One way to define a straight line in space is to define two points in space through which it passes. On the line below these two points are being taken as {0,1} defined by V₀ and V₁ respectively. The notation P₀ and P₁ is identical with this also, it is used to represent the position vectors of integer points only on the line, as a subset of all the numbers and is referenced separately later on in the context of factors and position vectors in vector cryptography.

An ordinary directed line

1) It can be seen here that the line has direction defined by (V₁-V₀).

2) The line has the general vector equation Vn = V0 + n(V₁-V₀) for all 'n' numbers.

3) The position vectors Pn defines the positions of the integer numbers 'n' only on the line and are particularly named Pn for this convenience.

4) 'n' to the right is always positive.

For the purposes of vector cryptography Alice takes a normal Vector N, so called because we are interested later in the plane that it defines, and she factorises it to get it's primary factors V₀ and V₁.

Factoring N

V₁ and V₀ are found by a customised factoring method described later elsewhere, they define each of the directed lines.

Alice's encryption path - (as a cryptic).

ASCII —› Decimal representation of the characters of ASCII as numbers —› Uses these together with her Digital Signature to index her encryption numbers-set that then becomes her encryption alphabet —› this establishes a set of integers on the directed lines that represent her plaintext —› the corresponding set of position vectors that represent these integers becomes her encryption transformation of plaintext.

The field of candidates for this subset of position vectors is in extent equal to:

10¹⁰⁰⁰x10³ x = 10¹⁰⁰³

The analysis of this vast domain breaks down to:

1000 normal vectors provide 1000 directed lines.

Alice's digital signature has 10¹⁰⁰⁰ possibilities of digit arrangements. This means there are 10¹⁰⁰⁰ encryption number-sets that can be formed according to how Alice's signature is configured. These number sets are then modelled on the directed lines.

If each number set that she configures goes on to each line in turn then it will provide 10¹⁰⁰⁰x10³ corresponding sets of position vectors. Each set of position vectors comprises one complete encryption alphabet.

Altogether, Alice has 10¹⁰⁰⁰x10³ encryption alphabets available to her at any time.

Note: the change-of-origin domain does not influence this logic since it simply goes on to the ciphertext after encryption.

Alice will use only one from this enormous domain of alphabets before discarding this line and creating a new one. The extent of these domains is the same each time but the order will be different in each case. They will have many gaps on the line in between the encryption numbers - imagine that only the nominated ones appear.

Notice, when a directed straight line is defined in this way, the distance between the integer points is equal to |V₁-V₀|. The significance of this last remark is to say that the distance is not arbitrary as in the traditional number line of number theory, where it is usually drawn to some scale (-› Scalar) and the number in question is the constant of proportionality between the sets of corresponding distances on all such similar number lines for that number.

Clearly,

Alice has an infinite domain of alphabets that she can use for vector cryptography. Vector cryptography -› Poly-alphabetic cryptography.

The reader will see that vector cryptography uses perfectly ordinary applied vector mathematics that uses directed straight lines in space to model the sets of integers. These lines have always used position vectors in the past in general vector mathematics to define points on the line, according to the equation of the line in question.

When Alice temporarily substitues an integer for a position vector and then works on the position vector using vector methods in lieu of the original scalar integer in her encryption transformation, this is no different to a comparable substitution in integral calculus say, that is very common also, as an example.