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Stefan–Boltzmann law

The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

For an ideal absorber/emitter or black body, the Stefan–Boltzmann law states that the total energy radiated per unit surface area per unit time (also known as the radiant exitance) is directly proportional to the fourth power of the black body's temperature, $T$: $$ M^{\circ} = \sigma\, T^{4}.$$

The constant of proportionality, $$\sigma$$, is called the Stefan–Boltzmann constant. It has the value

In the general case, the Stefan–Boltzmann law for radiant exitance takes the form: $$ M = \varepsilon\, M^{\circ} = \varepsilon\,\sigma\, T^4 ,$$ where $$\varepsilon$$ is the emissivity of the surface emitting the radiation. The emissivity is generally between zero and one. An emissivity of one corresponds to a black body.

Detailed explanation
The radiant exitance (previously called radiant emittance), $$M$$, has dimensions of energy flux (energy per unit time per unit area), and the SI units of measure are joules per second per square metre (J⋅s$−1$⋅m$−2$), or equivalently, watts per square metre (W⋅m$−2$). The SI unit for absolute temperature, $T$, is the kelvin (K).

To find the total power, $$P$$, radiated from an object, multiply the radiant exitance by the object's surface area, $$A$$: $$ P = A \cdot M = A \, \varepsilon\,\sigma\, T^{4}.$$

Matter that does not absorb all incident radiation emits less total energy than a black body. Emissions are reduced by a factor $$\varepsilon$$, where the emissivity, $$\varepsilon$$, is a material property which, for most matter, satisfies $$0 \leq \varepsilon \leq 1$$. Emissivity can in general depend on wavelength, direction, and polarization. However, the emissivity which appears in the non-directional form of the Stefan–Boltzmann law is the hemispherical total emissivity, which reflects emissions as totaled over all wavelengths, directions, and polarizations.

The form of the Stefan–Boltzmann law that includes emissivity is applicable to all matter, provided that matter is in a state of local thermodynamic equilibrium (LTE) so that its temperature is well-defined. (This is a trivial conclusion, since the emissivity, $$\varepsilon$$, is defined to be the quantity that makes this equation valid. What is non-trivial is the proposition that $$\varepsilon \leq 1$$, which is a consequence of Kirchhoff's law of thermal radiation. )

A so-called grey body is a body for which the spectral emissivity is independent of wavelength, so that the total emissivity, $$\varepsilon$$, is a constant. In the more general (and realistic) case, the spectral emissivity depends on wavelength. The total emissivity, as applicable to the Stefan–Boltzmann law, may be calculated as a weighted average of the spectral emissivity, with the blackbody emission spectrum serving as the weighting function. It follows that if the spectral emissivity depends on wavelength then the total emissivity depends on the temperature, i.e., $$\varepsilon = \varepsilon(T)$$. However, if the dependence on wavelength is small, then the dependence on temperature will be small as well.

Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to have an emissivity greater than 1.

In national and international standards documents, the symbol $$M$$ is recommended to denote radiant exitance; a superscript circle (°) indicates a term relate to a black body. (A subscript "e" is added when it is important to distinguish the energetic (radiometric) quantity radiant exitance, $$M_\mathrm{e}$$, from the analogous human vision (photometric) quantity, luminous exitance, denoted $$M_\mathrm{v}$$. ) In common usage, the symbol used for radiant exitance (often called radiant emittance) varies among different texts and in different fields.

The Stefan–Boltzmann law may be expressed as a formula for radiance as a function of temperature. Radiance is measured in watts per square metre per steradian (W⋅m$−2$⋅sr$−1$). The Stefan–Boltzmann law for the radiance of a black body is: $$ L^\circ_\Omega = \frac{M^{\circ}}\pi = \frac\sigma\pi\, T^{4}.$$

The Stefan–Boltzmann law expressed as a formula for radiation energy density is: $$ w^\circ_\mathrm{e} = \frac{4}{c} \, M^\circ = \frac{4}{c} \, \sigma\, T^{4} ,$$ where $$c$$ is the speed of light.

History
In 1864, John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament. The proportionality to the fourth power of the absolute temperature was deduced by Josef Stefan (1835–1893) in 1877 on the basis of Tyndall's experimental measurements, in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (On the relationship between thermal radiation and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.

A derivation of the law from theoretical considerations was presented by Ludwig Boltzmann (1844–1906) in 1884, drawing upon the work of Adolfo Bartoli. Bartoli in 1876 had derived the existence of radiation pressure from the principles of thermodynamics. Following Bartoli, Boltzmann considered an ideal heat engine using electromagnetic radiation instead of an ideal gas as working matter.

The law was almost immediately experimentally verified. Heinrich Weber in 1888 pointed out deviations at higher temperatures, but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535 K by 1897. The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and the Planck constant, is a direct consequence of Planck's law as formulated in 1900.

Stefan–Boltzmann constant
The Stefan–Boltzmann constant, $σ$, is derived from other known physical constants: $$\sigma = \frac{2 \pi^5 k^4}{15 c^2 h^3} $$ where $k$ is the Boltzmann constant, the $h$ is the Planck constant, and $c$ is the speed of light in vacuum.

As of the 2019 redefinition of SI base units, which establishes exact fixed values for $k$, $h$, and $c$, the Stefan–Boltzmann constant is exactly: $$ \sigma = \left[\frac{2 \pi^5 \left( 1.380\ 649 \times 10^{-23} \right)^4}{15 \left(2.997\ 924\ 58 \times 10^8 \right)^2 \left(6.626\ 070\ 15 \times 10^{-34} \right)^3}\right]\,\frac{\mathrm{W}}{\mathrm{m}^2{\cdot}\mathrm{K}^4} $$ Thus,

Prior to this, the value of $$\sigma$$ was calculated from the measured value of the gas constant.

The numerical value of the Stefan–Boltzmann constant is different in other systems of units, as shown in the table below.

Temperature of the Sun
With his law, Stefan also determined the temperature of the Sun's surface. He inferred from the data of Jacques-Louis Soret (1827–1890) that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same angular diameter as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 °C to 2000 °C. Stefan surmised that 1/3 of the energy flux from the Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5.

Precise measurements of atmospheric absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.574 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K. This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as $5.67 W⋅m^{−2}⋅K^{−4}$ were claimed. The lower value of 1800 °C was determined by Claude Pouillet (1790–1868) in 1838 using the Dulong–Petit law. Pouillet also took just half the value of the Sun's correct energy flux.

Temperature of stars
The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation. So: $$L = 4 \pi R^2 \sigma T^4$$ where $L$ is the luminosity, $σ$ is the Stefan–Boltzmann constant, $R$ is the stellar radius and $T$ is the effective temperature. This formula can then be rearranged to calculate the temperature: $$T = \sqrt[4]{\frac{L}{4 \pi R^2 \sigma}}$$ or alternatively the radius: $$R = \sqrt{\frac{L}{4 \pi \sigma T^4}}$$

The same formulae can also be simplified to compute the parameters relative to the Sun: $$\begin{align} \frac{L}{L_\odot} &= \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4 \\[1ex] \frac{T}{T_\odot} &= \left(\frac{L}{L_\odot}\right)^{1/4} \left(\frac{R_\odot}{R}\right)^{1/2} \\[1ex] \frac{R}{R_\odot} &= \left(\frac{T_\odot}{T}\right)^2 \left(\frac{L}{L_\odot}\right)^{1/2} \end{align}$$ where $$R_\odot$$ is the solar radius, and so forth. They can also be rewritten in terms of the surface area A and radiant exitance $$ M^{\circ}$$: $$\begin{align} L &= A M^{\circ} \\[1ex] M^{\circ} &= \frac{L}{A} \\[1ex] A &= \frac{L}{M^{\circ}} \end{align}$$ where $$A = 4 \pi R^2$$ and $$M^{\circ} = \sigma T^{4}.$$

With the Stefan–Boltzmann law, astronomers can easily infer the radii of stars. The law is also met in the thermodynamics of black holes in so-called Hawking radiation.

Effective temperature of the Earth
Similarly we can calculate the effective temperature of the Earth T⊕ by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation (Earth's own production of energy being small enough to be negligible). The luminosity of the Sun, L⊙, is given by: $$ L_\odot = 4\pi R_\odot^2 \sigma T_\odot^4 $$

At Earth, this energy is passing through a sphere with a radius of a0, the distance between the Earth and the Sun, and the irradiance (received power per unit area) is given by $$ E_\oplus = \frac{L_\odot}{4\pi a_0^2} $$

The Earth has a radius of R⊕, and therefore has a cross-section of $$\pi R_\oplus^2$$. The radiant flux (i.e. solar power) absorbed by the Earth is thus given by: $$ \Phi_\text{abs} = \pi R_\oplus^2 \times E_\oplus $$

Because the Stefan–Boltzmann law uses a fourth power, it has a stabilizing effect on the exchange and the flux emitted by Earth tends to be equal to the flux absorbed, close to the steady state where: $$ \begin{align} 4\pi R_\oplus^2 \sigma T_\oplus^4 &= \pi R_\oplus^2 \times E_\oplus \\ &= \pi R_\oplus^2 \times \frac{4\pi R_\odot^2\sigma T_\odot^4}{4\pi a_0^2} \\ \end{align} $$

T⊕ can then be found: $$ \begin{align} T_\oplus^4 &= \frac{R_\odot^2 T_\odot^4}{4 a_0^2} \\ T_\oplus &= T_\odot \times \sqrt\frac{R_\odot}{2 a_0} \\ & = 5780 \; {\rm K} \times \sqrt{6.957 \times 10^{8} \; {\rm m} \over 2 \times 1.495\ 978\ 707 \times 10^{11} \; {\rm m} } \\ & \approx 279 \; {\rm K} \end{align} $$ where T⊙ is the temperature of the Sun, R⊙ the radius of the Sun, and a0 is the distance between the Earth and the Sun. This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.

The Earth has an albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving 255 K.

The above temperature is Earth's as seen from space, not ground temperature but an average over all emitting bodies of Earth from surface to high altitude. Because of the greenhouse effect, the Earth's actual average surface temperature is about 288 K, which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.

In the above discussion, we have assumed that the whole surface of the earth is at one temperature. Another interesting question is to ask what the temperature of a blackbody surface on the earth would be assuming that it reaches equilibrium with the sunlight falling on it. This of course depends on the angle of the sun on the surface and on how much air the sunlight has gone through. When the sun is at the zenith and the surface is horizontal, the irradiance can be as high as 1120 W/m2. The Stefan–Boltzmann law then gives a temperature of $$T=\left(\frac{1120\text{ W/m}^2}\sigma\right)^{1/4}\approx 375\text{ K}$$ or 102 C. (Above the atmosphere, the result is even higher: 394 K.) We can think of the earth's surface as "trying" to reach equilibrium temperature during the day, but being cooled by the atmosphere, and "trying" to reach equilibrium with starlight and possibly moonlight at night, but being warmed by the atmosphere.

Thermodynamic derivation of the energy density
The fact that the energy density of the box containing radiation is proportional to $$T^{4}$$ can be derived using thermodynamics. This derivation uses the relation between the radiation pressure p and the internal energy density $$u$$, a relation that can be shown using the form of the electromagnetic stress–energy tensor. This relation is: $$ p = \frac{u}{3}.$$

Now, from the fundamental thermodynamic relation $$ dU = T \, dS - p \, dV, $$ we obtain the following expression, after dividing by $$ dV $$ and fixing $$ T $$: $$ \left(\frac{\partial U}{\partial V}\right)_T = T \left(\frac{\partial S}{\partial V}\right)_T - p = T \left(\frac{\partial p}{\partial T}\right)_V - p. $$

The last equality comes from the following Maxwell relation: $$ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V. $$

From the definition of energy density it follows that $$ U = u V $$ where the energy density of radiation only depends on the temperature, therefore $$ \left(\frac{\partial U}{\partial V}\right)_T = u \left(\frac{\partial V}{\partial V}\right)_T = u. $$

Now, the equality is $$ u = T \left(\frac{\partial p}{\partial T}\right)_V - p, $$ after substitution of $$ \left(\frac{\partial U}{\partial V}\right)_{T}.$$

Meanwhile, the pressure is the rate of momentum change per unit area. Since the momentum of a photon is the same as the energy divided by the speed of light, $$ u = \frac{T}{3} \left(\frac{\partial u}{\partial T}\right)_V - \frac{u}{3}, $$ where the factor 1/3 comes from the projection of the momentum transfer onto the normal to the wall of the container.

Since the partial derivative $$ \left(\frac{\partial u}{\partial T}\right)_V $$ can be expressed as a relationship between only $$ u $$ and $$ T $$ (if one isolates it on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes $$ \frac{du}{4u} = \frac{dT}{T}, $$ which leads immediately to $$ u = A T^4 $$, with $$ A $$ as some constant of integration.

Derivation from Planck's law
The law can be derived by considering a small flat black body surface radiating out into a half-sphere. This derivation uses spherical coordinates, with θ as the zenith angle and φ as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where θ = $\pi$/2.

The intensity of the light emitted from the blackbody surface is given by Planck's law, $$I(\nu,T) =\frac{2 h\nu^3}{c^2}\frac{1}{ e^{h\nu/(kT)}-1},$$ where
 * $$I(\nu,T)$$ is the amount of power per unit surface area per unit solid angle per unit frequency emitted at a frequency $$\nu $$ by a black body at temperature T.
 * $$h $$ is the Planck constant
 * $$c $$ is the speed of light, and
 * $$k $$ is the Boltzmann constant.

The quantity $$I(\nu,T) ~A \cos \theta ~d\nu ~d\Omega$$ is the power radiated by a surface of area A through a solid angle $dΩ$ in the frequency range between $5.67$ and $ν + dν$.

The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body, $$\frac{P}{A} = \int_0^\infty I(\nu,T) \, d\nu \int \cos \theta \, d\Omega $$

Note that the cosine appears because black bodies are Lambertian (i.e. they obey Lambert's cosine law), meaning that the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle. To derive the Stefan–Boltzmann law, we must integrate $d\Omega = \sin \theta\, d\theta \, d\varphi$ over the half-sphere and integrate $$\nu$$ from 0 to ∞.

$$ \begin{align} \frac{P}{A} & = \int_0^\infty I(\nu,T) \, d\nu \int_0^{2\pi} \, d\varphi \int_0^{\pi/2} \cos \theta \sin \theta \, d\theta \\ & = \pi \int_0^\infty I(\nu,T) \, d\nu \end{align} $$

Then we plug in for I: $$\frac{P}{A} = \frac{2 \pi h}{c^2} \int_0^\infty \frac{\nu^3}{ e^{\frac{h\nu}{kT}} - 1} \, d\nu $$

To evaluate this integral, do a substitution, $$ \begin{align} u & = \frac{h \nu}{k T} \\[6pt] du & = \frac{h}{k T} \, d\nu \end{align} $$ which gives: $$\frac{P}{A} = \frac{2 \pi h }{c^2} \left(\frac{k T}{h} \right)^4 \int_0^\infty \frac{u^3}{ e^u - 1} \, du .$$

The integral on the right is standard and goes by many names: it is a particular case of a Bose–Einstein integral, the polylogarithm, or the Riemann zeta function $$ \zeta(s) $$. The value of the integral is $$ \Gamma(4)\zeta(4) = \frac{\pi^4}{15} $$ (where $$\Gamma(s)$$ is the Gamma function), giving the result that, for a perfect blackbody surface: $$M^\circ = \sigma T^4 ~, \sigma = \frac{2 \pi^5 k^4 }{15 c^2 h^3} = \frac{\pi^2 k^4}{60 \hbar^3 c^2}. $$

Finally, this proof started out only considering a small flat surface. However, any differentiable surface can be approximated by a collection of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all convex blackbodies, too, so long as the surface has the same temperature throughout. The law extends to radiation from non-convex bodies by using the fact that the convex hull of a black body radiates as though it were itself a black body.

Energy density
The total energy density U can be similarly calculated, except the integration is over the whole sphere and there is no cosine, and the energy flux (U c) should be divided by the velocity c to give the energy density U: $$U = \frac{1}{c} \int_0^\infty I(\nu,T) \, d\nu \int \, d\Omega $$ Thus $\int_0^{\pi/2} \cos \theta \sin \theta \, d\theta $ is replaced by $ \int_0^{\pi} \sin \theta \, d\theta $, giving an extra factor of 4.

Thus, in total: $$U = \frac{4}{c} \, \sigma \, T^4 $$ The product $$\frac{4}{c} \sigma$$ is sometimes known as the radiation constant or radiation density constant.

Decomposition in terms of photons
The Stephan–Boltzmann law can be expressed as $$M^{\circ} = \sigma\, T^4 = N_\mathrm{phot} \, \langle E_\mathrm{phot} \rangle$$ where the flux of photons, $$N_\mathrm{phot}$$, is given by $$N_\mathrm{phot} = \pi \int_0^\infty \frac{B_\nu}{h\nu}\,\mathrm{d}\nu$$ $$N_\mathrm{phot} = \left({ 1.5205\times10^{15}} \; \textrm{photons}{\cdot}\textrm{s}^{-1}{\cdot}\textrm{m}^{-2}{\cdot}\mathrm{K}^{-3}\right)\cdot T^3$$ and the average energy per photon,$$\langle E_\textrm{phot}\rangle$$, is given by $$\langle E_\textrm{phot}\rangle = \frac{\pi^4}{30\,\zeta(3)} k\,T= \left({3.7294 \times 10^{-23}} \mathrm{J}{\cdot}\mathrm{K}^{-1}\right) \cdot T\,.$$

Marr and Wilkin (2012) recommend that students be taught about $$\langle E_\textrm{phot}\rangle$$ instead of being taught Wien's displacement law, and that the above decomposition be taught when the Stefan–Boltzmann law is taught.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Stefan–Boltzmann law


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Josef Stefan

Josef Stefan (Jožef Štefan; 24 March 1835 – 7 January 1893) was a Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire.

Life and work
Stefan was born in the village of St. Peter (Slovene: Sveti Peter) on the outskirts of Klagenfurt) to Aleš (Aleksander) Stefan (1805-1872) and Marija Startinik (1815-1863). His parents, both ethnic Slovenes, did not marry until Josef was eleven. The Stefans were of modest means; his father was a milling assistant and his mother served as a maidservant. Josef was their only child.

Stefan attended elementary school in Klagenfurt, where he showed talent, and was recommended for enrollment at the in 1845. At thirteen, he experienced the revolutionary year of 1848, which inspired him to show sympathy toward the Slovene literary and national movement.

After having graduated top of his class in high school, he briefly considered joining the Benedictine Order, but his great interest in physics prevailed. He left for Vienna in 1853 to study mathematics and physics. His professor of physics in the gymnasium was Karel Robida, who wrote the first Slovene physics textbook. Stefan then earned his habilitation in mathematical physics at the University of Vienna in 1858. During his student years, he also wrote and published a number of poems in Slovene.

Stefan taught physics at the University of Vienna, was Director of the Physical Institute from 1866, Vice-President of the Vienna Academy of Sciences, and member of several scientific institutions in Europe. He died in Vienna, Austria-Hungary. His life and work were extensively documented by the physicist Janez Strnad.

Work
Stefan published nearly 80 scientific articles, mostly in the Bulletins of the Vienna Academy of Sciences. He is best known for originating Stefan's law in 1879, a physical power law stating that the total radiation from a black body is proportional to the fourth power of its thermodynamic temperature T:

 * $$ j^{\star} = \sigma T^{4}$$

He derived this law from the measurements of the French physicists Dulong and Petit. As both incident radiation and blackbody emission are always equal, this equation applies equally to the temperature of any ideal body subject to incident radiation across its surface. In 1884, the law was extended to apply to grey-body emissions by Stefan's student Ludwig Boltzmann and hence is known as Stefan–Boltzmann law. Boltzmann treated a heat engine with light as a working matter. This law is the only physical law of nature named after a Slovene physicist. Today, the law is derived from Planck's law of black-body radiation:

 * $$ j^{\star} = \int_{0}^{\infty} \left( {dj^{\star}\over d\lambda} \right) d\lambda $$

With his law, Stefan determined the temperature of the Sun's surface, which he calculated to be 5430 °C. This was the first sensible value for the temperature of the Sun.

Stefan provided the first measurements of the thermal conductivity of gases, treated evaporation, and among others studied diffusion, heat conduction in fluids. For his treatise on optics, the University of Vienna bestowed the Lieben Prize on him. Because of his early work in calculating evaporation and diffusion rates, flow from a droplet or particle that is induced by evaporation or sublimation at the surface is now called the Stefan flow.

Very important are also his electromagnetic equations, defined in vector notation, and works in the kinetic theory of heat. Stefan was among the first physicists in Europe who fully understood Maxwell's electromagnetic theory and one of the few outside England who expanded on it. He calculated inductivity of a coil with a quadratic cross-section, and he corrected Maxwell's miscalculation. He also researched a phenomenon called the skin effect, where high-frequency electric current is greater on the surface of a conductor than in its interior.

In mathematics, the Stefan problems or Stefan's tasks with movable boundary are well known. The problem was first studied by Lamé and Clapeyron in 1831. Stefan solved the problem when he was calculating how quickly a layer of ice on water grows (Stefan's equation ).

Eponymous terms
Several concepts in physics and mathematics are named after Joseph Stefan:
 * Stefan–Boltzmann law
 * Stefan–Boltzmann constant σ
 * Stefan adhesion
 * Stefan problem
 * Stefan's equation
 * Stefan's formula
 * Stefan flow
 * Stefan number
 * Stefan tube
 * Maxwell–Stefan diffusion

The Jožef Stefan Institute, Slovenia's premier scientific establishment, is also named after him, and also:
 * Stefan (crater) on the Moon

title: Josef Stefan


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WebCrawler

WebCrawler is a search engine, and one of the oldest surviving search engines on the web today. For many years, it operated as a metasearch engine. WebCrawler was the first web search engine to provide full text search.

History
Brian Pinkerton first started working on WebCrawler, which was originally a desktop application, on January 27, 1994 at the University of Washington. On March 15, 1994, he generated a list of the top 25 websites.

WebCrawler launched on April 21, 1994, with more than 4,000 different websites in its database and on November 14, 1994, WebCrawler served its 1 millionth search query for "nuclear weapons design and research".

On December 1, 1994, WebCrawler acquired two sponsors, DealerNet and Starwave, which provided money to keep WebCrawler operating. Starting on October 3, 1995, WebCrawler was fully supported by advertising, but separated the adverts from search results.

On June 1, 1995, America Online (AOL) acquired WebCrawler. After being acquired by AOL, the website introduced its mascot "Spidey" on September 1, 1995.

Starting in April 1996, WebCrawler also included the human-edited internet guide GNN Select, which was also under AOL ownership.

On April 1, 1997, Excite acquired WebCrawler from AOL for $12.3 million.

WebCrawler received a redesign on June 16, 1997, adding WebCrawler Shortcuts, which suggested alternative links to material related to a search topic.

WebCrawler was maintained by Excite as a separate search engine with its own database until 2001, when it started using Excite's own database, effectively putting an end to WebCrawler as an independent search engine. Later that year, Excite (then called Excite@Home) went bankrupt and WebCrawler was bought by InfoSpace in 2001.

Pinkerton, WebCrawler's creator, led the Amazon A9.com search division as of 2012.

In July 2016, InfoSpace was sold by parent company Blucora to OpenMail for $45 million, putting WebCrawler under the ownership of OpenMail. OpenMail was later renamed System1.

In 2018, WebCrawler was redesigned from scratch and its logo was changed.

Traffic
WebCrawler was highly successful early on. At one point, it was unusable during peak times due to server overload. It was the second most visited website on the internet in February 1996, but it quickly dropped below rival search engines and directories such as Yahoo!, Infoseek, Lycos, and Excite in 1997.

title: WebCrawler


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Network address translation

Network address translation (NAT) is a method of mapping an IP address space into another by modifying network address information in the IP header of packets while they are in transit across a traffic routing device. The technique was originally used to bypass the need to assign a new address to every host when a network was moved, or when the upstream Internet service provider was replaced, but could not route the network's address space. It has become a popular and essential tool in conserving global address space in the face of IPv4 address exhaustion. One Internet-routable IP address of a NAT gateway can be used for an entire private network.

As network address translation modifies the IP address information in packets, NAT implementations may vary in their specific behavior in various addressing cases and their effect on network traffic. The specifics of NAT behavior are not commonly documented by vendors of equipment containing NAT implementations.

History
IPv4 uses 32-bit addresses, capable of uniquely addressing about 4.4 billion devices. By 1992 it became evident that that would not be enough. The 1994 describes NAT as a "short-term solution" to the two most compelling problems facing the IP Internet at that time: IP address depletion and scaling in routing. By 2004, NAT had become widespread.

Basic NAT
The simplest type of NAT provides a one-to-one translation of IP addresses (RFC 1631). refers to this type of NAT as basic NAT; it is also called a one-to-one NAT. In this type of NAT, only the IP addresses, IP header checksum, and any higher-level checksums that include the IP address are changed. Basic NAT can be used to interconnect two IP networks that have incompatible addressing.

One-to-many NAT

The majority of network address translators map multiple private hosts to one publicly exposed IP address.

Here is a typical configuration:

 * 1) A local network uses one of the designated private IP address subnets (RFC 1918 ).
 * 2) The network has a router having both a private and a public address. The private address is used by the router for communicating with other devices in the private local network. The public address (typically assigned by an Internet service provider) is used by the router for communicating with the rest of the Internet.
 * 3) As traffic passes from the network to the Internet, the router translates the source address in each packet from a private address to the router's own public address. The router tracks basic data about each active connection (particularly the destination address and port). When the router receives inbound traffic from the Internet, it uses the connection tracking data it stored during the outbound phase to determine to which private address (if any) it should forward the reply.

All IP packets have a source IP address and a destination IP address. Typically, packets passing from the private network to the public network will have their source address modified, while packets passing from the public network back to the private network will have their destination address modified. To avoid ambiguity in how replies are translated, further modifications to the packets are required. The vast bulk of Internet traffic uses Transmission Control Protocol (TCP) or User Datagram Protocol (UDP). For these protocols, the port numbers are changed so that the combination of IP address (within the IP header) and port number (within the Transport Layer header) on the returned packet can be unambiguously mapped to the corresponding private network destination. RFC 2663 uses the term network address and port translation (NAPT) for this type of NAT. Other names include port address translation (PAT), IP masquerading, NAT overload and many-to-one NAT. This is the most common type of NAT and has become synonymous with the term "NAT" in common usage.

This method allows communication through the router only when the conversation originates in the private network, since the initial originating transmission is what establishes the required information in the translation tables. Thus a web browser within the private network would be able to browse websites that are outside the network, whereas web browsers outside the network would be unable to browse a website hosted within. Protocols not based on TCP and UDP require other translation techniques.

An additional benefit of one-to-many NAT is that it mitigates IPv4 address exhaustion by allowing entire networks to be connected to the Internet using a single public IP address.

Methods of translation
Network address and port translation may be implemented in several ways. Some applications that use IP address information may need to determine the external address of a network address translator. This is the address that its communication peers in the external network detect. Furthermore, it may be necessary to examine and categorize the type of mapping in use, for example when it is desired to set up a direct communication path between two clients both of which are behind separate NAT gateways.

For this purpose, RFC 3489 specified a protocol called Simple Traversal of UDP over NATs (STUN) in 2003. It classified NAT implementations as full-cone NAT, (address) restricted-cone NAT, port-restricted cone NAT or symmetric NAT, and proposed a methodology for testing a device accordingly. However, these procedures have since been deprecated from standards status, as the methods are inadequate to correctly assess many devices. RFC 5389 standardized new methods in 2008 and the acronym STUN now represents the new title of the specification: Session Traversal Utilities for NAT.

Many NAT implementations combine these types, so it is better to refer to specific individual NAT behavior instead of using the Cone/Symmetric terminology. RFC 4787 attempts to alleviate confusion by introducing standardized terminology for observed behaviors. For the first bullet in each row of the above table, the RFC would characterize Full-Cone, Restricted-Cone, and Port-Restricted Cone NATs as having an Endpoint-Independent Mapping, whereas it would characterize a Symmetric NAT as having an Address- and Port-Dependent Mapping. For the second bullet in each row of the above table, RFC 4787 would also label Full-Cone NAT as having an Endpoint-Independent Filtering, Restricted-Cone NAT as having an Address-Dependent Filtering, Port-Restricted Cone NAT as having an Address and Port-Dependent Filtering, and Symmetric NAT as having either an Address-Dependent Filtering or Address and Port-Dependent Filtering. Other classifications of NAT behavior mentioned in the RFC include whether they preserve ports, when and how mappings are refreshed, whether external mappings can be used by internal hosts (i.e., its hairpinning behavior), and the level of determinism NATs exhibit when applying all these rules. Specifically, most NATs combine symmetric NAT for outgoing connections with static port mapping, where incoming packets addressed to the external address and port are redirected to a specific internal address and port.

NAT mapping vs NAT filtering
RFC 4787 makes a distinction between NAT mapping and NAT filtering.

Section 4.1 of the RFC covers NAT mapping and specifies how an external IP address and port number should be translated into an internal IP address and port number. It defines Endpoint-Independent Mapping, Address-Dependent Mapping and Address and Port-Dependent Mapping, explains that these three possible choices do not relate to the security of the NAT as security is determined by the filtering behavior and then specifies 'A NAT MUST have an "Endpoint-Independent Mapping" behavior.'

Section 5 of the RFC covers NAT filtering and describes what criteria are used by the NAT to filter packets originating from specific external endpoints. The options are Endpoint-Independent Filtering, Address-Dependent Filtering and Address and Port-Dependent Filtering. Endpoint-Independent Filtering is recommended where maximum application transparency is required while Address-Dependent Filtering is recommended where more stringent filtering behavior is most important.

Some NAT devices are not yet compliant with RFC 4787 as they treat NAT mapping and filtering in the same way so that their configuration option for changing the NAT filtering method also changes the NAT mapping method (e.g. Netgate TNSR). The PF firewall has a patch available to enable RFC 4787 support but this has not yet been merged.

Type of NAT and NAT traversal, role of port preservation for TCP
The NAT traversal problem arises when peers behind different NATs try to communicate. One way to solve this problem is to use port forwarding. Another way is to use various NAT traversal techniques. The most popular technique for TCP NAT traversal is TCP hole punching.

TCP hole punching requires the NAT to follow the port preservation design for TCP. For a given outgoing TCP communication, the same port numbers are used on both sides of the NAT. NAT port preservation for outgoing TCP connections is crucial for TCP NAT traversal because, under TCP, one port can only be used for one communication at a time, so programs bind distinct TCP sockets to ephemeral ports for each TCP communication, rendering NAT port prediction impossible for TCP.

On the other hand, for UDP, NATs do not need port preservation. Indeed, multiple UDP communications (each with a distinct endpoint) can occur on the same source port, and applications usually reuse the same UDP socket to send packets to distinct hosts. This makes port prediction straightforward, as it is the same source port for each packet.

Furthermore, port preservation in NAT for TCP allows P2P protocols to offer less complexity and less latency because there is no need to use a third party (like STUN) to discover the NAT port since the application itself already knows the NAT port.

However, if two internal hosts attempt to communicate with the same external host using the same port number, the NAT may attempt to use a different external IP address for the second connection or may need to forgo port preservation and remap the port.

, roughly 70% of the clients in P2P networks employed some form of NAT.

Establishing two-way communication
Every TCP and UDP packet contains a source port number and a destination port number. Each of those packets is encapsulated in an IP packet, whose IP header contains a source IP address and a destination IP address. The IP address/protocol/port number triple defines an association with a network socket.

For publicly accessible services such as web and mail servers the port number is important. For example, port 80 connects through a socket to the web server software and port 25 to a mail server's SMTP daemon. The IP address of a public server is also important, similar in global uniqueness to a postal address or telephone number. Both IP address and port number must be correctly known by all hosts wishing to successfully communicate.

Private IP addresses as described in RFC 1918 are usable only on private networks not directly connected to the internet. Ports are endpoints of communication unique to that host, so a connection through the NAT device is maintained by the combined mapping of port and IP address. A private address on the inside of the NAT is mapped to an external public address. Port address translation (PAT) resolves conflicts that arise when multiple hosts happen to use the same source port number to establish different external connections at the same time.

Telephone number extension analogy
A NAT device is similar to a phone system at an office that has one public telephone number and multiple extensions. Outbound phone calls made from the office all appear to come from the same telephone number. However, an incoming call that does not specify an extension cannot be automatically transferred to an individual inside the office. In this scenario, the office is a private LAN, the main phone number is the public IP address, and the individual extensions are unique port numbers.

Translation process
With NAT, all communications sent to external hosts actually contain the external IP address and port information of the NAT device instead of internal host IP addresses or port numbers. NAT only translates IP addresses and ports of its internal hosts, hiding the true endpoint of an internal host on a private network.

When a computer on the private (internal) network sends an IP packet to the external network, the NAT device replaces the internal source IP address in the packet header with the external IP address of the NAT device. PAT may then assign the connection a port number from a pool of available ports, inserting this port number in the source port field. The packet is then forwarded to the external network. The NAT device then makes an entry in a translation table containing the internal IP address, original source port, and the translated source port. Subsequent packets from the same internal source IP address and port number are translated to the same external source IP address and port number. The computer receiving a packet that has undergone NAT establishes a connection to the port and IP address specified in the altered packet, oblivious to the fact that the supplied address is being translated.

Upon receiving a packet from the external network, the NAT device searches the translation table based on the destination port in the packet header. If a match is found, the destination IP address and port number is replaced with the values found in the table and the packet is forwarded to the inside network. Otherwise, if the destination port number of the incoming packet is not found in the translation table, the packet is dropped or rejected because the PAT device doesn't know where to send it.

Applications

 * Routing: Network address translation can be used to mitigate IP address overlap. Address overlap occurs when hosts in different networks with the same IP address space try to reach the same destination host. This is most often a misconfiguration and may result from the merger of two networks or subnets, especially when using RFC 1918 private network addressing. The destination host experiences traffic apparently arriving from the same network, and intermediate routers have no way to determine where reply traffic should be sent to. The solution is either renumbering to eliminate overlap or network address translation.

 * Load balancing: In client–server applications, load balancers forward client requests to a set of server computers to manage the workload of each server. Network address translation may be used to map a representative IP address of the server cluster to specific hosts that service the request.

Related techniques
IEEE Reverse Address and Port Translation (RAPT or RAT) allows a host whose real IP address changes from time to time to remain reachable as a server via a fixed home IP address. Cisco's RAPT implementation is PAT or NAT overloading and maps multiple private IP addresses to a single public IP address. Multiple addresses can be mapped to a single address because each private address is tracked by a port number. PAT uses unique source port numbers on the inside global IP address to distinguish between translations. PAT attempts to preserve the original source port. If this source port is already used, PAT assigns the first available port number starting from the beginning of the appropriate port group 0–511, 512–1023, or 1024–65535. When there are no more ports available and there is more than one external IP address configured, PAT moves to the next IP address to try to allocate the original source port again. This process continues until it runs out of available ports and external IP addresses.

Mapping of Address and Port is a Cisco proposal that combines Address plus Port translation with tunneling of the IPv4 packets over an ISP provider's internal IPv6 network. In effect, it is an (almost) stateless alternative to carrier-grade NAT and DS-Lite that pushes the IPv4 address/port translation function (and the maintenance of NAT state) entirely into the existing customer premises equipment NAT implementation. Thus avoiding the NAT444 and statefulness problems of carrier-grade NAT, and also provides a transition mechanism for the deployment of native IPv6 at the same time with very little added complexity.

Issues and limitations
Hosts behind NAT-enabled routers do not have end-to-end connectivity and cannot participate in some internet protocols. Services that require the initiation of TCP connections from the outside network, or that use stateless protocols such as those using UDP, can be disrupted. Unless the NAT router makes a specific effort to support such protocols, incoming packets cannot reach their destination. Some protocols can accommodate one instance of NAT between participating hosts ("passive mode" FTP, for example), sometimes with the assistance of an application-level gateway (see ), but fail when both systems are separated from the internet by NAT. The use of NAT also complicates tunneling protocols such as IPsec because NAT modifies values in the headers which interfere with the integrity checks done by IPsec and other tunneling protocols.

End-to-end connectivity has been a core principle of the Internet, supported, for example, by the Internet Architecture Board. Current Internet architectural documents observe that NAT is a violation of the end-to-end principle, but that NAT does have a valid role in careful design. There is considerably more concern with the use of IPv6 NAT, and many IPv6 architects believe IPv6 was intended to remove the need for NAT.

An implementation that only tracks ports can be quickly depleted by internal applications that use multiple simultaneous connections such as an HTTP request for a web page with many embedded objects. This problem can be mitigated by tracking the destination IP address in addition to the port thus sharing a single local port with many remote hosts. This additional tracking increases implementation complexity and computing resources at the translation device.

Because the internal addresses are all disguised behind one publicly accessible address, it is impossible for external hosts to directly initiate a connection to a particular internal host. Applications such as VOIP, videoconferencing, and other peer-to-peer applications must use NAT traversal techniques to function.

Fragmentation and checksums
Pure NAT, operating on IP alone, may or may not correctly parse protocols with payloads containing information about IP, such as ICMP. This depends on whether the payload is interpreted by a host on the inside or outside of the translation. Basic protocols as TCP and UDP cannot function properly unless NAT takes action beyond the network layer.

IP packets have a checksum in each packet header, which provides error detection only for the header. IP datagrams may become fragmented and it is necessary for a NAT to reassemble these fragments to allow correct recalculation of higher-level checksums and correct tracking of which packets belong to which connection.

TCP and UDP, have a checksum that covers all the data they carry, as well as the TCP or UDP header, plus a pseudo-header that contains the source and destination IP addresses of the packet carrying the TCP or UDP header. For an originating NAT to pass TCP or UDP successfully, it must recompute the TCP or UDP header checksum based on the translated IP addresses, not the original ones, and put that checksum into the TCP or UDP header of the first packet of the fragmented set of packets.

Alternatively, the originating host may perform path MTU Discovery to determine the packet size that can be transmitted without fragmentation and then set the don't fragment (DF) bit in the appropriate packet header field. This is only a one-way solution, because the responding host can send packets of any size, which may be fragmented before reaching the NAT.

DNAT
Destination network address translation (DNAT) is a technique for transparently changing the destination IP address of a routed packet and performing the inverse function for any replies. Any router situated between two endpoints can perform this transformation of the packet.

DNAT is commonly used to publish a service located in a private network on a publicly accessible IP address. This use of DNAT is also called port forwarding, or DMZ when used on an entire server, which becomes exposed to the WAN, becoming analogous to an undefended military demilitarized zone (DMZ).

SNAT
The meaning of the term SNAT varies by vendor:
 * source NAT is a common expansion and is the counterpart of destination NAT (DNAT). This is used to describe one-to-many NAT; NAT for outgoing connections to public services.
 * stateful NAT is used by Cisco Systems
 * static NAT is used by WatchGuard
 * secure NAT is used by F5 Networks and by Microsoft (in regard to the ISA Server)

Secure network address translation (SNAT) is part of Microsoft's Internet Security and Acceleration Server and is an extension to the NAT driver built into Microsoft Windows Server. It provides connection tracking and filtering for the additional network connections needed for the FTP, ICMP, H.323, and PPTP protocols as well as the ability to configure a transparent HTTP proxy server.

Dynamic network address translation
Dynamic NAT, just like static NAT, is not common in smaller networks but is found within larger corporations with complex networks. Where static NAT provides a one-to-one internal to public static IP address mapping, dynamic NAT uses a group of public IP addresses.

NAT hairpinning
NAT hairpinning, also known as NAT loopback or NAT reflection, is a feature in many consumer routers where a machine on the LAN is able to access another machine on the LAN via the external IP address of the LAN/router (with port forwarding set up on the router to direct requests to the appropriate machine on the LAN). This notion is officially described in 2008,.

The following describes an example network:
 * Public address: . This is the address of the WAN interface on the router.
 * Internal address of router:
 * Address of the server:
 * Address of a local computer:

If a packet is sent to by a computer at, the packet would normally be routed to the default gateway (the router) A router with the NAT loopback feature detects that  is the address of its WAN interface, and treats the packet as if coming from that interface. It determines the destination for that packet, based on DNAT (port forwarding) rules for the destination. If the data were sent to port 80 and a DNAT rule exists for port 80 directed to, then the host at that address receives the packet.

If no applicable DNAT rule is available, the router drops the packet. An ICMP Destination Unreachable reply may be sent. If any DNAT rules were present, address translation is still in effect; the router still rewrites the source IP address in the packet. The local computer sends the packet as coming from, but the server  receives it as coming from. When the server replies, the process is identical to an external sender. Thus, two-way communication is possible between hosts inside the LAN network via the public IP address.

NAT in IPv6
Network address translation is not commonly used in IPv6 because one of the design goals of IPv6 is to restore end-to-end network connectivity. The large addressing space of IPv6 obviates the need to conserve addresses and every device can be given a unique globally routable address. Use of unique local addresses in combination with network prefix translation can achieve results similar to NAT.

The large addressing space of IPv6 can still be defeated depending on the actual prefix length given by the carrier. It is not uncommon to be handed a /64 prefix – the smallest recommended subnet – for an entire home network, requiring a variety of techniques to be used to manually subdivide the range for all devices to remain reachable. Even actual IPv6-to-IPv6 NAT, NAT66, can turn out useful at times: the APNIC blog outlines a case where the author was only provided a single address (/128).

Applications affected by NAT
Some application layer protocols, such as File Transfer Protocol (FTP) and Session Initiation Protocol (SIP), send explicit network addresses within their application data. File Transfer Protocol in active mode, for example, uses separate connections for control traffic (commands) and for data traffic (file contents). When requesting a file transfer, the host making the request identifies the corresponding data connection by its network layer and transport layer addresses. If the host making the request lies behind a simple NAT firewall, the translation of the IP address or TCP port number makes the information received by the server invalid. SIP commonly controls voice over IP calls, and suffer the same problem. SIP and its accompanying Session Description Protocol may use multiple ports to set up a connection and transmit voice stream via Real-time Transport Protocol. IP addresses and port numbers are encoded in the payload data and must be known before the traversal of NATs. Without special techniques, such as STUN, NAT behavior is unpredictable and communications may fail. Application Layer Gateway (ALG) software or hardware may correct these problems. An ALG software module running on a NAT firewall device updates any payload data made invalid by address translation. ALGs need to understand the higher-layer protocol that they need to fix, and so each protocol with this problem requires a separate ALG. For example, on many Linux systems, there are kernel modules called connection trackers that serve to implement ALGs. However, ALG cannot work if the protocol data is encrypted.

Another possible solution to this problem is to use NAT traversal techniques using protocols such as STUN or Interactive Connectivity Establishment (ICE), or proprietary approaches in a session border controller. NAT traversal is possible in both TCP- and UDP-based applications, but the UDP-based technique is simpler, more widely understood, and more compatible with legacy NATs. In either case, the high-level protocol must be designed with NAT traversal in mind, and it does not work reliably across symmetric NATs or other poorly behaved legacy NATs.

Other possibilities are Port Control Protocol (PCP), NAT Port Mapping Protocol (NAT-PMP), or Internet Gateway Device Protocol but these require the NAT device to implement that protocol.

Most client–server protocols (FTP being the main exception), however, do not send layer 3 contact information and do not require any special treatment by NATs. In fact, avoiding NAT complications is practically a requirement when designing new higher-layer protocols today.

NATs can also cause problems where IPsec encryption is applied and in cases where multiple devices such as SIP phones are located behind a NAT. Phones that encrypt their signaling with IPsec encapsulate the port information within an encrypted packet, meaning that NAT devices cannot access and translate the port. In these cases, the NAT devices revert to simple NAT operations. This means that all traffic returning to the NAT is mapped onto one client, causing service to more than one client behind the NAT to fail. There are a couple of solutions to this problem: one is to use TLS, which operates at layer 4 and does not mask the port number; another is to encapsulate the IPsec within UDP – the latter being the solution chosen by TISPAN to achieve secure NAT traversal, or a NAT with "IPsec Passthru" support; another is to use a session border controller to help traverse the NAT.

Interactive Connectivity Establishment is a NAT traversal technique that does not rely on ALG support.

The DNS protocol vulnerability announced by Dan Kaminsky on July 8, 2008, is indirectly affected by NAT port mapping. To avoid DNS cache poisoning, it is highly desirable not to translate UDP source port numbers of outgoing DNS requests from a DNS server behind a firewall that implements NAT. The recommended workaround for the DNS vulnerability is to make all caching DNS servers use randomized UDP source ports. If the NAT function de-randomizes the UDP source ports, the DNS server becomes vulnerable.

Examples of NAT software

 * Internet Connection Sharing (ICS): NAT & DHCP implementation included with Windows desktop operating systems
 * IPFilter: included with (OpenSolaris, FreeBSD and NetBSD, available for many other Unix-like operating systems
 * ipfirewall (ipfw): FreeBSD-native packet filter
 * Netfilter with iptables/nftables: the Linux packet filter
 * NPF: NetBSD-native packet filter
 * PF: OpenBSD-native packet filter
 * Routing and Remote Access Service (RRAS): routing implementation included with Windows Server operating systems
 * VPP: user space packet forwarding implementation for Linux
 * WinGate: third-party routing implementation for Windows

title: Network address translation


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Key (cryptography)

A key in cryptography is a piece of information, usually a string of numbers or letters that are stored in a file, which, when processed through a cryptographic algorithm, can encode or decode cryptographic data. Based on the used method, the key can be different sizes and varieties, but in all cases, the strength of the encryption relies on the security of the key being maintained. A key's security strength is dependent on its algorithm, the size of the key, the generation of the key, and the process of key exchange.

Scope
The key is what is used to encrypt data from plaintext to ciphertext. There are different methods for utilizing keys and encryption.

Symmetric cryptography
Symmetric cryptography refers to the practice of the same key being used for both encryption and decryption.

Asymmetric cryptography
Asymmetric cryptography has separate keys for encrypting and decrypting. These keys are known as the public and private keys, respectively.

Purpose
Since the key protects the confidentiality and integrity of the system, it is important to be kept secret from unauthorized parties. With public key cryptography, only the private key must be kept secret, but with symmetric cryptography, it is important to maintain the confidentiality of the key. Kerckhoff's principle states that the entire security of the cryptographic system relies on the secrecy of the key.

Key sizes
Key size is the number of bits in the key defined by the algorithm. This size defines the upper bound of the cryptographic algorithm's security. The larger the key size, the longer it will take before the key is compromised by a brute force attack. Since perfect secrecy is not feasible for key algorithms, researches are now more focused on computational security.

In the past, keys were required to be a minimum of 40 bits in length, however, as technology advanced, these keys were being broken quicker and quicker. As a response, restrictions on symmetric keys were enhanced to be greater in size.

Currently, 2048 bit RSA is commonly used, which is sufficient for current systems. However, current key sizes would all be cracked quickly with a powerful quantum computer.

“The keys used in public key cryptography have some mathematical structure. For example, public keys used in the RSA system are the product of two prime numbers. Thus public key systems require longer key lengths than symmetric systems for an equivalent level of security. 3072 bits is the suggested key length for systems based on factoring and integer discrete logarithms which aim to have security equivalent to a 128 bit symmetric cipher.”

Key generation
To prevent a key from being guessed, keys need to be generated randomly and contain sufficient entropy. The problem of how to safely generate random keys is difficult and has been addressed in many ways by various cryptographic systems. A key can directly be generated by using the output of a Random Bit Generator (RBG), a system that generates a sequence of unpredictable and unbiased bits. A RBG can be used to directly produce either a symmetric key or the random output for an asymmetric key pair generation. Alternatively, a key can also be indirectly created during a key-agreement transaction, from another key or from a password.

Some operating systems include tools for "collecting" entropy from the timing of unpredictable operations such as disk drive head movements. For the production of small amounts of keying material, ordinary dice provide a good source of high-quality randomness.

Establishment scheme
The security of a key is dependent on how a key is exchanged between parties. Establishing a secured communication channel is necessary so that outsiders cannot obtain the key. A key establishment scheme (or key exchange) is used to transfer an encryption key among entities. Key agreement and key transport are the two types of a key exchange scheme that are used to be  remotely exchanged between entities. In a key agreement scheme, a secret key, which is used between the sender and the receiver to encrypt and decrypt information, is set up to be sent indirectly. All parties exchange information (the shared secret) that permits each party to derive the secret key material. In a key transport scheme, encrypted keying material that is chosen by the sender is transported to the receiver. Either symmetric key or asymmetric key techniques can be used in both schemes.

The Diffie–Hellman key exchange and Rivest-Shamir-Adleman (RSA) are the most two widely used key exchange algorithms. In 1976, Whitfield Diffie and Martin Hellman constructed the Diffie–Hellman algorithm, which was the first public key algorithm. The Diffie–Hellman key exchange protocol allows key exchange over an insecure channel by electronically generating a shared key between two parties. On the other hand, RSA is a form of the asymmetric key system which consists of three steps: key generation, encryption, and decryption.

Key confirmation delivers an assurance between the key confirmation recipient and provider that the shared keying materials are correct and established. The National Institute of Standards and Technology recommends key confirmation to be integrated into a key establishment scheme to validate its implementations.

Management
Key management concerns the generation, establishment, storage, usage and replacement of cryptographic keys. A key management system (KMS) typically includes three steps of establishing, storing and using keys. The base of security for the generation, storage, distribution, use and destruction of keys depends on successful key management protocols.

Key vs password
A password is a memorized series of characters including letters, digits, and other special symbols that are used to verify identity. It is often produced by a human user or a password management software to protect personal and sensitive information or generate cryptographic keys. Passwords are often created to be memorized by users and may contain non-random information such as dictionary words. On the other hand, a key can help strengthen password protection by implementing a cryptographic algorithm which is difficult to guess or replace the password altogether. A key is generated based on random or pseudo-random data and can often be unreadable to humans.

A password is less safe than a cryptographic key due to its low entropy, randomness, and human-readable properties. However, the password may be the only secret data that is accessible to the cryptographic algorithm for information security in some applications such as securing information in storage devices. Thus, a deterministic algorithm called a key derivation function (KDF) uses a password to generate the secure cryptographic keying material to compensate for the password's weakness. Various methods such as adding a salt or key stretching may be used in the generation.

title: Key (cryptography)


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Symmetric-key algorithm

Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between the two keys. The keys, in practice, represent a shared secret between two or more parties that can be used to maintain a private information link. The requirement that both parties have access to the secret key is one of the main drawbacks of symmetric-key encryption, in comparison to public-key encryption (also known as asymmetric-key encryption). However, symmetric-key encryption algorithms are usually better for bulk encryption. With exception of the one-time pad they have a smaller key size, which means less storage space and faster transmission. Due to this, asymmetric-key encryption is often used to exchange the secret key for symmetric-key encryption.

Types
Symmetric-key encryption can use either stream ciphers or block ciphers.

Stream ciphers encrypt the digits (typically bytes), or letters (in substitution ciphers) of a message one at a time. An example is ChaCha20. Substitution ciphers are well-known ciphers, but can be easily decrypted using a frequency table.

Block ciphers take a number of bits and encrypt them in a single unit, padding the plaintext to achieve a multiple of the block size. The Advanced Encryption Standard (AES) algorithm, approved by NIST in December 2001, uses 128-bit blocks.

Implementations
Examples of popular symmetric-key algorithms include Twofish, Serpent, AES (Rijndael), Camellia, Salsa20, ChaCha20, Blowfish, CAST5, Kuznyechik, RC4, DES, 3DES, Skipjack, Safer, and IDEA.

Use as a cryptographic primitive
Symmetric ciphers are commonly used to achieve other cryptographic primitives than just encryption.

Encrypting a message does not guarantee that it will remain unchanged while encrypted. Hence, often a message authentication code is added to a ciphertext to ensure that changes to the ciphertext will be noted by the receiver. Message authentication codes can be constructed from an AEAD cipher (e.g. AES-GCM).

However, symmetric ciphers cannot be used for non-repudiation purposes except by involving additional parties. See the ISO/IEC 13888-2 standard.

Another application is to build hash functions from block ciphers. See one-way compression function for descriptions of several such methods.

Construction of symmetric ciphers
Many modern block ciphers are based on a construction proposed by Horst Feistel. Feistel's construction makes it possible to build invertible functions from other functions that are themselves not invertible.

Security of symmetric ciphers
Symmetric ciphers have historically been susceptible to known-plaintext attacks, chosen-plaintext attacks, differential cryptanalysis and linear cryptanalysis. Careful construction of the functions for each round can greatly reduce the chances of a successful attack. It is also possible to increase the key length or the rounds in the encryption process to better protect against attack. This, however, tends to increase the processing power and decrease the speed at which the process runs due to the amount of operations the system needs to do.

Most modern symmetric-key algorithms appear to be resistant to the threat of post-quantum cryptography. Quantum computers would exponentially increase the speed at which these ciphers can be decoded; notably, Grover's algorithm would take the square-root of the time traditionally required for a brute-force attack, although these vulnerabilities can be compensated for by doubling key length. For example, a 128 bit AES cipher would not be secure against such an attack as it would reduce the time required to test all possible iterations from over 10 quintillion years to about six months. By contrast, it would still take a quantum computer the same amount of time to decode a 256 bit AES cipher as it would a conventional computer to decode a 128 bit AES cipher. For this reason, AES-256 is believed to be "quantum resistant".

Key establishment
Symmetric-key algorithms require both the sender and the recipient of a message to have the same secret key. All early cryptographic systems required either the sender or the recipient to somehow receive a copy of that secret key over a physically secure channel.

Nearly all modern cryptographic systems still use symmetric-key algorithms internally to encrypt the bulk of the messages, but they eliminate the need for a physically secure channel by using Diffie–Hellman key exchange or some other public-key protocol to securely come to agreement on a fresh new secret key for each session/conversation (forward secrecy).

Key generation
When used with asymmetric ciphers for key transfer, pseudorandom key generators are nearly always used to generate the symmetric cipher session keys. However, lack of randomness in those generators or in their initialization vectors is disastrous and has led to cryptanalytic breaks in the past. Therefore, it is essential that an implementation use a source of high entropy for its initialization.

Reciprocal cipher
A reciprocal cipher is a cipher where, just as one enters the plaintext into the cryptography system to get the ciphertext, one could enter the ciphertext into the same place in the system to get the plaintext. A reciprocal cipher is also sometimes referred as self-reciprocal cipher.

Practically all mechanical cipher machines implement a reciprocal cipher, a mathematical involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.

Examples of reciprocal ciphers include:
 * Atbash
 * Beaufort cipher
 * Enigma machine
 * Marie Antoinette and Axel von Fersen communicated with a self-reciprocal cipher.
 * the Porta polyalphabetic cipher is self-reciprocal.
 * Purple cipher
 * RC4
 * ROT13
 * XOR cipher
 * Vatsyayana cipher

The majority of all modern ciphers can be classified as either a stream cipher, most of which use a reciprocal XOR cipher combiner, or a block cipher, most of which use a Feistel cipher or Lai–Massey scheme with a reciprocal transformation in each round.

title: Symmetric-key algorithm


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Lenin Peak

Lenin Peak or Ibn Sina (Avicenna) Peak (Ленин Чокусу, لەنىن چوقۇسۇ; Пик Ленина; қуллаи Ленин, renamed қуллаи Абӯалӣ ибни Сино (qulla‘i Abûalî ibni Sino) in July 2006 ), rises to 7,134 m in Gorno-Badakhshan (GBAO) on the Kyrgyzstan–Tajikistan border, and is the second-highest point of both countries. It is considered one of the less technical 7,000 m peaks in the world to climb and it has by far the most ascents of any 7,000 m or higher peak on Earth, with every year seeing hundreds of mountaineers make their way to the summit. Lenin Peak is the highest mountain in the Trans-Alay Range of Central Asia, and in the Pamir Mountains in Tajikistan it is exceeded only by Ismoil Somoni Peak (7,495 m). It was thought to be the highest point in the Pamirs in Tajikistan until 1933, when Ismoil Somoni Peak (known as Stalin Peak at the time) was climbed and found to be more than 300 metres higher. Two mountains in the Pamirs in China, Kongur Tagh (7,649 m) and Muztagh Ata (7,546 m), are higher than the Tajik summits.

Names
Lenin Peak was originally named Mount Kaufmann after Konstantin Kaufman, the first Governor-General of the Russian Turkestan. In 1928, the mountain was renamed Lenin Peak after the Russian revolutionary and first leader of the Soviet Union, Vladimir Lenin. In Tajikistan, the peak was renamed again in July 2006, and today it is officially called in Tajik Qullai Abuali ibni Sino (қуллаи Абӯалӣ ибни Сино, Ibn Sina Peak or, alternatively, Avicenna Peak) after Abu Ali ibn Sina (Avicenna).

In Kyrgyzstan, the peak is still officially called Lenin Chokusu (Ленин Чокусу, Lenin Peak). However, in October 2017, Kyrgyz president Almazbek Atambayev called for renaming the peak "Manas Peak", after the hero of the Epic of Manas. A peak named "Manas Peak" already exists in Kyrgyzstan; it is a mountain of 4488 m in the Talas Alatau range in Talas Region.

Local Kyrgyz names include Jel-Aidar ("Wind's God") and Achyk-Tash ("Open Rock"). Achik-Tash, is also the name of a plateau and a base camp at an elevation of 3,600 m on a popular northern climbing route to Lenin Peak, which starts in the southern Kyrgyz city of Osh, a day's drive north of the border. Another suggested local name, Pik Kaman ("wild pig",  "boar") probably belongs to an officially unnamed peak west of Lenin Peak.

Climbing history and routes
Initial exploration of this part of Central Asia occurred in the period 1774–82. Arguably the first recorded travel through the region is the involuntary journey of the slave Filipp Efremov (an ethnic Russian), who escaped from slavery in Bukhara. He crossed the Fergana valley, then via Osh, the Chigirik Pass and Terekdavan Pass he reached the Kashgar and finally came over the Karakorum. He was the first European who crossed the Alai Mountains.

Scientific expeditions to the Alai Mountains began in 1871, when Alexei Pavlovich Fedchenko discovered the Trans-Alai (Zaalayskiy) Range and its main peak. The first geographical expedition which came nearest to the base of the future Lenin Peak in the early 20th century was arguably the expedition of Nikolai Leopol'dovich Korzhenevskiy.

In September 1928, three mountaineers—the Germans and Karl Wien, and the Austrian —from the Soviet-German scientific expedition made the first attempt to reach the highest point of the Trans-Alai Range, which at that time had the name Kaufman Peak. They started climbing upstream of the Saukdara river along the South slope of Trans-Alai Range also Trans-Alay Range. From the river head they continued climbing along the Greater Saukdara Glacier towards a saddle at an elevation of 5820 m (this saddle is also known as the Krilenko Pass). On September 25, 1928, they started climbing from the saddle along the NE Ridge and at 15.30 they reached the summit. At the time, Kaufman Peak was the highest summit reached by men.

The title Lenin Peak was first applied to the highest point of the Trans-Alai Range in the same year (1928). When it was renamed after Lenin it was believed to be the highest point in the USSR.

On September 8, 1934, at 16:20 Kasian Chernuha, Vitaly Abalakov and Ivan Lukin, three members of a Soviet expedition, reached the summit at an elevation of 7134 m. Their attempt lasted for four days with three camps (5,700 m, 6,500 m and 7,000 m). The expedition started climbing from the Achik-Tash canyon in the Alai valley. The summit attempt itself was started along the Western ice slope of the Lenin glacier. They continued climbing along the North Face, passing the rocks that were later given the name Lipkin's Rocks. At the end of the second day they reached the crest of the NE ridge at an elevation of about 6500 m. During the following day and a half they climbed along the NE Ridge and, utterly exhausted, reached the summit.

The third ascent was three years later, in 1937, when eight Soviet climbers under the direction of Lev Barkhash reached the summit by the same route. This was at the beginning of mass political repressions in the Soviet Union and many of the most prominent Soviet climbers, including Lev Barkhash, were brought to trial.

Subsequent attempts to climb Lenin Peak could not begin until 1950, when the USSR began to recover from the Second World War. On August 14, 1950, twelve climbers (V. Aksenov, K, Zaporojchenko, Y. Izrael, V. Kovalev, A. Kormshikov, Y. Maslov, E. Nagel, V. Narishkin, V. Nikonov, V. Nozdryuhin, I. Rojkov) under the direction of Vladimir Racek reached the summit for the fourth time.

All three Soviet expeditions including Racec's expedition of 1950 were by almost the same route via the NE Ridge.

The route which now is known as the classic route, via the Razdelnaya Peak and NW Ridge, was first climbed in 1954 by the team of Soviet climbers under the direction of V. Kovalev (P. Karpov, E. Nagel, V. Narishkin, V. Nozdryuhin).

In 1960, a group of eight Soviet climbers made a successful direct climb along the North Face (15.08.1960).

There are 16 established routes, nine on the southern side and seven on the northern slopes. The peak is quite popular with climbers due to its easy access and some uncomplicated routes. However, the peak is not without its share of disasters.

In August 1974, an entire team of eight Russian women climbers died high on the mountain in a storm. Elwira Szatajewa maintained radio contact with base camp as her team-mates perished around her. In her final words she said, 'I'm alone now, with just a few minutes left to live. See you in eternity.'

The first high-speed ascent of Lenin Peak was completed in 1987. The team, led by Valery Khrichtchatyi, included G. Lunyakov, Z. Khalitov, Anatoli Boukreev, V. Suviga, Y. Moiseev and A. Tselischev. The climb beginning from ABC base camp (4400 m) to the summit and return to ABC base camp took 12 hours.

Under the leadership of Leonid Troshchinenko the team, composed of 19 mountaineers from Leningrad and three from Kazakhstan, made the first winter ascent of Lenin Peak in 1988. With temperatures hovering at -45 °C, only six men reached the summit: including Valery Khrichtchatyi, Balyberdin, S. Arsentiev, U. Moiseev, I. Tulaiev and V. Dedi.

In 1987 the first solo ascent was achieved by the great high altitude mountaineer Anatoli Boukreev. An avalanche triggered by an earthquake killed 43 climbers in 1990.

As it is now, considering the existing infrastructure and BC/ABC locations, there are three most attractive routes from the North (approximately indicated on the Scheme): Lipkin's rocks route and NE Ridge; the classic North Face route; the route via Razdelnaya Peak and the NW Ridge.

title: Lenin Peak