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Mathematical function used in cryptography
In cryptography, a T-function is a bijective mapping that updates every bit of the state in a way that can be described as x i ′ = x i + f ( x 0 , ⋯ ,
T-function
Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by T(h,a)=12π∫0ae−12h2(1+x2)1+x2dx(−∞<h,a<+∞). The function was first
Owen's_T_function
Extension of the factorial function
}t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function:
Gamma_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
normalized boxcar function) is defined as rect ( t T ) = Π ( t T ) = { 0 , if | t | > T 2 1 2 , if | t | = T 2 1 , if | t | < T 2 . {\displaystyle
Rectangular_function
actuarial mathematics, the accumulation function a(t) is a function of time t expressing the ratio of the value at time t (future value) and the initial investment
Accumulation_function
Hyperbolic analogues of trigonometric functions
hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin
Hyperbolic_functions
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Mathematical function with no sudden changes
example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the
Continuous_function
Probability distribution
cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function. For t > 0 , F ( t ) = ∫ − ∞ t f ( u ) d u =
Student's_t-distribution
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Mathematical function
Riemann–Siegel theta function and the Riemann zeta function by Z ( t ) = e i θ ( t ) ζ ( 1 2 + i t ) . {\displaystyle Z(t)=e^{i\theta (t)}\zeta \left({\frac
Z_function
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
probability density function, Φ ( x ) = ∫ − ∞ x φ ( t ) d t = 1 2 [ 1 + erf ( x 2 ) ] {\displaystyle \Phi (x)=\int _{-\infty }^{x}\varphi (t)\,dt={\frac
List of integrals of Gaussian functions
List_of_integrals_of_Gaussian_functions
Type of white blood cell
On the other hand, CD4+ T cells function as "helper cells." Unlike CD8+ killer T cells, the CD4+ helper T (TH) cells function by further activating memory
T_cell
Symbol representing a mathematical concept
Similarly, if T {\displaystyle T} is some term in the language, F ( T ) {\displaystyle F(T)} is also a term. As such, the interpretation of a function symbol
Function_symbol
Dynamical system whose system function is not directly dependent on time
time-dependent output function y ( t ) {\displaystyle y(t)} , and a time-dependent input function x ( t ) {\displaystyle x(t)} , the system will
Time-invariant_system
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Maximized objective function of an optimization problem
value function represents the optimal payoff of the system over the interval [ t , t 1 ] {\displaystyle [t,t_{1}]} when started at the time- t {\displaystyle
Value_function
Function that only depends on time
for each value of t. In the more general case, any nonhomogeneous source function in any variable can be described as a forcing function, and the resulting
Forcing function (differential equations)
Forcing_function_(differential_equations)
Theorem of convex functions
convex function (for t ∈ [0,1]), tf(x1)+(1−t)f(x2), while the graph of the function is the convex function of the weighted means, f(tx1+(1−t)x2). Thus
Jensen's_inequality
Periodic distribution ("function") of "point-mass" Dirac delta sampling
as sha function, impulse train or sampling function) is a periodic generalized function with the formula Ш T ( t ) := ∑ k = − ∞ ∞ δ ( t − k T ) {\displaystyle
Dirac_comb
Economic model which weighs rewards based on when they are received
the discount function f(t) having a negative first derivative and with ct (or c(t) in continuous time) defined as consumption at time t, total utility
Discount_function
First known wavelet basis
wavelet function ψ ( t ) {\displaystyle \psi (t)} can be described as ψ ( t ) = { 1 0 ≤ t < 1 2 , − 1 1 2 ≤ t < 1 , 0 otherwise. {\displaystyle \psi (t)={\begin{cases}1\quad
Haar_wavelet
Generalized function whose value is zero everywhere except at zero
function multiplied by the time-delayed Dirac delta δ T ( t ) = δ ( t − T ) {\displaystyle \delta _{T}(t){=}\delta (t{-}T)} is ∫ − ∞ ∞ f ( t ) δ ( t −
Dirac_delta_function
Model of thermodynamic properties
specified temperature T and pressure P. Common departure functions include those for enthalpy, entropy, and internal energy. Departure functions are used to calculate
Departure_function
Genetically engineered T cell
activating functions into a single receptor. CAR T cell therapy is a cell therapy that uses T cells engineered with CARs to treat cancer. T cells are modified
CAR_T_cell
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Integral transform and linear operator
singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given
Hilbert_transform
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
mimic function changes a file A {\displaystyle A} so it assumes the statistical properties of another file B {\displaystyle B} . That is, if p ( t , A )
Mimic_function
Function in condensed matter physics
scattering function is the spatial Fourier transform of the van Hove function G ( r → , t ) {\displaystyle G({\vec {r}},t)} : F ( k → , t ) ≡ ∫ G ( r → , t ) exp
Dynamic_structure_factor
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Special case of the polylogarithm
dilogarithm function is sometimes defined as ∫ 1 v ln t 1 − t d t = Li 2 ( 1 − v ) . {\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname
Dilogarithm
Class of mathematical functions
Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at
Subharmonic_function
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Branch of functional analysis
to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s →
Borel_functional_calculus
S-shaped curve
constant function, and one has that 1 T ∫ 0 T c ( t ) d t > r → lim t → + ∞ x ( t ) = 0 , {\displaystyle {\frac {1}{T}}\int _{0}^{T}c(t)\,dt>r\to \lim _{t\to
Logistic_function
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists
Inverse_function
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Mathematical function
Riemann–Siegel theta function is defined in terms of the gamma function as θ ( t ) = arg ( Γ ( 1 4 + i t 2 ) ) − log π 2 t {\displaystyle \theta (t)=\arg \left(\Gamma
Riemann–Siegel_theta_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions
Bochner_measurable_function
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Special case of the short-time Fourier transform
analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by
Gabor_transform
Computer communications authentication algorithm
type of message authentication code (MAC) involving a cryptographic hash function and a secret cryptographic key. As with any MAC, it may be used to simultaneously
HMAC
Growth of quantities at rate proportional to the current amount
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size
Exponential_growth
Parameter in atmospheric modeling
The Exner function is a parameter used in atmospheric modeling. Depending on the application, the Exner function may be defined as Π = c p ( p p 0 ) R
Exner_function
Information used for message authentication and integrity checking
MAC functions are similar to cryptographic hash functions, they possess different security requirements. To be considered secure, a MAC function must
Message_authentication_code
Variation of the Wigner distribution function
distribution function is abbreviated here as WD rather than WDF as used at Wigner distribution function A Modified Wigner distribution function is a variation
Modified Wigner distribution function
Modified_Wigner_distribution_function
characteristic function satisfies d 0 | t | β 0 exp ( − | t | β / γ ) ≤ | φ X ( t ) | ≤ d 1 | t | β 1 exp ( − | t | β / γ ) as t → ∞ {\displaystyle d_{0}|t|^{\beta
Smoothness (probability theory)
Smoothness_(probability_theory)
Mathematical operation
of a function F {\displaystyle F} is a real function f {\displaystyle f} that is piecewise-continuous, exponentially-restricted (that is, | f ( t ) | ≤
Inverse_Laplace_transform
Mathematical function relating circular and hyperbolic functions
{gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s
Gudermannian_function
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Special function defined by an integral
d t ln t . {\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}.} Here, ln denotes the natural logarithm. The function 1/(ln t) has
Logarithmic_integral_function
Effect in signal processing
The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear
Spectral_leakage
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
Mathematical function
the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ(x) or θ(x)
Chebyshev_function
product's excess supply function is the negative of the excess demand function—it is the product's supply function minus its demand function. In most cases the
Excess_demand_function
Integral transform useful in probability theory, physics, and engineering
transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain) to a function of a complex variable s {\displaystyle
Laplace_transform
Special function defined by an integral
Fresnel integrals S(x) and C(x), and their auxiliary functions F(x) and G(x) are transcendental functions named after Augustin-Jean Fresnel that are used in
Fresnel_integral
Complex complementary error function
_{0}^{z}e^{t^{2}}{\text{d}}t\right).} It is related to the Fresnel integral, to Dawson's integral, and to the Voigt function. The function arises in various
Faddeeva_function
Branch of functional analysis
holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which
Holomorphic functional calculus
Holomorphic_functional_calculus
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Mathematical function common in physics
The stretched exponential function f β ( t ) = e − t β {\displaystyle f_{\beta }(t)=e^{-t^{\beta }}} is obtained by inserting a fractional power law into
Stretched exponential function
Stretched_exponential_function
Subset of T lymphocytes
Memory T cells are a subset of T lymphocytes that might have some of the same functions as memory B cells. Their lineage is unclear. Antigen-specific memory
Memory_T_cell
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Electrical engineering concept
real-valued function s(t), it is determined from the function's analytic representation, sa(t): φ ( t ) = arg { s a ( t ) } = arg { s ( t ) + j s ^ ( t ) }
Instantaneous phase and frequency
Instantaneous_phase_and_frequency
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Mathematical operation
Laplace transform. If f ( t ) {\displaystyle f(t)} is a real- or complex-valued function of the real variable t {\displaystyle t} defined for all real
Two-sided_Laplace_transform
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Stream cipher
lookup tables are required to implement the cipher. The mixing function uses a g-function based on arithmetical squaring, and the ARX operations – logical
Rabbit_(cipher)
Concept in mathematics
In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of
K-function
Special mathematical function
mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special
Polylogarithm
Special mathematical function defined as sin(x)/x
In mathematics, physics and engineering, the sinc function (/ˈsɪŋk/), denoted by sinc(x), is defined as either sinc ( x ) = sin x x {\displaystyle
Sinc_function
Negative of a convex function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to
Concave_function
Meromorphic function
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m +
Polygamma_function
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
Function describing equilibrium states of a system
path: Φ ( t 0 , t 1 ) = ∫ t 0 t 1 P d V d t d t + ∫ t 0 t 1 V d P d t d t = ∫ t 0 t 1 d ( P V ) d t d t = P ( t 1 ) V ( t 1 ) − P ( t 0 ) V ( t 0 ) . {\displaystyle
State_function
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the
Inverse_function_rule
Computing algorithm
PJW hash function is a non-cryptographic hash function created by Peter J. Weinberger of AT&T Bell Labs. A variant of PJW hash had been used to create
PJW_hash_function
Asymmetric sigmoid function
or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes
Gompertz_function
Family of South Korean jet aircraft
and multi-function display by Samsung Thales. Other South Korean subcontractors such as Elemech, Dawin Friction, and Withus cooperate in T-50 components
KAI_T-50_Golden_Eagle
Growth curve model
The von Bertalanffy growth function (VBGF), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy
Von_Bertalanffy_function
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Property of functions which is weaker than continuity
Briefly, a function on a domain X {\displaystyle X} is lower semi-continuous if its epigraph { ( x , t ) ∈ X × R : t ≥ f ( x ) } {\displaystyle \{(x,t)\in X\times
Semi-continuity
Theorem relating continuity to graphs
continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. A blog post by T. Tao lists
Closed_graph_theorem
Derivative of a function with respect to time
derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time
Time_derivative
Base of natural logarithms
mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352 e plays
E_(mathematical_constant)
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Scorer's functions can also be defined in terms of Airy functions: G i ( x ) = B i ( x ) ∫ x ∞ A i ( t ) d t + A i ( x ) ∫ 0 x B i ( t ) d t , H i ( x
Scorer's_function
Polynomial function of degree 4
In algebra, a quartic function is a function of the form f ( x ) = a x 4 + b x 3 + c x 2 + d x + e , {\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,} where
Quartic_function
travel, tourism, insurance
T FUNCTION
T FUNCTION
T FUNCTION
T FUNCTION
T FUNCTION
T FUNCTION
T FUNCTION
T FUNCTION
T FUNCTION
travel, tourism, insurance