Information theory and cryptography

[Mike Johnson <exabyte!smtplink!mikej@uunet.UU.NET>]

Robert W. Clark asks:
> I asked this one a while back and got no response.  sci.crypt
> was equally unresponsive.  It concerns the possibly obscure
> relation between cryptography, number theory and information
> theory.
> 
> Is there considered to be a one-to-one isomorphism between
> the units in a plaintext-cyphertext pair?  By this, I mean,
> are they considered to contain the same information?
>
> Is this affected by whether or not the key is known?  If the
> key has been irretrievably lost, does this lessen the amount
> of information, or does the 'potential' informational content
> remain the same?

If you have the proper algorithm and key, then the same information can be
derived from both.  There may be additional information added to the
cipher text that is not relevant to the plain text if _noise addition_ is
used in the encryption.  The noise is stripped out on decryption.  Likewise,
data compression may be involved, so the cipher text may be smaller or larger
in physical size, but will contain the same information.  Note that strictly
speaking, the information in the plain text is conveyed by the combination of
the ciphertext, the key, and the algorithm used.  Take any of these away, and
you lose information.  In most cases, loss of the algorithm is not an issue,
but if you forget, lose, or damage either the key or the ciphertext, you have
reduced the useful information content, possibly to zero.

> Is cryptography considered to be as simple as, say, Huffman
> coding, for purposes of informational content?  That is, is
> the relationship between the units of a plaintext-cyphertext
> pair considered to be more or less 'transparent,' or entirely
> isomorphic?

Sort of.  If I encrypt my communications with Alice, and you don't have the
key, then the information content of the message for Alice (who has the key)
is the same as the plain text, but for you the information content is zero
unless you crack the encryption scheme.

> Does the Second Law of Thermodynamics enter into this?  Is there
> a minimum amount of energy required to extract information from
> cyphertext, or a minimum amount of waste of energy?

Decryption _always_ consumes energy (electrical, mechanical, biological or
whatever).  Do you have a computer that uses no electricity, heat, light, or
sound, does not eat, and involves no mechanical work?  If so, patent it quick
and sell it!  Don't confuse thermodynamic and information theory entropy,
though.  They are mathmatically similar, but measure different things.

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