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Pseudorandom Generator Theorem
In computational complexity theory and cryptography, the existence of pseudorandom generators is related to the existence of one-way functions through a number of theorems, collectively referred to as the pseudorandom generator theorem. Introduction Pseudorandomness A distribution is considered pseudorandom if no efficient computation can distinguish it from the true uniform distribution by a non-negligible advantage. Formally, a family of distributions ''Dn'' is ''pseudorandom'' if for any polynomial size circuit ''C'', and any ''ε'' inversely polynomial in ''n'' :, Prob''x''∈''U'' 'C''(''x'')=1− Prob''x''∈''D'' 'C''(''x'')=1nbsp;, ≤ ''ε''. Pseudorandom generators A function ''Gl'': ''l'' → ''m'', where ''l'' 'C' ''(''u1,>,...,ui,y'') = 1, ≥ ''ε / m'', where ''u'' is a string of ''i'' uniformly random bits, ''s'' is a string of ''l'' uniformly random bits, and ''y'' is a string of ''l''+1 uniformly random bits. Then, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. Cryptography prior to the modern age was effectively synony ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Generator
In theoretical computer science and cryptography, a pseudorandom generator (PRG) for a class of statistical tests is a deterministic procedure that maps a random seed to a longer pseudorandom string such that no statistical test in the class can distinguish between the output of the generator and the uniform distribution. The random seed itself is typically a short binary string drawn from the uniform distribution. Many different classes of statistical tests have been considered in the literature, among them the class of all Boolean circuits of a given size. It is not known whether good pseudorandom generators for this class exist, but it is known that their existence is in a certain sense equivalent to (unproven) circuit lower bounds in computational complexity theory. Hence the construction of pseudorandom generators for the class of Boolean circuits of a given size rests on currently unproven hardness assumptions. Definition Let \mathcal A = \ be a class of functions. These ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-way Function
In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient for a function to be called one-way (see Theoretical definition, below). The existence of such one-way functions is still an open conjecture. Their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science.Oded Goldreich (2001). Foundations of Cryptography: Volume 1, Basic Tools,draft availablefrom author's site). Cambridge University Press. . (see als The converse is not known to be true, i.e. the existence of a proof that P≠NP would not directly imply the existence of one-way functions. In applied contexts, the terms "easy" and "hard" are us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Background The generation of random numbers has many uses, such as for random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, people use pseudorandom numbers, which ideally have the unpredictability of a truly random sequence, despite being generated by a deterministic process. In many applications, the deterministic process is a computer algorithm called a pseudorandom number generator, which must first be provided ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (discrete)
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair dice. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negligible Function
In mathematics, a negligible function is a function \mu:\mathbb\to\mathbb such that for every positive integer ''c'' there exists an integer ''N''''c'' such that for all ''x'' > ''N''''c'', :, \mu(x), 0 such that for all ''x'' > ''N''poly : , \mu(x), 0, there exists a positive number \delta>0 such that , x-x_0, N_\varepsilon ::, \mu(x), 0 by the functions 1/x^c where c>0 or by 1/\operatorname(x) where \operatorname(x) is a positive polynomial. This leads to the definitions of negligible functions given at the top of this article. Since the constants \varepsilon>0 can be expressed as 1/\operatorname(x) with a constant polynomial this shows that negligible functions are a subset of the infinitesimal functions. Use in cryptography In complexity-based modern cryptography, a security scheme is ''provably secure'' if the probability of security failure (e.g., inverting a one-way function, distinguishing cryptographically strong pseudorandom bits from truly r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advantage (cryptography)
In cryptography, an adversary's advantage is a measure of how successfully it can attack a cryptographic algorithm, by distinguishing it from an idealized version of that type of algorithm. Note that in this context, the "adversary" is itself an algorithm and not a person. A cryptographic algorithm is considered secure if no adversary has a non-negligible advantage, subject to specified bounds on the adversary's computational resources (see concrete security). "Negligible" usually means "within O(2−p)" where p is a security parameter associated with the algorithm. For example, p might be the number of bits in a block cipher's key. Description of concept Let F be an oracle for the function being studied, and let G be an oracle for an idealized function of that type. The adversary A is a probabilistic algorithm, given F or G as input, and which outputs 1 or 0. A's job is to distinguish F from G, based on making queries to the oracle that it's given. We say: Adv(A) = , \Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid Approach
Hybrid may refer to: Science * Hybrid (biology), an offspring resulting from cross-breeding ** Hybrid grape, grape varieties produced by cross-breeding two ''Vitis'' species ** Hybridity, the property of a hybrid plant which is a union of two different genetic parent strains * Hybrid (particle physics), a valence quark-antiquark pair and one or more gluons * Hybrid solar eclipse, a rare solar eclipse type Technology Transportation * Hybrid vehicle, a vehicle using more than one power source or an engine sourced from a different chassis ** Hybrid electric vehicle, a vehicle using both internal combustion and electric power sources *** Plug-in hybrid, whose battery can be recharged by a charging cable * Hybrid bicycle, a bicycle with features of road and mountain bikes * Hybrid train, a locomotive, railcar, or train that uses an onboard rechargeable energy storage system * Hybrid motorcycle, a motorcycle built using components from more than one original-manufacturer products, suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. It is an example of the weaker logical refutation '' reductio ad absurdum''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, ''Q'' and ''¬Q'', and appealing to the Law of noncontradiction. #Since assuming ''P'' to be false leads to a contradiction, it is concluded that ''P'' is in fact true. An important special case is the existence proof by contradiction: in order to demonstrate the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hard-core Predicate
In cryptography, a hard-core predicate of a one-way function ''f'' is a predicate ''b'' (i.e., a function whose output is a single bit) which is easy to compute (as a function of ''x'') but is hard to compute given ''f(x)''. In formal terms, there is no probabilistic polynomial-time (PPT) algorithm that computes ''b(x)'' from ''f(x)'' with probability significantly greater than one half over random choice of ''x''. Goldwasser, S. and Bellare, M.br>"Lecture Notes on Cryptography". Summer course on cryptography, MIT, 1996-2001 In other words, if ''x'' is drawn uniformly at random, then given ''f(x)'', any PPT adversary can only distinguish the hard-core bit ''b(x)'' and a uniformly random bit with negligible advantage over the length of ''x''. A hard-core function can be defined similarly. That is, if ''x'' is chosen uniformly at random, then given ''f(x)'', any PPT algorithm can only distinguish the hard-core function value ''h(x)'' and uniformly random bits of length '', h(x), '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
