security for a stream cipher . Where the same key is used for more than one message then a different nonce is used to ensure that the keystream is different for different messages encrypted with that key. Often the message number is used. Some also refer to initialization vector s as nonces for the above reasons. To ensure that a nonce is used only once, it should be time variant including a suitably fine grained timestamp in its value , or generated with enough random bits to ensure a probabilistically insignificant chance of repeating a previously generated value. Some authors define pseudorandomness or unpredictability as a requirement for a nonce. ref http www.cs.ucdavis.edu rogaway papers nonce.pdf Nonce Based Symmetric Encryption ref Another example is the bitcoin protocol. Each block in the block chain is signed by a nonce which must be found by trial and error by a miner such that the hash of the block, including the nonce and the prior block hash string, has a specified number of leading zeros. This nonce is computationally non trivial to find and serves to prevent counterfeiting and double spending. See a ... more details
In cryptography , a keystream is a Stream computing stream of Randomness random or Pseudorandomness pseudorandom characters that are combined with a plaintext message to produce an encrypted message the ciphertext . The characters in the keystream can be bit s, byte s, numbers or actual characters like A Z depending on the usage case. Usually each character in the keystream is either added, subtracted or XOR cipher XORed with a character in the plaintext to produce the ciphertext, using modular arithmetic . Keystreams are used in the one time pad cipher and in most stream cipher s. Block cipher s can also be used to produce keystreams. For instance, CTR mode is a Block cipher modes of operation block mode that makes a block cipher produce a keystream and thus turns the block cipher into a stream cipher. Example In this simple example we use the English alphabet of 26 characters from a z. Thus we can not encrypt numbers, commas, spaces and other symbols. The random numbers in the keystream then have to be at least between 0 and 25. To encrypt we add the keystream numbers to the plaintext. And to decrypt we subtract the same keystream numbers from the ciphertext to get the plaintext. If a ciphertext number becomes larger than 25 we wrap it to a value between 0 25. Thus 26 becomes 0 and 27 becomes 1 and so on. Such wrapping is called modular arithmetic . Here the plaintext message attack at dawn is combined by addition with the keystream kjcngmlhylyu and produces the ciphertext kcvniwlabluh . class wikitable align center Plaintext a t t a c k a t d a w n align right Plaintext as numbers 0 19 19 0 2 10 0 19 3 0 22 13 align center Keystream k j c n g m l h y l y u align right Keystream as numbers 10 9 2 13 6 12 11 7 24 11 24 20 align right Ciphertext as numbers 10 28 21 13 8 22 11 26 27 11 46 33 align right Ciphertext as numbers br wrapped to 0 25 10 2 21 13 8 22 11 0 1 11 20 7 align center Ciphertext as text k c v n i w l a b l u h References http www.cacr.math.uwaterloo.ca ... more details
BLP sources date January 2010 Infobox scientist name Salil Vadhan image Replace this image male.svg only free content images are allowed for depicting living people see WP NONFREE image size 150px caption Salil Vadhan birth date birth place death date death place residence citizenship United States nationality ethnicity field Computational complexity theory , Cryptography work institution Harvard University alma mater Massachusetts Institute of Technology doctoral advisor Shafi Goldwasser doctoral students known for author abbreviation bot author abbreviation zoo prizes G del Prize , 2009 religion footnotes Salil Vadhan is Vicky Joseph Professor of Computer Science and Applied Mathematics at Harvard University . ref http www.seas.harvard.edu directory salil Harvard faculty directory . ref He obtained his PhD in Applied Mathematics from Massachusetts Institute of Technology in 1999, where his advisor was Shafi Goldwasser . ref mathgenealogy name Salil Vadhan id 37013 . ref His research centers around the interface between computational complexity theory and cryptography . He focuses on the topics of pseudorandomness and zero knowledge proofs. His work on zig zag product , with Omer Reingold and Avi Wigderson , was awarded the 2009 G del Prize . ref http www.eatcs.org index.php component content article 1 news 563 2009 goedel prize 2009 G del Prize , European Association for Theoretical Computer Science . ref Contributions Zig zag Graph Product for Constructing Expander Graphs One of the main contribution of his work is a new type of graph product, called the zig zag product . Taking a product of a large graph with a small graph, the resulting graph inherits roughly its size from the large one, its degree from the small one, and its expansion properties from both Iteration yields simple explicit constructions of constant degree expanders of every size, starting from one constant size expander. Crucial to the intuition and simple analysis of the properties of the zig z ... more details
Modulation techniques Multiplex techniques In telecommunication s, direct sequence spread spectrum DSSS is a modulation technique. As with other spread spectrum technologies, the transmitted signal takes up more Bandwidth signal processing bandwidth than the information signal that is being modulated. The name spread spectrum comes from the fact that the carrier signals occur over the full bandwidth spectrum of a device s transmitting frequency. Certain IEEE 802.11 standards use DSSS signaling. Features DSSS Phase modulation phase modulates a sine wave Pseudorandomness pseudorandom ly with a continuous string computer science string of pseudorandom noise pseudonoise PN code symbols called Chip CDMA chips , each of which has a much shorter duration than an information bit . That is, each information bit is modulated by a sequence of much faster chips. Therefore, the Chip CDMA chip rate is much higher than the information signal Baud bit rate . DSSS uses a signaling telecommunication signal structure in which the sequence of chips produced by the transmitter is already known by the receiver. The receiver can then use the same PN Sequences PN sequence to counteract the effect of the PN sequence on the received signal in order to reconstruct the information signal. Transmission method Direct sequence spread spectrum transmissions multiply the data being transmitted by a noise signal. This noise signal is a pseudorandom sequence of code 1 code and code 1 code values, at a frequency much higher than that of the original signal. The resulting signal resembles white noise , like an audio recording of static . However, this noise like signal can be used to exactly reconstruct the original data at the receiving end, by multiplying it by the same pseudorandom sequence because 1 1 1, and 1 1 1 . This process, known as de spreading , mathematically constitutes a correlation of the transmitted PN sequence with the PN sequence that the receiver believes the transmitter is using. T ... more details
See also algorithmic randomness A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities sequences such as the results of an ideal dice dice roll , or the digits of pi exhibit statistical randomness. ref http news.uns.purdue.edu UNS html4ever 2005 050426.Fischbach.pi.html Pi seems a good random number generator but not always the best , Chad Boutin, Purdue University ref Statistical randomness does not necessarily imply true randomness , i.e., objective unpredictability. Pseudorandomness is sufficient for many uses, such as statistics, hence the name statistical randomness. Global randomness and local randomness are different. Most philosophical conceptions of randomness are global&mdash because they are based on the idea that in the long run a sequence looks truly random, even if certain sub sequences would not look random. In a truly random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but zeros, though on the whole the sequence might be random. Local randomness refers to the idea that there can be minimum sequence lengths in which random distributions are approximated. Long stretches of the same digits, even those generated by truly random processes, would diminish the local randomness of a sample it might only be locally random for sequences of 10,000 digits taking sequences of less than 1,000 might not appear random at all, for example . A sequence exhibiting a pattern is not thereby proved not statistically random. According to principles of Ramsey theory , sufficiently large objects must necessarily contain a given substructure complete disorder is impossible . Legislation concerning gambling imposes certain standards of statistical randomness to slot machine s. Tests The first tests for random numbers were published by M.G. Kendall and Bernard Babington Smith in the Journal of the Royal Statistical Society in 1938. They were built on stat ... more details
RadioGat n is a Cryptographic hash function cryptographic hash primitive created by Guido Bertoni, Joan Daemen , Micha l Peeters, and Gilles Van Assche. It was first publicly presented at the NIST Second Cryptographic Hash Workshop, held in Santa Barbara, California , on August 24 25, 2006, as part of the NIST hash function competition . Although RadioGat n is a derivative of Panama cryptography Panama , a stream cipher and hash construction from the late 1990s whose hash construction has been broken, RadioGat n does not have Panama s weaknesses when used as a hash function. RadioGat n is actually a family of 64 different hash functions, distinguished by a single parameter, the word width in bit s w , adjustable between 1 and 64. The algorithm uses 58 words, each of size w , to store its internal state. Thus, for example, the 32 bit version of needs 232 bytes to store its state and the 64 bit version 464 bytes. RadioGat n can be used either as a hash function or a stream cipher it can output an arbitrarily long stream of Pseudorandomness pseudo random numbers . The same team that developed RadioGat n went on to make considerable revisions to this cryptographic primitive , leading to the Keccak SHA3 submission. ref cite web url http drops.dagstuhl.de opus volltexte 2009 1958 title The Road from Panama to Keccak via RadioGat n author Bertoni, Guido Daemen, Joan Peeters, Micha l Van Assche, Gilles accessdate 2009 10 20 ref Claimed strength The algorithm s designers claim that the first 19 w bits where w is the word width used of RadioGat n s output is a cryptographically secure hash function. In other words, the first 608 bits of the 32 bit version and 1216 bits of the 64 bit version of RadioGat n can be used as a cryptographic hash value. In light of the birthday attack , this means that for a given word width w , RadioGat n is designed to have no attack with complexity greater than 2 sup 9.5 w sup . This corresponds to 2 sup 304 sup for the 32 bit version and 2 sup 6 ... more details
Complexity Lower Bounds and Pseudorandomness year 2003 edition first publisher Birkh user Basel isbn ... and Pseudorandomness year 1998 edition first publisher Springer isbn 978 3540647669 Category Pseudorandom ... more details
ProbabilityTopicsTOC This is a list of probability topics , by Wikipedia page. It overlaps with the alphabetical list of statistical topics . There are also the outline of probability and catalog of articles in probability theory . For distributions, see List of probability distributions . For journals, see list of probability journals . For contributors to the field, see list of mathematical probabilists and list of statisticians . General aspects Probability Randomness , Pseudorandomness , Quasirandom Randomization , hardware random number generator Random number generator Random sequence Coin flipping tossing checking if a coin is biased Uncertainty Statistical dispersion Observational error Equiprobable Equipossible Average Probability interpretations Markovian Statistical regularity Central tendency Bean machine Relative frequency Frequency probability Maximum likelihood Bayesian probability Principle of indifference Cox s theorem Principle of maximum entropy Information entropy Urn problem s Extractor mathematics Extractor Aleatoric , aleatoric music Free probability Exotic probability Schr dinger method Empirical measure Glivenko Cantelli theorem Zero one law Kolmogorov s zero one law Hewitt Savage zero one law Law of Truly Large Numbers Littlewood s law Infinite monkey theorem Littlewood Offord problem Inclusion exclusion principle Impossible event Information geometry Talagrand s concentration inequality Foundations of probability theory Probability theory Probability space Sample space Standard probability space Random element Random compact set Dynkin system Probability axioms Normalizing constant Event probability theory Complementary event Elementary event Mutually exclusive Boole s inequality Probability density function Cumulative distribution function Law of total cumulance Law of total expectation Law of total probability Law of total variance Almost surely Cox s theorem Bayesianism Prior probability Posterior probability Borel s paradox Bertrand s ... more details
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