For example, if such that . Then a cyclic permutation on has the form
In the usual permutation notation, it looks like
Remark. For every finite set of cardinality , there are cyclic permutations. Each non-trivial cyclic permutation has order . Furthermore, if is a prime number, the set of cyclic permutations forms a cyclic group.
Cyclic permutations on words
Given a word on a set (may or may not be finite), a cyclic conjugate of is a word derived from based on a cyclic permutation. In other words, for some cyclic permutation on . Equivalently, and are cyclic conjugates of one another iff and for some words .
For example, the cyclic conjugates of the word over are
Strictly speaking, is a cyclic permutation on the multiset , which can be thought of as a cyclic permutation on the set . Furthermore, can be extended to a function on : for every word , , where is a permutation on .
Given any word on , two cyclic permutations on are said to be the same if . For example, with the word , then the cyclic permutation
Cyclic permutations were used as a ciphering scheme by Julius Caesar. Given an alphabet with letters, say , messages in letters are encoded so that each letter is shifted by three places. For example, the name
“Julius Caesar” becomes “Mxolxv Fdhvdu”.
A ciphering scheme based on cyclic permutations is therefore also known as a Caesar shift cipher.
|Date of creation||2013-03-22 17:33:54|
|Last modified on||2013-03-22 17:33:54|
|Last modified by||CWoo (3771)|
|Defines||Caesar shift cipher|