# Boolean function

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In mathematics, a Boolean function is a function whose arguments, as well as the function itself, assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}).   Alternative names are switching function, used especially in older computer science literature,   and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory. 

## Contents

A Boolean function takes the form $f:\{0,1\}^{k}\to \{0,1\}$ , where $\{0,1\}$ is known as the Boolean domain and $k$ is a non-negative integer called the arity of the function. In the case where $k=0$ , the "function" is a constant element of $\{0,1\}$ . A Boolean function with multiple outputs, $f:\{0,1\}^{k}\to \{0,1\}^{m}$ with $m>1$ is a vectorial or vector-valued Boolean function (an S-box in cryptography). 

There are $2^{2^{k}}$ different Boolean functions with $k$ arguments; equal to the number of different truth tables with $2^{k}$ entries.

Every $k$ -ary Boolean function can be expressed as a propositional formula in $k$ variables $x_{1},...,x_{k}$ , and two propositional formulas are logically equivalent if and only if they express the same Boolean function.

## Examples

The rudimentary symmetric Boolean functions (logical connectives or logic gates) are:

• NOT, negation or complement - which receives one input and returns true when that input is false ("not")

An example of a more complicated function is the majority function (of an odd number of inputs).

## Representation

A Boolean function may be specified in a variety of ways:

• Truth table: explicitly listing its value for all possible values of the arguments
• Marquand diagram: truth table values arranged in a two-dimensional grid (used in a Karnaugh map)
• Binary decision diagram, listing the truth table values at the bottom of a binary tree
• Venn diagram, depicting the truth table values as a colouring of regions of the plane

Algebraically, as a propositional formula using rudimentary boolean functions:

Boolean formulas can also be displayed as a graph:

In order to optimize electronic circuits, Boolean formulas can be minimized using the Quine–McCluskey algorithm or Karnaugh map.

## Analysis

### Properties

A Boolean function can have a variety of properties: 

• Constant: Is always true or always false regardless of its arguments.
• Monotone: for every combination of argument values, changing an argument from false to true can only cause the output to switch from false to true and not from true to false. A function is said to be unate in a certain variable if it is monotone with respect to changes in that variable.
• Linear: for each variable, flipping the value of the variable either always makes a difference in the truth value or never makes a difference (a parity function).
• Symmetric: the value does not depend on the order of its arguments.
• Read-once: Can be expressed with conjunction, disjunction, and negation with a single instance of each variable.
• Balanced: if its truth table contains an equal amount of zeros and ones. The Hamming weight of the function is the number of ones in the truth table.
• Bent: its derivatives are all balanced (the autocorrelation spectrum is zero)
• Correlation immune to mth order: if the output is uncorrelated with all (linear) combinations of at most m arguments
• Evasive: if evaluation of the function always requires the value of all arguments
• A Boolean function is a Sheffer function if it can be used to create (by composition) any arbitrary Boolean function (see functional completeness)
• The algebraic degree of a function is the order of the highest order monomial in its algebraic normal form

Circuit complexity attempts to classify Boolean functions with respect to the size or depth of circuits that can compute them.

### Derived functions

A Boolean function may be decomposed using Boole's expansion theorem in positive and negative Shannoncofactors (Shannon expansion), which are the (k-1)-ary functions resulting from fixing one of the arguments (to zero or one). The general (k-ary) functions obtained by imposing a linear constraint on a set of inputs (a linear subspace) are known as subfunctions. 

The Boolean derivative of the function to one of the arguments is a (k-1)-ary function that is true when the output of the function is sensitive to the chosen input variable; it is the XOR of the two corresponding cofactors. A derivative and a cofactor are used in a Reed–Muller expansion. The concept can be generalized as a k-ary derivative in the direction dx, obtained as the difference (XOR) of the function at x and x + dx. 

The Möbius transform (or Boole-Möbius transform) of a Boolean function is the set of coefficients of its polynomial (algebraic normal form), as a function of the monomial exponent vectors. It is a self-inverse transform. It can be calculated efficiently using a butterfly algorithm ("Fast Möbius Transform"), analogous to the Fast Fourier Transform.  Coincident Boolean functions are equal to their Möbius transform, i.e. their truth table (minterm) values equal their algebraic (monomial) coefficients.  There are 2^2^(k−1) coincident functions of k arguments. 

### Cryptographic analysis

The Walsh transform of a Boolean function is a k-ary integer-valued function giving the coefficients of a decomposition into linear functions (Walsh functions), analogous to the decomposition of real-valued functions into harmonics by the Fourier transform. Its square is the power spectrum or Walsh spectrum. The Walsh coefficient of a single bit vector is a measure for the correlation of that bit with the output of the Boolean function. The maximum (in absolute value) Walsh coefficient is known as the linearity of the function.  The highest number of bits (order) for which all Walsh coefficients are 0 (i.e. the subfunctions are balanced) is known as resiliency, and the function is said to be correlation immune to that order.  The Walsh coefficients play a key role in linear cryptanalysis.

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 ouput. For a given bit vector it is related to the Hamming weight of the derivative in that direction. The maximal autocorrelation coefficient (in absolute value) is known as the absolute indicator.   If all autocorrelation coefficients are 0 (i.e. the derivatives are balanced) for a certain number of bits then the function is said to satisfy the propagation criterion to that order; if they are all zero then the function is a bent function.  The autocorrelation coefficients play a key role in differential cryptanalysis.

The Walsh coefficients of a Boolean function and its autocorrelation coefficients are related by the equivalent of the Wiener–Khinchin theorem, which states that the autocorrelation and the power spectrum are a Walsh transform pair. 

These concepts can be extended naturally to vectorial Boolean functions by considering their output bits (coordinates) individually, or more thoroughly, by looking at the set of all linear functions of output bits, known as its components.  The set of Walsh transforms of the components is known as a Linear Approximation Table (LAT)   or correlation matrix;   it describes the correlation between different linear combinations of input and output bits. The set of autocorrelation coefficients of the components is the autocorrelation table,  related by a Walsh transform of the components  to the more widely used Difference Distribution Table (DDT)   which lists the correlations between differences in input and output bits (see also: S-box).

## Real polynomial form

### On the unit hypercube

Any Boolean function $f(x):\{0,1\}^{n}\rightarrow \{0,1\}$ can be uniquely extended (interpolated) to the real domain by a multilinear polynomial in $\mathbb {R} ^{n}$ , constructed by summing the truth table values multiplied by indicator polynomials:

$f^{*}(x)=\sum _{a\in {\{0,1\}}^{n}}f(a)\prod _{i:a_{i}=1}x_{i}\prod _{i:a_{i}=0}(1-x_{i})$ For example, the extension of the binary XOR function $x\oplus y$ is

$0(1-x)(1-y)+1x(1-y)+1(1-x)y+0xy$ which equals

$x+y-2xy$ Some other examples are negation ($1-x$ ), AND ($xy$ ) and OR ($x+y-xy$ ). When all operands are independent (share no variables) a function's polynomial form can be found by repeatedly applying the polynomials of the operators in a Boolean formula. When the coefficients are calculated modulo 2 one obtains the algebraic normal form (Zhegalkin polynomial). Direct expressions for the coefficients of the polynomial can be derived by taking an appropriate derivative:

${\begin{array}{lcl}f^{*}(00)&=&(f^{*})(00)&=&f(00)\\f^{*}(01)&=&(\partial _{1}f^{*})(00)&=&-f(00)+f(01)\\f^{*}(10)&=&(\partial _{2}f^{*})(00)&=&-f(00)+f(10)\\f^{*}(11)&=&(\partial _{1}\partial _{2}f^{*})(00)&=&f(00)-f(01)-f(10)+f(11)\\\end{array}}$ this generalizes as the Möbius inversion of the partially ordered set of bit vectors:

$f^{*}(m)=\sum _{a\subseteq m}(-1)^{|a|+|m|}f(a)$ where $|a|$ denotes the weight of the bit vector $a$ . Taken modulo 2, this is the Boolean Möbius transform, giving the algebraic normal form coefficients:

${\hat {f}}(m)=\bigoplus _{a\subseteq m}f(a)$ In both cases, the sum is taken over all bit-vectors a covered by m, i.e. the "one" bits of a form a subset of the one bits of m.

When the domain is restricted to the n-dimensional hypercube $[0,1]^{n}$ , the polynomial $f^{*}(x):[0,1]^{n}\rightarrow [0,1]$ gives the probability of a positive outcome when the Boolean function f is applied to n independent random (Bernoulli) variables, with individual probabilities x. A special case of this fact is the piling-up lemma for parity functions. The polynomial form of a Boolean function can also be used as its natural extension to fuzzy logic.

### On the symmetric hypercube

Often, the Boolean domain is taken as $\{-1,1\}$ , with false ("0") mapping to 1 and true ("1") to -1 (see Analysis of Boolean functions). The polynomial corresponding to $g(x):\{-1,1\}^{n}\rightarrow \{-1,1\}$ is then given by:

$g^{*}(x)=\sum _{a\in {\{-1,1\}}^{n}}g(a)\prod _{i:a_{i}=-1}{\frac {1-x_{i}}{2}}\prod _{i:a_{i}=1}{\frac {1+x_{i}}{2}}$ Using the symmetric Boolean domain simplifies certain aspects of the analysis, since negation corresponds to multiplying by -1 and linear functions are monomials (XOR is multiplication). This polynomial form thus corresponds to the Walsh transform (in this context also known as Fourier transform) of the function (see above). The polynomial also has the same statistical interpretation as the one in the standard Boolean domain, except that it now deals with the expected values $E(X)=P(X=1)-P(X=-1)\in [-1,1]$ (see piling-up lemma for an example).

## Applications

Boolean functions play a basic role in questions of complexity theory as well as the design of processors for digital computers, where they are implemented in electronic circuits using logic gates.

The properties of Boolean functions are critical in cryptography, particularly in the design of symmetric key algorithms (see substitution box).

In cooperative game theory, monotone Boolean functions are called simple games (voting games); this notion is applied to solve problems in social choice theory.

## Related Research Articles Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals. In mathematics, convolution is a mathematical operation on two functions that produces a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. The integral is evaluated for all values of shift, producing the convolution function.

In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation. In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 1.

Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor, and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.

A maximum length sequence (MLS) is a type of pseudorandom binary sequence.

In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems.

In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in is statistically independent of the value of .

Zhegalkinpolynomials are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, x2 = x. Hence a polynomial such as 3x2y5z is congruent to, and can therefore be rewritten as, xyz.

In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones in the input. For this reason they are also known as Boolean counting functions.

In Boolean algebra, a parity function is a Boolean function whose value is 1 if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function.

In computational complexity the decision tree model is the model of computation in which an algorithm is considered to be basically a decision tree, i.e., a sequence of queries or tests that are done adaptively, so the outcome of the previous tests can influence the test is performed next. In the mathematical field of combinatorics, a bent function is a special type of Boolean function which is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables. Concretely, this means the maximum correlation between the output of the function and a linear function is minimal. In addition, the derivatives of a bent function are a balanced Boolean functions, so for any change in the input variables there is a 50 percent chance that the output value will change.

In cryptography, SWIFFT is a collection of provably secure hash functions. It is based on the concept of the fast Fourier transform (FFT). SWIFFT is not the first hash function based on FFT, but it sets itself apart by providing a mathematical proof of its security. It also uses the LLL basis reduction algorithm. It can be shown that finding collisions in SWIFFT is at least as difficult as finding short vectors in cyclic/ideal lattices in the worst case. By giving a security reduction to the worst-case scenario of a difficult mathematical problem, SWIFFT gives a much stronger security guarantee than most other cryptographic hash functions.

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively. Instead of elementary algebra, where the values of the variables are numbers and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction (and) denoted as ∧, the disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. It is thus a formalism for describing logical operations, in the same way that elementary algebra describes numerical operations.

1. "Boolean function - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-05-03.
2. Weisstein, Eric W. "Boolean Function". mathworld.wolfram.com. Retrieved 2021-05-03.
3. "switching function". TheFreeDictionary.com. Retrieved 2021-05-03.
4. Davies, D. W. (December 1957). "Switching Functions of Three Variables". IRE Transactions on Electronic Computers. EC-6 (4): 265–275. doi:10.1109/TEC.1957.5222038. ISSN   0367-9950.
5. McCluskey, Edward J. (2003-01-01), "Switching theory", Encyclopedia of Computer Science, GBR: John Wiley and Sons Ltd., pp. 1727–1731, ISBN   978-0-470-86412-8 , retrieved 2021-05-03
6. Carlet, Claude. "Vectorial Boolean Functions for Cryptography" (PDF). University of Paris.
7. "Boolean functions — Sage 9.2 Reference Manual: Cryptography". doc.sagemath.org. Retrieved 2021-05-01.
8. Tarannikov, Yuriy; Korolev, Peter; Botev, Anton (2001). Boyd, Colin (ed.). "Autocorrelation Coefficients and Correlation Immunity of Boolean Functions". Advances in Cryptology — ASIACRYPT 2001. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. 2248: 460–479. doi:. ISBN   978-3-540-45682-7.
9. Carlet, Claude (2010), "Boolean Functions for Cryptography and Error-Correcting Codes" (PDF), Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, pp. 257–397, ISBN   978-0-521-84752-0 , retrieved 2021-05-17
10. Pieprzyk, Josef; Wang, Huaxiong; Zhang, Xian-Mo (2011-05-01). "Mobius transforms, coincident Boolean functions and non-coincidence property of Boolean functions". International Journal of Computer Mathematics. 88 (7): 1398–1416. doi:10.1080/00207160.2010.509428. ISSN   0020-7160.
11. Nitaj, Abderrahmane; Susilo, Willy; Tonien, Joseph (2017-10-01). "Dirichlet product for boolean functions". Journal of Applied Mathematics and Computing. 55 (1): 293–312. doi:10.1007/s12190-016-1037-4. ISSN   1865-2085.
12. Canteaut, Anne; Carlet, Claude; Charpin, Pascale; Fontaine, Caroline (2000-05-14). "Propagation characteristics and correlation-immunity of highly nonlinear boolean functions". Proceedings of the 19th International Conference on Theory and Application of Cryptographic Techniques. EUROCRYPT'00. Bruges, Belgium: Springer-Verlag: 507–522. ISBN   978-3-540-67517-4.
13. Heys, Howard M. "A Tutorial on Linear and Differential Cryptanalysis" (PDF).
14. "S-Boxes and Their Algebraic Representations — Sage 9.2 Reference Manual: Cryptography". doc.sagemath.org. Retrieved 2021-05-04.
15. Daemen, Joan; Govaerts, René; Vandewalle, Joos (1995). Preneel, Bart (ed.). "Correlation matrices". Fast Software Encryption. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. 1008: 275–285. doi:. ISBN   978-3-540-47809-6.
16. Daemen, Joan (10 June 1998). "Chapter 5: Propagation and Correlation - Annex to AES Proposal Rijndael" (PDF). NIST.
17. Nyberg, Kaisa (December 1, 2019). "The Extended Autocorrelation and Boomerang Tables and Links Between Nonlinearity Properties of Vectorial Boolean Functions" (PDF).