Editing Finite field (section)

== Applications ==
In [[cryptography]], the difficulty of the [[discrete logarithm problem]] in finite fields or in [[elliptic curves]] is the basis of several widely used protocols, such as the [[Diffie–Hellman]] protocol. For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol ([[ECDHE]]) over a large finite field.<ref>This can be verified by looking at the information on the page provided by the browser.</ref> In [[coding theory]], many codes are constructed as [[linear subspace|subspace]]s of [[vector space]]s over finite fields.

Finite fields are used by many [[error correction code]]s, such as [[Reed–Solomon error correction|Reed–Solomon error correction code]] or [[BCH code]].  The finite field almost always has characteristic of {{math|2}}, since computer data is stored in binary.  For example, a byte of data can be interpreted as an element of {{math|GF(2<sup>8</sup>)}}. One exception is [[PDF417]] bar code, which is {{math|GF(929)}}. Some CPUs have special instructions that can be useful for finite fields of characteristic {{math|2}}, generally variations of [[carry-less product]].

Finite fields are widely used in [[number theory]], as many problems over the integers may be solved by reducing them [[modular arithmetic|modulo]] one or several [[prime number]]s. For example, the fastest known algorithms for [[polynomial factorization]] and [[linear algebra]] over the field of [[rational number]]s proceed by reduction modulo one or several primes, and then reconstruction of the solution by using [[Chinese remainder theorem]], [[Hensel lifting]] or the [[LLL algorithm]].

Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example, ''[[Hasse principle]]''. Many recent developments of [[algebraic geometry]] were motivated by the need to enlarge the power of these modular methods. [[Wiles' proof of Fermat's Last Theorem]] is an example of a deep result involving many mathematical tools, including finite fields.

The [[Weil conjectures]] concern the number of points on [[Algebraic variety|algebraic varieties]] over finite fields and the theory has many applications including [[Exponential sum|exponential]] and [[character sum]] estimates.

Finite fields have widespread application in [[combinatorics]], two well known examples being the definition of [[Paley graph|Paley Graphs]] and the related construction for [[Paley construction|Hadamard Matrices]]. In [[arithmetic combinatorics]] finite fields<ref>{{Citation|last=Shparlinski|first=Igor E.|chapter=Additive Combinatorics over Finite Fields: New Results and Applications|publisher=DE GRUYTER|isbn=9783110283600|doi=10.1515/9783110283600.233|title=Finite Fields and Their Applications| year=2013|pages=233–272}}</ref> and finite field models<ref>{{Citation|last=Green|first=Ben|chapter=Finite field models in additive combinatorics|pages=1–28|publisher=Cambridge University Press|isbn=9780511734885| doi=10.1017/cbo9780511734885.002| title=Surveys in Combinatorics 2005|year=2005|arxiv=math/0409420|s2cid=28297089}}</ref><ref>{{Cite journal|last=Wolf|first=J.| date=March 2015|title=Finite field models in arithmetic combinatorics – ten years on|journal=Finite Fields and Their Applications | volume=32|pages=233–274|doi=10.1016/j.ffa.2014.11.003|issn=1071-5797|doi-access=free|hdl=1983/d340f853-0584-49c8-a463-ea16ee51ce0f|hdl-access=free}}</ref> are used extensively, such as in [[Szemerédi's theorem]] on arithmetic progressions.