The Elegance of the Humble Cross-Product in Vector Cryptography.



Running through the algorithm quickly, Alice ‘guesses’ (virtually) a
normal vector that she uses to define a plane in space and also to
define a line in that plane that she then commissions as a directed
number line. Her eventual encryption transformation of a plaintext is
that it evolves as a number on this number line that is represented
now by a position vector called Pn i.e the position of ‘n’ on the
line. In collusion with Bob she claps a large change-of-origin onto
to this Pn and sends it as ciphertext to Bob who removes the
obfuscating change-of-origin to get the Pn in its true from and
decrypts it back to ‘n’ before then decoding n back to its plaintext
equivalent.

The Overview in detail.

The geometry of these lines is such that Alice’s encryption line is
inextricably related to the direction of the normal vector that she
guessed at the beginning. It is apart from that line however in the
sense that a) they don’t physically intersect and b) by the same token
is not parallel to it either – in fact it is at right angles to it.
So, the lines are skew by a) and secondly, they are orthogonal by
b).

That they are “orthogonal skew lines” is a valid description – this is
all incidental to what I want to say about the artistic beauty of the
algorithm.

The Elegance of Bob’s Decryption.

The model being used in this cryptography boils down to a pair of
orthogonal skew lines as explained. Let us extend the model now by
drawing a longer line along i.e. coincident with, the normal vector
(extending it by using the normal as the parallel direction of the
line in other words) and we will now call this Bob’s number line as a
compliment to Alice’s number line.

We can now talk unambiguously about Alice’s number line and Bob’s
number line as a pair of orthogonal skew lines. Alice encrypts on her
line and Bob decrypts on his line.

The way it works.

Saying it as quickly as possible, the vector cross product methodology
enables any number on Alice’s number line to be transformed by
rotation onto Bob’s number line where he is able to proceed with its
decryption back into plaintext. Bob performs this ‘transformation of
n by rotation’ using Pn x V0 on Alice’s line to define n’s position
exactly on his own line.

This means that Bob is able to transform a number from a line that he
doesn’t need to know anything about (although he probably does on most
occasions) onto to his own line that he knows everything about and
then decode it back to its plaintext value.

The salient property of vectors that makes this possible is the
perpendicularity of the *cross product methodology (i.e. the cross
product of any two vectors is another, third vector, that is mutually
at right angles to each of the pair being multiplied).

This perpendicularity of the cross product is used widely in mechanics
but its use here in cryptography to decrypt numbers is to my mind
quite beautiful.

As a famous American golfer once said “we should stop to smell the
flowers more often”.

I’m sure readers will agree that the saying above is figuratively
pertinent to cryptography also. It is appropriate to highlight this
elegant application of the cross-product in cryptography by that same
token I think.

Appendix.

– the algebraic corollary of Bob’s “decryption by rotation” is
Pn x V 0 / N = n (dividing corresponding coefficients on the left
enables this equation to be solved although division is not generally
defined in vector arithmetic).

Euphemism.

The model of Alice and Bob’s encryption lines being a pair of
orthogonal skew lines is a bit like a pair of chop-sticks being held
at right angles to each other - Chop stick Cryptography?

Cheers - adacrypt


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