The Sturmian Word

A Sturmian Word is a mathematical construct used to generate a string of binary digits, a sequence. Sturmian Word sequences can be defined strictly in terms of their combinatoric properties or geometrically as cutting sequences for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences on the alphabet of the two symbols 0 and 1. This string may be recursive or non-recursive. A recursive string means that sequential binary digital patterns will repeat at known frequencies. A non-recursive string means that no sequential binary digital pattern will ever occur and the the entire string will be unique. This uniqueness lends the use of non-recursive Sturmian Words in generating a binary digital string that may be used to generate a cypher. Any number of iterations may then be chosen as the starting reference point for the encoding sequence, some arbitrary prime number of iterations.

One means of generating a non-recursive Sturmian Word string is the Coding of Irrational Rotation. Another means is the Cutting Sequence of Irrational (http://en.wikipedia.org/wiki/Sturmian_word). The parameters, theta and x, absolutely define the Sturmian Word string generated for the Irrational Rotation just as slope and intercept absolutely define the Cutting Sequence of Irrational.

The generated Sturmian Word string is converted to cyphertext according to required usage, i.e., 64 bits, 128 bits, 256 bits, to generate the required security. Each sequential element of the Sturmian Word string, 64 bits for example, is assigned to 1 character of the ASCII breakdown. More on this subject later in the description.

The Cartesian Space Coordinates

Cartesian Space Coordinates refers to a mathematical construct used to define a point uniquely in space. This construct is usually referred to as the Cartesian Coordinate System. Three dimensional space is characterized relationally by 3 lines intersecting at a point. Each line is designated as an axis, commonly noted as the x, y and z axes. For the purposes of this description, we need only move 1 unit from the point (0,0,0). We may move 1 unit from 0,0,0 in either direction along any axis. Thus we move in a designated (+) or (-) 1 unit from 0,0,0. Any 2 axes will define a plane in space. Three axes will define 8 distinct and unique fields positioned about the point at 0,0,0 and hereafter referred to as the Block or Cube. Plus 1 is designated as a 1 for our alphabet. Minus 1 is designated as a 0 for our alphabet. Thus, each field maps to 1 of 8 different 3 bit strings: 000, 001, 010, 100, 011, 101, 111 and 110. This represents the functional matrix of the Block or Cube.

ASCII

It is required that the cleartext be in ASCII format, all this means is that 'plain' text with no formatting such as tabs, bold or underscoring, the raw format that any computer can understand. This is usually done so that the 'plain' text can be easily imported into a file and inserted into any application without issues. The ASCII has 2 elements of 128 bits with 256 bits overall. The first is referred to as the ASCII and the second is referred to as the Extended ASCII (From Wikipedia - http://www.asciitable.com/). The ASCII character table includes descriptions of the first 32 non-printing characters. ASCII was actually designed for use with teletypes and so the descriptions are presently somewhat obscure. More on this topic later in the description. The Extended ASCII is used for other language characters and scientific notation and may be required upon textual and functional requirements.

Thus, the 256 characters of the ASCII may be amended in any manner of choice that is suited for any particular purpose, especially those characters dealing with teletype terminology. In addition, the entire ASCII table may be re-arranged in any manner required to become the elements of any 1 of the 8 fields of the Block. This rearrangement or breakdown may then be changed according to any frequency required. Thus, a predetermined set of Block fields may be designated as the matrix to be chosen for the use of encipherment according to the frequency at hand at the time of encryption. This predetermined set must be known by both the instigator and the recipient of the cypher. If a change in the ASCII is not required, a standard set of 8 field elements for the matrix is all that is needed.

Frequency for change of the 8 Field matrix elements

The frequency of the change in the 8 Field matrix is known beforehand by both the instigator of the encipherment and the recipient of the transmission of the cypher. There are innumerable schemas for possible frequencies. Examples are by the minute, hour, day, week, month, year. The frequency schema changes in use at the time of the original encryption of information must be known by both the instigator and the recpient.

The Header

The header may consist of 2 parts, both clear text originally. The first part is the 3 bit field matrix designator for the Sturmian Word parameters, predetermined and known by both the instigator and the recipient. The second part is the 3 bit field matrix designator for the ASCII brakedown schema, predetermined and known by both the instigator and the recipient.

The Cypher

Finally, the cypher will consist of the header followed by the Sturmian Word cypher text. Again, the header and the Sturmian Word cyphertext may be re-encoded by any preexisting cypher in use. Decypherment is essentially the reverse process.