PlanetMath: cyclic permutation

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cyclic permutation

(Definition)

Let $A=\lbrace a_0,a_1,\ldots, a_{n-1}\rbrace$ be a finite set indexed by $i=0,\ldots, n-1$ . A cyclic permutation on is a permutation $\pi$ on such that, for some integer ,

$\displaystyle \pi(a_i)=a_{(i+k)\!\!\!\!\!\! \pmod n},$

where $a\!\!\pmod b:= a - \lfloor a/b \rfloor b$ , the remainder of

when divided by

, and $\lfloor \cdot \rfloor$ is the floor function.

For example, if $A=\lbrace 1,2,\ldots,m\rbrace$ such that . Then a cyclic permutation $\pi$ on has the form

$\displaystyle \pi(1)$	$\displaystyle =$	$\displaystyle r$
$\displaystyle \pi(2)$	$\displaystyle =$	$\displaystyle r+1$
	$\displaystyle \vdots$
$\displaystyle \pi(m-r+1)$	$\displaystyle =$	$\displaystyle m$
$\displaystyle \pi(m-r+2)$	$\displaystyle =$	$\displaystyle 1$
	$\displaystyle \vdots$
$\displaystyle \pi(m)$	$\displaystyle =$	$\displaystyle r-1.$

In the usual permutation notation, it looks like

$\displaystyle \pi=\left( \begin{array}{ccccccc} 1 & 2 & \cdots & m-r+1 & m-r+2 & \cdots & m \ r & r+1 & \cdots & m & 1 & \cdots & r-1 \end{array} \right)$

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 $w=a_1a_2\cdots a_n$ on a set $\Sigma$ (may or may not be finite), a cyclic conjugate of is a word derived from based on a cyclic permutation. In other words, $v=\pi(a_1)\pi(a_2)\cdots \pi(a_n)$ for some cyclic permutation $\pi$ on $\lbrace a_1,\ldots, a_n\rbrace$ . Equivalently, and are cyclic conjugates of one another iff and for some words .

For example, the cyclic conjugates of the word over $\lbrace a,b\rbrace$ are

$\displaystyle baba^2,\quad aba^2b,\quad ba^2ba,\quad a^2bab,$ and itself $\displaystyle .$

Strictly speaking, $\pi$ is a cyclic permutation on the multiset $A=\lbrace a_1,\ldots, a_n\rbrace$ , which can be thought of as a cyclic permutation on the set $A'=\lbrace (1,a_1),\ldots, (n,a_n)\rbrace$ . Furthermore, $\pi$ can be extended to a function on

: for every word $w=a_{\phi(1)}\cdots a_{\phi(m)}$ , $\pi(w):=\pi(a_{\phi(1)})\cdots \pi(a_{\phi(m)})$ , where $\phi$ is a permutation on

Given any word $w=a_1a_2\cdots a_n$ on $\Sigma$ , two cyclic permutations $\pi_1,\pi_2$ on $\lbrace a_1,\ldots, a_n\rbrace$ are said to be the same if $\pi_1(w)=\pi_2(w)$ . For example, with the word , then the cyclic permutation

$\displaystyle \left( \begin{array}{cccc} 1 & 2 & 3 & 4 \ 3 & 4 & 1 & 2 \end{array} \right)$

is the same as the identity permutation. There is a one-to-one correspondence between the set of all cyclic conjugates of

and the set of all distinct cyclic permutations on $\lbrace a_0,a_1,\ldots, a_n\rbrace$ .

Remarks.

In a group , if two elements are cyclic conjugates of one another, then they are conjugates: for if and , then $v=t(st)t^{-1}=tut^{-1}$ .
Cyclic permutations were used as a ciphering scheme by Julius Caesar. Given an alphabet with letters, say $a,b,c, \ldots,x,y,z$ , 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.

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"cyclic permutation" is owned by CWoo.

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See Also: cyclic code

Other names:

Caesar cipher

Also defines:

Caesar shift cipher, cyclic conjugate

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Cross-references: places, alphabet, scheme, conjugates, group, one-to-one correspondence, identity, function, multiset, strictly, iff, finite, word, cyclic group, prime number, order, cardinality, permutation notation, floor function, remainder, integer, permutation, indexed by, finite set
There are 3 references to this entry.

This is version 10 of cyclic permutation, born on 2007-10-01, modified 2007-10-08.
Object id is 9974, canonical name is CyclicPermutation.
Accessed 930 times total.

Classification:

AMS MSC:	20B99 (Group theory and generalizations :: Permutation groups :: Miscellaneous)
	03-00 (Mathematical logic and foundations :: General reference works )
	05A05 (Combinatorics :: Enumerative combinatorics :: Combinatorial choice problems )
	11Z05 (Number theory :: Miscellaneous applications of number theory)
	94A60 (Information and communication, circuits :: Communication, information :: Cryptography)
	94B15 (Information and communication, circuits :: Theory of error-correcting codes and error-detecting codes :: Cyclic codes)

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