Editing Field (mathematics) (section)

=== Finite fields: cryptography and coding theory ===
[[File:ECClines.svg|thumb|The sum of three points {{math|''P''}}, {{math|''Q''}}, and {{math|''R''}} on an elliptic curve {{math|''E''}} (red) is zero if there is a line (blue) passing through these points.]]
A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing
: {{math|1=''a''<sup>''n''</sup> = ''a'' ⋅ ''a'' ⋅ ⋯ ⋅ ''a''}} ({{math|''n''}} factors, for an integer {{math|''n'' ≥ 1}})
in a (large) finite field {{math|'''F'''<sub>''q''</sub>}} can be performed much more efficiently than the [[discrete logarithm]], which is the inverse operation, i.e., determining the solution {{math|''n''}} to an equation
: {{math|1=''a''<sup>''n''</sup> = ''b''}}.
In [[elliptic curve cryptography]], the multiplication in a finite field is replaced by the operation of adding points on an [[elliptic curve]], i.e., the solutions of an equation of the form
: {{math|1=''y''<sup>2</sup> = ''x''<sup>3</sup> + ''ax'' + ''b''}}.

Finite fields are also used in [[coding theory]] and [[combinatorics]].