# Re: An Adaptation of the Traditional Number Line as a Derived Modular Line in Cryptography.

"adacrypt" <austin.obyrne@xxxxxxxxxxx> wrote in message news:94e86d9f-c22a-4aba-b15f-8b09cff2bc02@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Mar 17, 10:46 am, adacrypt <austin.oby...@xxxxxxxxxxx> wrote:
I am making a prediction here that a lot, if not all of future
cryptography will be number-theoretic in concept and that it will use
the traditional number line per se to model integer numbers as points
on a straight line that may have arbitrary direction. The
construction of this line i.e. the rule or equation of the line is
invariably taken for granted and I doubt if anybody ever actually
envisages a number as a point on that line when they use it, unless of
course you are teaching children to add and subtract line segments as
an introduction to arithmetical addition of numbers or else you are
talking about numbers at a much higher level say, in number theory.

A large part of mathematics depends on a basic assumption that the
traditional number line has unitary period and unitary scale i.e. the
integers are separated by 1 and the graphic scale of the interval
between integers is 1 also i.e. scale of the line is 1 = 1.

I have been labouring the point for over ten years that this is the
downfall of mathematics in cryptography in that the traditional number
line is too transparent to cryptanalysts and no matter what a
cryptographer will ever do to obfuscate a plaintext by mathematical
means the cryptanalyst will always find another mathematical means to
invert it. However accurate this prognosis is or is not, is
unimportant, the fact remains that mathematics based on the
traditional number line has failed in cryptography. There is nothing
to defend in that failure, cryptography is a nefarious business anyway
and our beautiful number system was never intended to misinform people
in the first place.

To get to the point therefore I am promoting “Scalable Key
Cryptography” as something that still uses mathematics but in such a
way that the transparency that a cryptanalyst has enjoyed up to now
while using the traditional number line is no longer available to
him. I am defining number lines in a different way that is compatible
and interchangeable with the traditional number line but quite
different at the same time. I am using lines that have unitary period
but with a scale that is not unitary as in the traditional number line
and is instead some thing else, it may be any positive integer taken
from a large domain of positive integers (only the computer capacity
to store large positive integers restrains this), then the scale of
the line is N = 1.

This ploy takes me into the realm of modular arithmetic that was
invented by the great mathematician Leonard Euler and a contemporary
Johann Carl Friedrich Gauss. I don’t wish to open a can of
argumentive worms here from readers, lets say they did it jointly.
The mathematics of modular arithmetic is often called clock arithmetic
because of the way it is modelled on rotations of the hands of a
clock.

For my purposes in scalable key cryptography I am inventing number-
lines that have non-unitary periodic scale. When these straight lines
are given variable non-unitary periodic scale it becomes tantamount to
the clock arithmetic of modular invention being portrayed as opened-
out straight lines now. The upshot is that if one of these modular
lines is superimposed on a standard traditional number line any
integer from the latter line can be mapped onto the modular line as a
residue modulo (some ‘N’) if N is known.

As applied to cryptography the trick is not to make ‘N’ known.

The upshot in cryptography is that while the ciphertext as a set of
integers may still be represented as points on the traditional number
line still bearing their iconic Arabic names and shapes, they can only
be decrypted by mapping them onto some modular line but which one? –
there are N possible coincident modular lines where N is numerically
equal to the message length. Clearly, the possibility space of these
derived modular lines is vast.

A different value of N enables a fresh modular line to be used for
each plaintext.

The upshot of this invention is that the items of ciphertext that are
nominally ordinary integers and are still related to each other by
periodic scale are now not inductively decipherable by any
mathematical means. It is not possible to deduce the operand from the
function in other words. This pulls the plug on all mathematical
cryptanalysis i.e. cryptanalysis that seeks to invert transformations
by inverse functional means.

I think I am entitled to claim the credit for this invention of
modular lines in cryptography and I hope it does not give offence to

PS : Look at the elephants here never mind the pedantic fleas.

How it Works - Summary.

trying to go through the steps - please be patient - o.k?

Alice reads in the next plaintext which is say capiatl P. 'P' has
denary value 80 in ASCII .

S1: P = plaintext char or byte?

She multiplies 80 by 10 so as to create an
empty units place => 80 now becomes 800.

S2: P' = P * k

I shall stop here and ask:
1. Where did 10 come from?
2. Is it randomly chosen?
3. Is it chosen by some other means?
4. Is it a secret? Or Is it published to all?
5. What is the possible range of k? if you see what I mean?

She next inserts a digit from her Digital Signature array -say 7, into
the empty units place - 800 becomes 807.

She uses 807 to index an array of non-repeating integers that range
from -526 to + 487.

She calls this indexed integer whatever it is, her representation of
capital P - lets say it is - 428 (minus 428). - Lets call this Alice's
Digital Signature.

She next calls on an array of change-of-origin integers that ranges
from 300 to 1300.

She next calls a value of 'N' from another array that might range from
14000 to 28000.

Then,
Encryption,

CipherText = "Change-of-Orign" integer + "Alice's Digital Signature"
integer + n multples of 'N'. = some fairly bulky integer.

Decryption : Bob simply reverses everything from his identical arrays
having mapped the ciphertext integer across to his relevant "modular
line" that has been described above.

Note : a variation of this algorithm configures this cipher to work
only between certain bounds that are assigned to it by Alice and Bob.
Certain situations might require that also.