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From: Mok-Kong Shen on 25 Feb 2010 06:07
In classical crypto there are the so-called fractional substitution
ciphers. A typical example is the German ADFGX, where a 5*5 matrix
containing a randomly ordered 25 letter alphabet (normally manually
obtained with a key and transposition) is read out with a column vector
at the left and a row vector above the matrix, which both consist of
ADFGX, whence the name of the cipher. Each plaintext letter is thus
'fractionalized' into a pair of letters in the reduced 5 letter
alphabet and the stream of this transcription is then processed by a
simple final transposition. (See
http://members.aon.at/cipherclerk/Doc/ADFGVX.html for other fractional
substitution schemes, e.g. bifid
and trifid.)

I wonder why does one transcribe the plaintext letters into an alphabet
of 'reduced' size. If one employs 3 matrices, each a (differently)
randomly ordered 25 letter alphabet, positioned in the form of a
mirrored L, then each plaintext letter can be transcribed into pairs
of letters in 25 different ways, i.e. one has homophony. Wouldn't
that be much better?

It seems evident that one could also advantageously combine this
generalization with Playfair, namely one first uses the middle matrix
to do Playfair of a pair of plaintext letters and fractionalize each
of the resulting letters as depicted above, which could then be subject
to a (preferably not too simple) final transposition process as usual.

M. K. Shen
From: Mok-Kong Shen on 25 Feb 2010 09:51
[Addendum] Another point I also wonder is that one seems to have
highly favoured the use of 5*5 matrices. If, while still confining
oneself to the 26 letter alphabet for the plaintext, one uses 6*6
matrices, incoporating the numerals 0-9, quite some qualitative boost
could result, I would think.

M. K. Shen
From: Mok-Kong Shen on 28 Feb 2010 05:43
Mok-Kong Shen wrote:
> [Addendum] Another point I also wonder is that one seems to have
> highly favoured the use of 5*5 matrices. If, while still confining
> oneself to the 26 letter alphabet for the plaintext, one uses 6*6
> matrices, incoporating the numerals 0-9, quite some qualitative boost
> could result, I would think.

[Addendum 2] While from literature it seems that square matrices have
been highly favoured, one can certainly also employ arbitrary matrices.
For an alphabet of e.g. 40 (26 letters + digits + some special symbols)
one could use a 5*8 matrix.

The homophony I depicted in the original post of using 3 matrices
(instead of one matrix and two vectors) exists by definition clearly
only as long as the ciphertext is longer than the plaintext. If the
stream of letters obtained from the fractional substitution is later
(after normally being subjected to a transposition) is recombined
for the purpose that the final ciphertext is of the same length as the
original plaintext, then of course no homophony can function.

Like all cipher schemes, fractional substitutions can be cascaded,
preferably with a transposition inbetween. If one exploits homophony as
described in my original post, then (using in the second pass again 3
matrices, preferably different from those of the first pass) one could
achieve e.g. a homophony of 1:(25)^3 =15625 in the case of 5*5 matrices
(in the first pass each plaintext letter has 25 possible pairs, each
letter of such a pair has 25 possible pairs in the second pass). The
disadvantage, of course, is that the ciphertext is now 4 times as long
as the plaintext. (One could, with reduced homophony of 1:(25)^2,
reduce this to 3 times as long, if in the second pass one treats only
every second symbol and leaves the others unchanged.) Doing a
transposition (could be e.g. a keyed columnar transposition) as a final
step is evidently always advantageous for homophonic transformations.

M. K. Shen

From: Maaartin on 28 Feb 2010 20:57
On Feb 25, 12:07 pm, Mok-Kong Shen <mok-kong.s...(a)t-online.de> wrote:
> I wonder why does one transcribe the plaintext letters into an alphabet
> of 'reduced' size. If one employs 3 matrices, each a (differently)
> randomly ordered 25 letter alphabet, positioned in the form of a
> mirrored L, then each plaintext letter can be transcribed into pairs
> of letters in 25 different ways, i.e. one has homophony. Wouldn't
> that be much better?

I don't even understand how you use your 3 matrices. And how do you
get 25 different ways?

Later you speak about matrices instead of vectors, this gives me an
idea, but still... I see that achieving homophony is important, but
lengthening the ciphertext means making more work and I wonder if
doing something else couldn't be more efficient.
From: Mok-Kong Shen on 1 Mar 2010 04:39
Maaartin wrote:
> Mok-Kong Shen wrote:
>> I wonder why does one transcribe the plaintext letters into an alphabet
>> of 'reduced' size. If one employs 3 matrices, each a (differently)
>> randomly ordered 25 letter alphabet, positioned in the form of a
>> mirrored L, then each plaintext letter can be transcribed into pairs
>> of letters in 25 different ways, i.e. one has homophony. Wouldn't
>> that be much better?
>
> I don't even understand how you use your 3 matrices. And how do you
> get 25 different ways?

A kind of schema like this:

b c
d a

c b a c
d a b d

Here a could be transcribed to cb, cd, bb, or bd.

> Later you speak about matrices instead of vectors, this gives me an
> idea, but still... I see that achieving homophony is important, but
> lengthening the ciphertext means making more work and I wonder if
> doing something else couldn't be more efficient.

Doing such kind of homophony that renders the ciphertext longer than
the plaintext can certainly be regarded as an essential minus point in
respect of efficiency, I fully agree. I believe one should always
examine in "any" given situation of encryption whether doing something
else could be more worthwhile for an equal or even less amount of
effort. In the present case, I would think that doing Playfair twice
(with different matrices and a displacement of 1 in the 2nd round or
even a transposition inbetween) instead of one Playfair and homophony
may be a better alternative. On the other hand, one could always
attempt to "additionally" do something more (here homophony) in
encryption, if one "feels" insecure for some justified or "unjustified"
(cf. hypochondria in health) reasons. I simply wanted to point out that
such a possibility by way of homophony exists in the present context.

Thanks,

M. K. Shen

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