Re: commuting?/non-group cipher?

From: Peter Fairbrother (zenadsl6186_at_zen.co.uk)
Date: 10/29/04 

Date: Fri, 29 Oct 2004 01:45:49 +0100

Brian McKeever wrote:

> Peter Fairbrother <zenadsl6186@zen.co.uk> wrote in message
> news:<BDA64883.6FF4A%zenadsl6186@zen.co.uk>...
>> Some ciphers have the property that a double encryption can always be
>> replaced by a single encryption, ie E(k3)[P] = E(k1)[E(k2)[P]]
>>
>> Does anyone know the correct name for this property? If there isn't one,
>> does anyone know a reason why "commuting (adj.)" cipher would not be okay?

For future reference, can I rewrite the property as:

Given a set S and a binary operation *;

the property that for all a, b in S there exists a c in S such that, for all
d in S, c*d = a*(b*d)

> Because commuting is already taken. It implies that order doesn't
> matter (ab=ba).

I thought that was "commutative", rather than "commuting".

> I would call it closure - that the set of encryption
> operations is closed under composition.

Isn't "closure" already taken - to mean: for all a and b in S, a*b is in S?

The property is different.

A set and binary operation with the property will have closure; but closure
doesn't imply the property - for instance many [1] ciphers have closure, but
do not have the property.

Closure and associativity together does imply the property:

a*(b*d) = (a*b)*d,
-> some c = a*b exists in S, such that c*d = a*(b*d)

so all ciphers which are groups have the property,

and Kristian showed that all ciphers that have the property are groups.

Whew. Is all of the above correct? Any mistakes? Thanks.

[1] (all?) I'll reply seperately to the other part of your post ('cos it's
very complicated!! for my poor little brain, and also seems to relate to
John M's post). 

-- 
Peter Fairbrother

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