Indistinguishability (other)
Indistinguishability may refer to: * Identical (indistinguishable) particles * Computational indistinguishability * Ciphertext indistinguishability * Indistinguishability obfuscation * Topological indistinguishability See also * Incomparable (other) Incomparable may refer to: * Comparability, in mathematics, with respect to a given relation over a set *HMS Incomparable, a proposal for a very large battlecruiser, suggested in 1915 *Incomparable (diamond), one of the largest diamonds ever found ...

Identical Particles
In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Quasiparticles also behave in this way. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which cannot (as described by the Pauli exclusion principle). Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei. The fact t ...
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Computational Indistinguishability
In computational complexity and cryptography, two families of distributions are computationally indistinguishable if no efficient algorithm can tell the difference between them except with negligible probability. Formal definition Let \scriptstyle\_ and \scriptstyle\_ be two distribution ensembles indexed by a security parameter ''n'' (which usually refers to the length of the input); we say they are computationally indistinguishable if for any non-uniform probabilistic polynomial time algorithm ''A'', the following quantity is a negligible function in ''n'': : \delta(n) = \left, \Pr_A(x) = 1- \Pr_A(x) = 1\. denoted D_n \approx E_n. In other words, every efficient algorithm ''As behavior does not significantly change when given samples according to ''D''''n'' or ''E''''n'' in the limit as n\to \infty. Another interpretation of computational indistinguishability, is that polynomial-time algorithms actively trying to distinguish between the two ensembles cannot do so: that any su ...
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Ciphertext Indistinguishability
Ciphertext indistinguishability is a property of many encryption schemes. Intuitively, if a cryptosystem possesses the property of indistinguishability, then an adversary will be unable to distinguish pairs of ciphertexts based on the message they encrypt. The property of indistinguishability under chosen plaintext attack is considered a basic requirement for most provably secure public key cryptosystems, though some schemes also provide indistinguishability under chosen ciphertext attack and adaptive chosen ciphertext attack. Indistinguishability under chosen plaintext attack is equivalent to the property of semantic security, and many cryptographic proofs use these definitions interchangeably. A cryptosystem is considered ''secure in terms of indistinguishability'' if no adversary, given an encryption of a message randomly chosen from a two-element message space determined by the adversary, can identify the message choice with probability significantly better than that of r ...
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Indistinguishability Obfuscation
In cryptography, indistinguishability obfuscation (abbreviated IO or iO) is a type of software obfuscation with the defining property that obfuscating any two programs that compute the same mathematical function results in programs that cannot be distinguished from each other. Informally, such obfuscation hides the implementation of a program while still allowing users to run it. Formally, IO satisfies the property that obfuscations of two circuits of the same size which implement the same function are computationally indistinguishable. Indistinguishability obfuscation has several interesting theoretical properties. Firstly, iO is the "best-possible" obfuscation (in the sense that any secret about a program that can be hidden by any obfuscator at all can also be hidden by iO). Secondly, iO can be used to construct nearly the entire gamut of cryptographic primitives, including both mundane ones such as public-key cryptography and more exotic ones such as deniable encryption and ...
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Topological Indistinguishability
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', and ''Ny'' is the set of all neighborhoods that contain ''y'', then ''x'' and ''y'' are "topologically indistinguishable" if and only if ''Nx'' = ''Ny''. (See Hausdorff's axiomatic neighborhood systems.) Intuitively, two points are topologically indistinguishable if the topology of ''X'' is unable to discern between the points. Two points of ''X'' are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct poin ...
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