Editing Exclusive or (section)

==Relation to modern algebra==

Although the [[Operation (mathematics)|operators]] <math>\wedge</math> ([[Logical conjunction|conjunction]]) and <math>\lor</math> ([[Logical disjunction|disjunction]]) are very useful in logic systems, they fail a more generalizable structure in the following way:

The systems <math>(\{T, F\}, \wedge)</math> and <math>(\{T, F\}, \lor)</math> are [[monoid]]s, but neither is a [[group (mathematics)|group]]. This unfortunately prevents the combination of these two systems into larger structures, such as a [[Ring (mathematics)|mathematical ring]].

However, the system using exclusive or <math>(\{T, F\}, \oplus)</math> ''is'' an [[abelian group]].  The combination of operators <math>\wedge</math> and <math>\oplus</math> over elements <math>\{T, F\}</math> produce the well-known [[GF(2)|two-element field <math>\mathbb{F}_2</math>]].  This field can represent any logic obtainable with the system <math>(\land, \lor)</math> and has the added benefit of the arsenal of algebraic analysis tools for fields.

More specifically, if one associates <math>F</math> with 0 and <math>T</math> with 1, one can interpret the logical "AND" operation as multiplication on <math>\mathbb{F}_2</math> and the "XOR" operation as addition on <math>\mathbb{F}_2</math>:
: <math>\begin{matrix}
   r = p \land q & \Leftrightarrow & r = p \cdot q \pmod 2 \\[3pt]
  r = p \oplus q & \Leftrightarrow & r = p + q \pmod 2 \\
\end{matrix}</math>

The description of a [[Boolean function]] as a [[polynomial]] in <math>\mathbb{F}_2</math>, using this basis, is called the function's [[algebraic normal form]].<ref>{{cite book|title=Algorithmic Cryptanalysis|first=Antoine|last=Joux|publisher=CRC Press|year=2009|isbn=9781420070033|contribution=9.2: Algebraic normal forms of Boolean functions|pages=285–286|contribution-url=https://books.google.com/books?id=buQajqt-_iUC&pg=PA285}}</ref>