The Inverse Function  

Brief Overview

There are four versions of the same basic cipher in this scheme called 'AdaCrypt Vector Cryptography'. These differ from each other mainly by the way in which Bob, the receiver of an encrypted message, decodes Alice's cipher-text. The inverse function takes the cipher-text that Bob receives from Alice, removes the change-of-origin vector from it by subtraction and retrieves Alice's computed position vector. In every case the modus operandi is to use this position vector in conjunction with the normal vector N of a plane that is known to contain this position vector, designated Pn, and use it to determine the 'n' that is the subscript of Pn, and decode that back into the plaintext that Alice wants Bob to know.

The four versions of the cipher only differ from each other in 1) the way in which Bob acquires a suitable normal vector N and 2) in the way he uses it to deduce the 'n' of Pn. Suffice it to say for the time being that Bob is in possession of 1) a Pn and 2) the normal vector N of a plane that is known to contain Pn. Then,

From previous work,

Let N = then, V₀ =

Pn x V₀ = n • N - n is a scalar

—> Pn x = n • N

—> Pn x = E_x • n • N = ∞ • N where ∞ = E_x • n.

Note: E_x and 'n' can vary from plane to plane according to differing N's, but ∞ remains constant meanwhile. To get the correct 'n' at decryption time Bob divides ∞ by Alice's E_x known at the time of encryption. He decodes this 'n' back to it's plaintext value according to the agreed encryption/decryption alphabet with Alice. In essence this describes the decryption process sufficiently for the time being. There is more to it than this but later. Contact austein.obyrne@btinternet.com

Two of the ciphers however, differ from the others quite sharply in that Bob does not use the explicit normal that Alice also used, but instead conjures up a different normal almost like a 'rib from the side of the Pn' in metaphoric terms. This is shown in the sketch that follows but it requires a bit of mental gymnastics by the reader to realise the idea.

Lemma

Every innocent-looking vector W, say, on a page, is the line of intersection of an infinite family of planes that all intersect along W, as a common line. They each contain the vector W and each plane has a defining normal vector that belongs in a 'fantail' of radials that radiate outwardly from the origin and are in fact the normal vectors of their respective planes, they exist in the plane of radial vectors to which Pn itself is the overall defining normal vector.

Any one of these ghostly radials will do very nicely as a decryption tool in Bob's decryption scheme as a substitute for the exact normal that Alice used at her end. Finding one of these tertiary normal vectors requires some convoluted working by Bob but the reader is well capable of discerning this. The sketch below may help.

It needs to be remembered therefore, that every vector in space is the line of intersection of an infinite family of planes (there to be found) that all contain that vector. This kind of latent geometry analysis is very characteristic of vector mathematics and will be of interest to cryptanalysts.

This is truly a mathematical function. There is only one value for the 'n' of Pn when it is decrypted and so there is no ambiguity about the result. That is very important. The diagram shows the convolutions of Bob's decryption process. This the most difficult part to understand in all of the theory of this cipher and the reader is advised to give it his best shot.

Any one of these radial vectors in the left-hand sketch is suitable as a normal vector that Bob can use as N. He then proceeds as described above earlier.


Back - to - Back Model of the Decryption Process.

This is simply another view that might also be useful to the reader.

Alice's Pn as the Line of Intersection of many Planes.

Yet another view that might be useful.

The N2's are tertiary normals that can be rotated round to replace N₁. Any one of these 'other' normal vectors designated N₂in the sketch, can ro-tated up to the vertical position and replace N₁ to suit Bob's Inverse Process.

That is part of his key. This requires the correct Pn that only he alone knows apart from Alice.

The reader is looking back along Alices Pn from the terminal end of the vector and in the background one can see some of the infinite set of all the defining 'Normals' of the 'family' that intersect along the line of Pn. These are found by factoring Pn after it is proposed as being the defining normal vector of that plane in the background that is at right angles to it. Any one of these Pn₂'s will serve Bob's purpose of finding a plane that will enable him to decrypt Pn according to the manner described already.

Home Contact Us