Editing Group theory (section)

===Cryptography===

[[File:Caesar3.svg|thumb|The [[cyclic group]] '''Z'''<sub>26</sub> underlies [[Caesar's cipher]].]]
Very large groups of prime order constructed in [[elliptic curve cryptography]] serve for [[public-key cryptography]]. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the [[discrete logarithm]] very hard to calculate. One of the earliest encryption protocols, [[Caesar cipher|Caesar's cipher]], may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular [[Diffie–Hellman key exchange]] uses finite [[cyclic group]]s. So the term [[group-based cryptography]] refers mostly to [[cryptographic protocol]]s that use infinite [[non-abelian group]]s such as a [[braid group]].