Cryptosystems & Algebraic Topologies

TCP/IP is primarily used to transport data of nodes within a given computer network. That being said, has an individual pondered the data that is being transported by the TCP/IP protocol? This is what we will explore further.

Typically, data transported across a network is encrypted. Now, how does the encryption process work? The answer is relegated to two primary types of encryption categories; specifically, we are referencing symmetric and asymmetric encryption and decryption techniques, respectively. Albeit, these two categories for encrypting data across a network, although distinct, share a common basis for their underlying mechanics. Specifically, both the symmetric and asymmetric techniques use "algebraic topologies" for the structure of performing encryption on data. Holding the prior statement in mind, I believe that we should take a pause to address what are, and what utility that algebraic topologies provide for sending data "across the wire."

In as much, an algebraic topology is in essence a computational matrix which is parameterized across a specific number of independent variables. For example, the number of columns of a matrix, which is the computational implementation of realizing an algebraic topology, would be the independent variables that "parameterize" the given algebraic series of computations wherein the given matrix facilitates; the number of rows of that matrix represent the algebraic operations performed within the given topology.

Providing an illustration, we will use the matrix diagram listed below as an illustrative exhibit:

Looking at the illustration above, what can you interpret? Albeit, one can think of the output of a series of matrix operations as a computational singleton where the output of the corresponding matrix operations, is a singular computational "object" (e.g., a "blob"). Here then, it is presumed that the output of the given operations is an n x n matrix, where the variable, n, is the number of rows and columns, respectively.

Clarifying, the series of matrix computations provides a singleton computational output as an exhibit of the Taylor Series. That being said, while the rows of the given computation matrices represent the discrete series of matrix computations leading to the corresponding singleton (encrypted) value, the columns of that series represent the parameterizations underlying those matrix operations (i.e., the independent variables of those matrix operations). For example, parameterizations include, but are not limited to:

1. An encryption key
2. A decryption key
3. A random-number-generator (i.e., a pseudo random-number-generator, or a proper random-number-generator, respectively)

Given these conditions, an individual may appreciate how difficult it would be to compromise data sent between computer networks due to the sheer wake of computational power necessary to perform cryptoanalysis on these algebraic topologies in the motivations to decrypt data that is sent between computer networks encrypted.

In as much, what is specifically difficult as a means to perform cryptoanalysis on these given algebraic topologies? The answer is not only to determine the static values within the set of independent variables that parameterize the given set of matrix operations. But in conjunction, one must determine those values under the parameterization of time being projected onto those given static variables. Thus, one must perform computations to reverse engineer the given algebraic topology singleton output under the gauze in which those static values embedded within the given matrix operations used to encrypt data being sent "across the wire" that will expire relative to the time constraints which a random-number-generator facilitates.