# Re: commuting?/non-group cipher?

From: Kristian Gjøsteen (kristiag+news_at_math.ntnu.no)
Date: 11/12/04

```Date: Fri, 12 Nov 2004 08:38:32 +0000 (UTC)

```

>Let me partly define what I mean - the "natural" group is not a permutation
>group. The "natural" group set is the set of texts, and perhaps keys. The
>members of a "natural" group set are not permutations.

Let me try some examples:

So you want the keys to have a group structure (your property), and the
group structure is "natural" if it is derived from some algebraic
structure on the set of texts.

For the Caesar cipher, the key structure is the additive group structure
on the alphabet.

For the multiplicative cipher, the key structure is derived from the ring
structure on the alphabet (the group of multiplicative units).

For the Pohlig-Hellman cipher, the key structure is derived from the
automorphism group of the alphabet group. A key is an automorphism on
G, and for a cyclic group of order n, Aut(G) is isomorphic to Z_n*. So
we can represent the key by an integer in {1,...,n-1}.

These are therefore all "natural".

For the substitution cipher on some alphabet, the key structure is that
of a permutation group, and does not have any natural relation to a
possible algebraic structure on the alphabet.

So it is therefore not "natural".

Is this what you want?

```--
Kristian Gjøsteen
```

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