In cryptography , a pseudorandomfunctionfamily , 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 pseudorandomfunctionfamily 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
pseudorandomfunctionfamilypseudorandomfunction . Let p and l be prime number s with l p ... coordinate vector vector a a sub 1 sub , ..., a sub n sub math mathbb F l n math they define the function ... The evaluation of function math f a x math in the Naor Reingold construction can be done very efficiently. Computing the value of the function math f a x math at any given point is comparable with one modular exponentiation and n modular multiplications. This function can be computed in parallel by threshold circuits of bounded depth and polynomial size. The Naor Reingold function can be used ... encryption , authentication and digital signatures . Security of the Function Assume that an attacker sees several outputs of the function, e.g. math f a 1 g a 1 , f a 2 g a 2 , f a 3 g a 1 a 2 math ... with a hard Diffie Hellman problem DHP . Example An attacker sees several outputs of the function ... 1 1 4 1 4 math . Then, the attacker wants to predict the next sequence element of this function, math ... f a 1 math and math f a 5 math . There are other attacks that would be very bad for a Pseudorandom ... mathcal A f math denote the algorithm math mathcal A math with access to an oracle for evaluating the function ... A R p,g to 1 math is Negligible function negligible . The first probability is taken over the choice ..., g by math mathcal I mathcal G n math , instance generator, and the random choice of the function ... of this function. ref name ShparlinskiUniformity Sequences in Elliptic Curve The elliptic curve version of this function is of interest as well. In particular, it may help to improve the cryptographic ... generator Generalized inversive congruential pseudorandom numbers Notes Reflist refs ref name ... Shparlinski, Igor E. Linear Complexity of the Naor Reingold pseudo random function, Inform ... of distribution of the Naor Reingold pseudo random function, Finite Fields and Their Applications ... and Pseudorandomness year 1998 edition first publisher Springer isbn 978 3540647669 Category Pseudorandom ... more details
In cryptography , the term pseudorandom permutation , abbreviated PRP , refers to a function that cannot be distinguished from a random permutation that is, a permutation selected at random with uniform probability, from the family of all permutations on the function s domain with practical effort. A pseudorandom permutation family is a collection of pseudorandom permutations, where a specific permutation may be chosen using a key. The idealized abstraction of a block cipher is a truly random permutation. If a distinguishing algorithm exists that achieves significant advantage cryptography advantage with less effort than specified by the block cipher s security parameter this usually means the effort required should be about the same as a brute force search through the cipher s key space , then the cipher is considered broken at least in a certificational sense, even if such a break doesn t immediately lead to a practical security failure. See also Block cipher pseudorandom permutation families operating on fixed size blocks of bits Format Preserving Encryption pseudorandom permutation families operating on arbitrary finite sets Pseudorandomfunction References cite book author Mihir Bellare , Phillip Rogaway date 2005 09 20 title Introduction to Modern Cryptography url http www cse.ucsd.edu mihir cse207 classnotes.html chapter Chapter 3 Pseudorandom functions accessdate 2007 09 30 Category Theory of cryptography Category Cryptographic primitives crypto stub ... more details
. A function G 0, 1 sup s sup 0, 1 sup n sup , where s n , is a pseudorandom generator against ... of cryptographic pseudorandom generators is equivalent to the existence of one way function ... generators, called pseudorandomfunction s. Pseudorandom generators and derandomization ... P . To perform such a simulation, it is sufficient to construct pseudorandom generators against the family ... of this term, see Pseudorandom number generator In theoretical computer science and cryptography , a pseudorandom ... procedure that maps a random seed to a longer pseudorandompseudorandom string such that no statistical ... pseudorandom generators for this class exist, but it is known that their existence is in a certain ... of pseudorandom generators for the class of Boolean circuits of a given size rests on currently ... the stretch of the pseudorandom generator. Functions from the class F sub n sub are sometimes called adversaries . A pseudorandom generator against a family of adversaries F F sub n sub with bias n is a collection of pseudorandom generators G sub n sub 0, 1 sup s n sup 0, 1 sup n sup , where G sub n sub is a pseudorandom generator against F sub n sub with bias n . In most applications, the family F represents some model of computation , and one is interested in desigining a pseudorandom generator that is computable in the same or some closely related model. Pseudorandom generators in cryptography main Cryptographically secure pseudorandom number generator In cryptography , the class ... bit output, and one is interested in designing pseudorandom generators that are computable by a polynomial time algorithm and whose bias is negligible in the circuit size. These pseudorandom generators are sometimes called cryptographically secure pseudorandom generators CSPRGs . It is not known if cryptographic pseudorandom generators exist. Their existence would imply that P NP . However, the existence of cryptographic pseudorandom generators is widely believed to be true citation ... more details
Orphan date February 2009 Notability Products date August 2011 A pseudorandom encoder is an electro mechanical device used for measuring distance and finding position. It is different from the conventional absolute and incremental encoder in its disk pattern, which gives it aspects of the other two designs. The disk of a pseudorandom encoder has an evenly spaced notch design just as an incremental encoder to measure distance. A second track of code is used to find the position just like an absolute encoder however, an absolute encoder tells position from reading from multiple parallel tracks while a pseudorandom encoder uses one single track. This aspect allows pseudorandom rotary encoders to have smaller diameters and larger through holes. To know its position, the encoder must first move a certain distance to read the code. This distance is generally small, about 1 2 degrees for rotary encoders and 1 mm for linear encoders, though this idiosyncrasy does hinder its usefulness in some applications such as trackball track balls since position is not immediately known when the system starts. ref cite web url http www.virtualabsolute.com VAoverview.htm title A New Kind of Encoder virtualabsolute.com ref See also Rotary encoder References reflist Category Dimensional instruments ... more details
In cryptography , pseudorandom noise PRN ref cite web url http www.gps.gov technical icwg meetings 2011 09 13 WAS IS FINAL PRN Expansion 4May2011.pdf title Change Topic Pseudorandom Noise PRN Expansion publisher GPS.GOV accessdate 13 July 2011 ref ref cite web url http www.losangeles.af.mil library factsheets factsheet.asp?id 8618 title Pseudorandom Noise PRN Code Assignments publisher Los Angeles Air Force Base accessdate 13 July 2011 ref is a Signalling telecommunication signal similar to noise physics noise which satisfies one or more of the standard tests for statistical randomness . Although it seems to lack any definite pattern , pseudorandom noise consists of a deterministic sequence of Pulse signal processing pulses that will repeat itself after its period. ref FS1037C MS188 ref In cryptography cryptographic device s, the pseudorandom noise pattern is determined by a key cryptography key and the repetition period can be very long, even millions of digits. Pseudorandom noise is used in some synthesizer electronic musical instruments , either by itself or as an input to subtractive synthesis , and in many white noise machine s. In spread spectrum systems, the receiver correlation correlates a locally generated signal with the received signal information theory signal . Such spread spectrum systems require a set of one or more codes or sequences such that Like random noise, the local sequence has a very low correlation with any other sequence in the set, or with the same sequence at a significantly different time offset, or with narrowband interference, or with thermal ... sequence. In a direct sequence spread spectrum system, each bit in the pseudorandom ... . In a frequency hopping spread spectrum sequence, each value in the pseudorandom sequence is known ... range finding applications involve two way transmissions. A local station generates a pseudorandom ... also Maximum length sequence Pseudorandom number generator Pseudorandomness White noise References ... 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
negligible function negligible advantage cryptography advantage . Formally, a family of distributions D sub n sub is pseudorandom if for any polynomial size circuit C , and any inversely polynomial ... . Pseudorandom generators A function G sub l sub 0,1 sup l sup 0,1 sup m sup , where l m is a pseudorandom generator if G sub l sub can be computed in time polynomial in l G sub l sub x is pseudorandom, when x is uniformly random. One additional pseudorandom bit implies polynomially more pseudorandom bits It can be shown that if there is a pseudorandom generator G sub l sub 0,1 sup l sup 0,1 sup l 1 sup , i.e. a generator that adds only one pseudorandom bit, then for any m poly l , there is a pseudorandom generator G sub l sub 0,1 sup l sup 0,1 sup m sup ... generators The existence of pseudorandom generators is related to the existence of one way function ... way permutation is a one way function that is also a permutation of the input bits. A pseudorandom generator ...In computational complexity theory and cryptography , the existence of pseudorandom generator s is related to the existence of one way function s through a number of theorems, collectively referred to as the pseudorandom generator theorem . Introduction Pseudorandomness A distribution is considered pseudorandom ... and used as the seed to the first instance of G sub l sub , which is known to be a pseudorandom generator ... of the output. Repeating this process for m times yields an output of m pseudorandom bits. It can be shown that such G sub l sub , that consists of m instances of G sub l sub , is indeed a pseudorandom ... number i 1 . Now, assume that G sub l sub is not a pseudorandom distribution that is, there exists ... l sub or a drawn from uniform distribution. Let s reuse the approach of building large pseudorandom generators out of instances of G sub l sub and construct a string of pseudorandom bits of length m&minus ... is not a pseudorandom generator, which is a contradiciton to the hypothesis. Q.E.D. Now, let s illustrate ... 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
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
Image VEST Core4 LowLevel.png thumbnail 320px right VEST 4 T function followed by a transposition layer In cryptography , a T function is a bijection bijective mapping that updates every bit of the state computer science state in a way that can be described as math x i x i f x 0, cdots, x i 1 math , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T function is called triangular . Thanks to their bijectivity no collisions, therefore no entropy loss regardless of the used Boolean function s and regardless of the selection of inputs as long as they all come from one side of the output bit , T functions are now widely used in cryptography to construct block cipher s, stream cipher s, PRNG s and cryptographic hash function hash functions . T functions were first proposed in 2002 by Alexander Klimov A. Klimov and Adi Shamir A. Shamir in their paper A New Class of Invertible Mappings . Ciphers such as TSC 1 , TSC 3 , TSC 4 , ABC stream cipher ABC , Mir 1 and VEST are built with different types of T functions. Because arithmetic operation s such as addition , subtraction and multiplication are also T functions triangular T functions , software efficient word based T functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T functions based on arithmetic operations is predictability of their period mathematics period , which is highly attractive to cryptographers. Although triangular T functions are naturally ... bit. Subsequent transposition of the output bits and iteration of the T function also do not affect ... see family keying . Hardware efficient lightweight T functions with identical widths of all the update ... and losing the T function bias of depending only on the less significant bits of the state. References ... more details
Citation style date September 2009 A pseudorandom number generator PRNG , also known as a deterministic random bit generator DRBG , ref http csrc.nist.gov publications nistpubs 800 57 sp800 57 Part1 revised2 Mar08 2007.pdf ref is an algorithm for generating a sequence of numbers that approximates the properties of random numbers. The sequence is not truly random in that it is completely determined by a relatively small set of initial values, called the PRNG s state , which includes a truly random seed . Although sequences that are closer to truly random can be generated using hardware random number generator s, pseudorandom numbers are important in practice for their speed in number generation and their reproducibility, and they are thus central in applications such as simulations e.g., of physical systems with the Monte Carlo method , in cryptography , and in procedural generation . Good statistical properties are a central requirement for the output of a PRNG, and common classes of suitable ... than its output this is particularly obvious with PRNGs with a 1 bit output. Most pseudorandom ... 165.pdf Cryptographically secure pseudorandom number generators Main Cryptographically secure pseudorandom ... has only negligible function negligible advantage cryptography advantage in distinguishing the generator ... s Cryptographic Application Programming Interface function CryptGenRandom , the Yarrow algorithm ... continuous uniform distribution PRNG and a function that relates the two distributions. First, one needs the cumulative distribution function math F b math of the target distribution math f ... Rayleigh and Poisson distribution Poisson . See also Portal Mathematics List of pseudorandom number generators Pseudorandom binary sequence Quasi random Random number generator attack Randomness Notes ... http eeyore.wu wien.ac.at src prng A collection of algorithms for generating pseudorandom numbers as a library ... Security DEFAULTSORT Pseudorandom Number Generator Category Pseudorandom number generators ca Generador ... more details
wiktionary functionFunction may refer to Diatonic function , a term in music theory Function E 40 song , a 2012 song by American rapper E 40 featuring YG rapper YG , iAmSu & Problem Function biology , explaining why a feature survived selection Function computer science , or subroutine, a portion of code within a larger program, performs a specific task Function engineering , related to the selected property of a system Function language , in linguistics, a way of achieving an aim using language Function mathematics , an abstract entity that associates an input to a corresponding output according to some rule Function model , a structured representation of the functions, activities or processes Function object , or functor or functionoid, a concept of object oriented programming Function Drinks , a beverage company based in Redondo Beach, California. An organised event such as a party or meeting See also Functionalism disambiguation Function hall Functional disambiguation Functionality in polymer chemistry see Structural unit Functor disambiguation bg bs Funkcija vor ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko id Fungsi it Funzione lt Funkcija lmo Funziun nl Functie ja no Funksjon nn Funksjon pl Funkcja ujednoznacznienie pt Fun o desambigua o ro Func ie dezambiguizare ru simple Function sk Funkcia sl Funkcija razlo itev sr sh Funkcija razvrstavanje sv Funktion olika betydelser th uk zh ... more details
In mathematics, S function may refer to sigmoid function Schur polynomials In physics, it may refer to Action physics action functional mathdab Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
generator Naor Reingold PseudorandomFunction References Reflist refs ref name one G. H. Hardy and E ...An approach to nonlinear congruential methods of Pseudorandom number generator generating uniform pseudorandom numbers in the interval 0,1 is the Inversive congruential generator with prime modulus. A generalization for arbitrary composite moduli math m p 1, dots p r math with arbitrary distinct Prime number primes math p 1, dots ,p r ge 5 math will be present here. Let math mathbb Z m 0,1,...,m 1 math .For integer s math a,b in mathbb Z m math with gcd a,m 1 a generalized inversive congruential sequence math y n n geqslant 0 math of elements of math mathbb Z m math is defined by math y 0 rm seed math math y n 1 equiv a y n varphi m 1 b pmod m text , n geqslant 0 math where math varphi m p 1 1 dots p r 1 math denotes the number of positive integers less than m which are Coprime relatively prime to m . Example Let take m 15 math 3 times 5 , a 2 , b 3 math and math y 0 1 math . Hence math varphi m 2 times 4 8 , math and the sequence math y n n geqslant 0 1,5,13,2,4,7,1, dots math is not maximum. The result below shows that these sequences are closely related to the following inversive congruential sequence with prime moduli. For math 1 le i le r math let math mathbb Z p i 0,1, dots ,p i 1 , m i m p i math and math a i ,b i in mathbb Z p i math be integers with math a equiv m i 2 a i ... 1 i pmod p i math , which implies the desired result. Generalized Inversive Congruential Pseudorandom ... independence properties is based on the discrepancy of s tuples of pseudorandom numbers ... Congruential Pseudorandom Numbers for math s ge 2 math . Higher bound Let math s ge 2 math Then the discrepancy ... Congruential Pseudo random Numbers model true random numbers very closely. See also Pseudorandom number ... Herrmann first J rgen title On Generalized Inversive Congruential Pseudorandom Numbers year 1994 edition ... Congruential Pseudorandom Numbers Category Pseudorandom number generators ... more details
speaking, are not pure pseudorandom number generators, as their output is not completely determined ... will be lost. A cryptographically secure cryptographic hash function hash of a counter might also act ... a pseudorandom stream of bits that are combined almost always XOR cipher XOR ed with the plaintext running the cipher on a counter will return a new pseudorandom stream, possibly with a longer period ..., which does not attempt to evaluate the entropic quality of its inputs. the function CryptGenRandom ... Press, 1996, http www.cacr.math.uwaterloo.ca hac about chap5.pdf Chapter 5 Pseudorandom Bits and Sequences ... designs A Statistical Test Suite for Random and Pseudorandom Number Generators , http csrc.nist.gov ... Cryptanalysis of the Dual Elliptic Curve Pseudorandom Generator , Berry Schoenmakers and Andrey Sidorenko, IACR ePrint 2006 190. http eprint.iacr.org 2006 321 Efficient Pseudorandom Generators Based ... of a cryptographically safe shrinking pseudorandom number generator. Interlang cryptography navbox DEFAULTSORT Cryptographically Secure Pseudorandom Number Generator Categories Category Cryptographic algorithms Category Cryptographically secure pseudorandom number generators Category Cryptographic ... more details
. Because the normal distribution is a location scale family , its quantile function for arbitrary parameters can be derived from a simple transformation of the quantile function of the standard normal ...See also quantile . In probability and statistics , the quantile function of the probability distribution ..., with that probability. The quantile function is one way of prescribing a probability distribution, and it is an alternative to the probability density function probability density or probability mass function mass function , the cumulative distribution function and the Characteristic function probability theory characteristic function . The quantile function, Q , of a probability distribution is the inverse function inverse of its cumulative distribution function cdf F . The derivative of the quantile function, namely the quantile density function, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function. Definition Assuming a continuous and strictly monotonic distribution function, math scriptstyle F colon R to 0,1 math , the quantile function returns the value below which random draws from the given ... of x such that math F x Pr X le x p. , math File Quantile distribution function.svg thumb Quantile function for a general distribution function If the probability distribution is discrete rather than ... weakly monotonic there may be flat spots in its range. In either case, the quantile function is math ... function returns the minimum value of x for which the previous probability statement holds. Simple example For example, the quantile function for Exponential i.e. intensity and expected ... sampling the quantile function see, e.g., http course.shufe.edu.cn jpkc jrjlx ref StaTable.pdf ... ref Monte Carlo simulations employ quantile functions to produce non uniform random or pseudorandom ... be obtained in principle by applying its quantile function to a sample from a uniform distribution ... more details
otheruses negligible onlyinclude In mathematics, a negligible function is a Function mathematics function math mu x mathbb N rightarrow mathbb R math such that for every positive integer c there exists an integer N sub c sub such that for all x N sub c sub , math mu x frac 1 x c . math onlyinclude Equivalently, we may also use the following definition. A function math mu x mathbb N rightarrow mathbb R math is negligible , if for every positive polynomial poly there exists an integer N sub poly sub 0 such that for all x N sub poly sub math mu x frac 1 text poly x . math History The concept of negligibility can find its trace back to sound models of analysis. Though the concepts of continuous function continuity and infinitesimal became important in mathematics during Isaac Newton Newton and Gottfried Leibniz Leibniz s time 1680s , they were not well defined until late 1810s. The first reasonably rigorous definition of continuity in mathematical analysis was due to Bernard Bolzano , who wrote in 1817 the modern definition of continuity. Lately Augustin Louis Cauchy Cauchy , Karl Weierstrass Weierstrass and Eduard Heine Heine also defined as follows with all numbers in the real number domain math mathbb R math Continuous function A function math f x mathbb R rightarrow mathbb R math is continuous at math x x 0 math if for every math epsilon 0 math , there exists a positive number math delta 0 math such that math x x 0 delta math implies math f ... math x 0 infty math with math f x 0 0 math , we must define the concept of infinitesimal function Infinitesimal A continuous function math mu x mathbb R rightarrow mathbb R math is infinitesimal ... security provably secure if the probability of security failure e.g., inverting a one way function , distinguishing Cryptographically secure pseudorandom number generator cryptographically strong pseudorandom bits from truly random bits is negligible in terms of the input math x math cryptographic ... more details
unreferenced date February 2008 In computer programming , a function computer science function may be described as pure if both these statements about the function hold The function always evaluates the same result value given the same argument value s . The function result value cannot depend on any hidden information or program state state that may change as program execution proceeds or between different executions of the program, nor can it depend on any external input from input output I O devices ... than the argument values. The function may return multiple result values and these conditions must apply to all returned values for the function to be considered pure. If an argument is Pass by reference ... an expression impure even if the function called is pure. clarify date April 2012 if the reference is also a pure function, wouldn t then the combined still be pure? Examples Pure functions code sin ... k Impure functions A hypothetical function code today code that returns the current day of the week .... Similarly, any function that uses global state or a static variable is potentially impure. code random code is impure because each call potentially yields a different value. This is because pseudorandom ... of a function subexpression applied to one or more argument subexpressions is pure if both these statements about the subexpressions hold The function and argument subexpressions are pure expressions. The function subexpression yields a pure function. Typically the function subexpression is simply a function identifier. Pure expressions are often referred to as being Referential transparency computer ... function or more generally a value which contains one or more impure functions . It is also possible for an expression to be pure even if one or more of the argument subexpressions yields an impure function or a value which contains one or more impure functions . In this case the impure function s in the argument ... allow to prove that a certain function does not have any side effects although it uses impure operations ... more details
, one important type of random function is studied under the name of stochastic process es, for which there are a variety of models describing systems where an observation is a random function of time ... a function is known and where the state of knowledge about the true function can be expressed by saying that it is an unknown realisation of a random function, for example in the Dirichlet process ... last3 M ller first4 Stephen G. last4 Walker year 2010 ref A special case of a random function is a random permutation , where a realisation can be interpreted as being in the form of a function on the set of integers describing the original location of an item, where the value of the function provides ... function can be a useful building block in enabling cryptographic protocol s. Definition A random function is a type of random element in which a single outcome is selected from some family of functions, where the family consists some class of all maps from the Domain of a function domain to the codomain . Clarify date June 2011 For example the class may be restricted to all continuous function s or to all step function . The values determined by a random function evaluated at different points from ... as independent. Applications Thus, a random function can be considered to map each input independently ... of a cryptographic hash function . A random function is a useful building block in enabling ... distrustful parties to agree on a random function i.e., coin flipping is impossible . Citation needed ... function or a related object. See random oracle model , common reference string model . Notes ... more details
Unreferenced date December 2009 In mathematics , antiholomorphic functions also called antianalytic functions are a family of Function mathematics function s closely related to but distinct from holomorphic function s. A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to z exists in the neighbourhood of each and every point in that set, where z is the complex conjugate . One can show that if f z is a holomorphic function on an open set D , then f z is an antiholomorphic function on D , where D is the reflection against the x axis of D , or in other words, D is the set of complex conjugates of elements of D . Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if it can be expanded in a power series in z in a neighborhood of each point in its domain. If a function is both holomorphic and antiholomorphic, then it is constant on any connected space connected component of its domain. DEFAULTSORT Antiholomorphic Function Category Complex analysis Category Types of functions eo Kontra holomorfa funkcio it Funzione antiolomorfa pl Funkcja antyholomorficzna ru tr Tersholomorf fonksiyon uk ... more details
In mathematics, omega function or function may refer to Pearson Cunningham function Lambert W function Wright Omega function Mathematical disambiguation ... more details
Barnes G function , related to the Gamma function Meijer G function , a generalization of the hypergeometric function Siegel G function , a class of functions in transcendence theory mathdab ... more details
In mathematics, by sigma function one can mean one of the following The Divisor function sum of divisors function sub a sub n , an arithmetic function Weierstrass sigma function , related to elliptic functions Kronecker s sigma function Rado s sigma function, see busy beaver See also sigmoid function . mathdab de Teilersumme fr Fonction sigma ... more details
N w i C x i,x j w j. math A function is a valid covariance function if and only if ref name Cressie ... , &hellip , w sub N sub . A function with this property is called Positive definite kernel ... random field , where math C x i,x j C x i h,x j h , math for any lag h , the covariance function can be represented by a one parameter function math C s h C 0,h C x,x h , math which is called a covariogram and also a covariance function . Implicitly the C x sub i sub , x sub j sub can be computed from C sub s sub h by math C x,y C s y x . , math The Positive definite function positive definiteness of this single argument version of the covariance function can be checked by Bochner s theorem ... covariance function is the exponential covariance function math C d exp d V math where V is a scaling ... covariance function are not smooth. The squared exponential covariance function math C d exp d 2 V math is a stationary covariance function with smooth sample paths. The Mat rn covariance function and rational quadratic covariance function are two parametric families of stationary covariance functions. The Mat rn family includes the exponential and squared exponential covariance ... function Correlation function References references Category Geostatistics Category Spatial data ... more details
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