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Boolean differential calculus (BDC) (German: Boolescher Differentialkalkül (BDK)) is a subject field of Boolean algebra discussing changes of Boolean variables and Boolean functions. Boolean differential calculus concepts are analogous to those of classical differential calculus , notably studying the changes in functions and variables with ...
In Boolean logic, a Reed–Muller expansion (or Davio expansion) is a decomposition of a Boolean function. ... Derivation of the second-order boolean derivative:
Boolean difference: The Boolean difference or Boolean derivative of the function F with respect to the literal x is defined as: = ′ Universal quantification: The universal quantification of F is defined as: = ′
The autocorrelation of a Boolean function is a k-ary integer-valued function giving the correlation between a certain set of changes in the inputs and the function output. For a given bit vector it is related to the Hamming weight of the derivative in that direction.
The total influence of a Boolean function is also the average sensitivity of the function. The sensitivity of a Boolean function at a given point is the number of coordinates such that if we flip the 'th coordinate, the value of the function changes. The average value of this quantity is exactly the total influence.
Balanced Boolean function; Bent function; Binary decision diagram; Bitwise operation; George Boole; Boole's expansion theorem; Boolean algebras canonically defined; Boolean conjunctive query; Boolean data type; Boolean differential calculus; Boolean domain; Boolean expression; Boolean function; Boolean matrix; Boolean prime ideal theorem ...
In addition, the derivatives of a bent function are balanced Boolean functions, so for any change in the input variables there is a 50 percent chance that the output value will change. The maximal nonlinearity means approximating a bent function by an affine (linear) function is hard, a useful property in the defence against linear cryptanalysis.
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}