Re: commuting?/non-group cipher?

From: Brian McKeever (brian.mckeever_at_gmail.com)
Date: 10/28/04 

Date: 28 Oct 2004 14:56:02 -0700

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?

Because commuting is already taken. It implies that order doesn't
matter (ab=ba). I would call it closure - that the set of encryption
operations is closed under composition.

> Can anyone think of an example of a cipher with this property that is not a
> group?

As Kristian Gjøsteen showed, for the cases you are probably interested
in, closure implies that it's a group. I say it that way because he
assumed the input is finite (like in a block cipher). But for
illustritive purposes (not as a proposed cipher) consider E(K)[P] = K
concat P, for arbitrary length strings K and P. Then these operations
are closed, but lack inverses.

Brian, pedantically

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