Boolean function explained
Boolean function should not be confused with Binary function.
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually, or).[1] [2] Alternative names are switching function, used especially in older computer science literature,[3] [4] and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory.
A Boolean function takes the form
, where
is known as the
Boolean domain and
is a non-negative integer called the
arity of the function. In the case where
, the function is a constant element of
. A Boolean function with multiple outputs,
with
is a
vectorial or
vector-valued Boolean function (an
S-box in symmetric
cryptography).
There are
different Boolean functions with
arguments; equal to the number of different
truth tables with
entries.
Every
-ary Boolean function can be expressed as a
propositional formula in
variables
, and two propositional formulas are
logically equivalent if and only if they express the same Boolean function.
Examples
See also: Truth table and Truth function. The rudimentary symmetric Boolean functions (logical connectives or logic gates) are:
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
See also: Analysis of Boolean functions.
Properties
A Boolean function can have a variety of properties:[5]
- 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 number 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 Shannon cofactors (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.[6]
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.[7] There are 2^2^(k−1) coincident functions of k arguments.[8]
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 output. 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.[9] 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.
Linear approximation table
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.[10] The set of Walsh transforms of the components is known as a Linear Approximation Table (LAT)[11] [12] or correlation matrix;[13] [14] 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[15] 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
can be uniquely extended (interpolated) to the
real domain by a
multilinear polynomial in
, constructed by summing the truth table values multiplied by
indicator polynomials:
For example, the extension of the binary XOR function
is
which equals
Some other examples are negation (
), AND (
) and OR (
). 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:this generalizes as the Möbius inversion of the partially ordered set of bit vectors:
f(a)where
denotes the weight of the bit vector
. Taken modulo 2, this is the
Boolean Möbius transform, giving the
algebraic normal form coefficients:
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.
, the polynomial
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
, 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 → \{-1,1\}
is then given by:
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.
See also
Further reading
- Janković . Dragan . Stanković . Radomir S. . Moraga . Claudio . November 2003 . Arithmetic expressions optimisation using dual polarity property . Serbian Journal of Electrical Engineering . 1 . 71-80, number 1 . 10.2298/SJEE0301071J . 71–80 . free .
- Book: Arnold, Bradford Henry . 1 January 2011 . Logic and Boolean Algebra . Courier Corporation . 978-0-486-48385-6 .
Notes and References
- Web site: Boolean function - Encyclopedia of Mathematics. 2021-05-03. encyclopediaofmath.org.
- Web site: Weisstein. Eric W.. Boolean Function. 2021-05-03. mathworld.wolfram.com. en.
- Web site: switching function. 2021-05-03. TheFreeDictionary.com.
- Davies. D. W.. December 1957. Switching Functions of Three Variables. IRE Transactions on Electronic Computers. EC-6. 4. 265–275. 10.1109/TEC.1957.5222038. 0367-9950.
- Web site: Boolean functions — Sage 9.2 Reference Manual: Cryptography. 2021-05-01. doc.sagemath.org.
- Book: Tarannikov. Yuriy. Korolev. Peter. Botev. Anton. Advances in Cryptology — ASIACRYPT 2001 . Autocorrelation Coefficients and Correlation Immunity of Boolean Functions . 2001. Boyd. Colin. Lecture Notes in Computer Science. 2248. en. Berlin, Heidelberg. Springer. 460–479. 10.1007/3-540-45682-1_27. 978-3-540-45682-7. free.
- 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. 10.1080/00207160.2010.509428. 9580510 . 0020-7160.
- Nitaj. Abderrahmane. Susilo. Willy. Tonien. Joseph. 2017-10-01. Dirichlet product for boolean functions. Journal of Applied Mathematics and Computing. en. 55. 1. 293–312. 10.1007/s12190-016-1037-4. 16760125 . 1865-2085.
- 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. 978-3-540-67517-4.
- Web site: Carlet. Claude. Vectorial Boolean Functions for Cryptography. live. University of Paris. https://web.archive.org/web/20160117102533/http://www.math.univ-paris13.fr:80/~carlet/chap-vectorial-fcts-corr.pdf . 2016-01-17 .
- Web site: Heys. Howard M.. A Tutorial on Linear and Differential Cryptanalysis. live. https://web.archive.org/web/20170517014157/http://www.cs.bc.edu:80/~straubin/crypto2017/heys.pdf . 2017-05-17 .
- Web site: S-Boxes and Their Algebraic Representations — Sage 9.2 Reference Manual: Cryptography. 2021-05-04. doc.sagemath.org.
- Daemen . Joan . Govaerts . René . Vandewalle . Joos . Preneel . Bart . Correlation matrices . 10.1007/3-540-60590-8_21 . 275–285 . Springer . Lecture Notes in Computer Science . Fast Software Encryption: Second International Workshop. Leuven, Belgium, 14-16 December 1994, Proceedings . 1008 . 1994. free .
- Web site: Daemen. Joan. 10 June 1998. Chapter 5: Propagation and Correlation - Annex to AES Proposal Rijndael. live. NIST. https://web.archive.org/web/20180723015757/https://csrc.nist.gov/CSRC/media/Projects/Cryptographic-Standards-and-Guidelines/documents/aes-development/PropCorr.pdf . 2018-07-23 .
- Web site: Nyberg. Kaisa. December 1, 2019. The Extended Autocorrelation and Boomerang Tables and Links Between Nonlinearity Properties of Vectorial Boolean Functions. live. https://web.archive.org/web/20201102023321/https://eprint.iacr.org/2019/1381.pdf . 2020-11-02 .