It is time for us to pay attention to non-deterministic polynomial time (NP-complete) problems. Non-deterministic polynomial time references non-deterministic Turing machines for problems and solutions dealing with decisions. It is still unknown if an NP-complete problem can be solved in polynomial time on a deterministic Turing machine. What if the later statement is NOT true? There are well known problems that can be solved in polynomial time. Many other problems will never check out for as long as, this fleeting Mathematical truth exists. Modern day cryptographic algorithms must find a better application. Cryptographic algorithms (technology) of the day are built on the axiom that; “Insolvability of NP problems in polynomial times gives validity to the integrity of most algorithms”.
This has that implication that, all along we have constructed our solutions to fit the arbitrary requirement of all computation in polynomial time. -Irreducibility of problems will no longer suffice.
This is our self-imposed limitation in computer science because not all problems fit in either polynomial time or realistic time. It is possible that such absolute classification is one of the reasons why the problem of internet security persists today. We have super computers now.
One question that comes up all the time as desiderata of NP-Completeness is this: “Given a knight in position (x) and dimension; column (m) and row (n) generalization of a chessboard, will black guarantee a win if it moves first”? Our question is, “Given a knight in position (x) and dimension; column(m) and row (n) generalization of a chessboard, will the black or white knight land on all the squares (change state) -in a sweeping move without a retrace of movement to the original position (x)”? If the answer to the later question is true. Assuming that (m) and (n) expand in size geometrically. Can anyone perform the above mentioned exercise in a polynomial time (very quickly)? I will say that the possibility is there; in both polynomial and exponential time because the bigger the dimension of the chessboard; the greater the number of moves one will make. We based the answer on the fact that we do not know how quickly this maneuvers could be accomplished.
Another question is, “Given the knight’ legal moves what kind of Turing machine will you call this chess board event? Non-deterministic or deterministic? If it is non-deterministic, it simply means that modern encryption will have long tenure in this era as the insolvability of NP Complete problems reigns supreme. Really! Is this statement accurate at all? That is what had been programmed in our minds for too long now. Well, we know that the polynomials used in the AES are irreducible ones. Galois field (GF ( 2^8 )) elements are created from these polynomials. Let’s re-write this field equation to GF(2^P); where p <=8. Will this maxim of AES hold still in the wake of quantum computing, Cloud, AI and the new era –Industrie 4.0? Has anyone thought about the requirement of the Galois Field when p > 8? These question raises other questions while piquing our minds in the direction we must go.
Whichever way the equation leans in your eyes. There is a possibility that what is deterministic today could appear non-deterministic tomorrow and vice-verse. Especially, with the rate at which technology advances, anything seems possible. It is imperative to know that earlier on we realized that there must be an introduction of an external context into symmetric cryptography (encryption). This is to keep the system firmly grounded and stabilized in the new era. This system is achieved by using the numbers generated from a knight’s tour solution.
LokDon invested much time (17+ years) to develop a unique knight’s template (KT) which is a scope of 16*16 or 256 integers. It is about 680 digits long string. A character set called standard state (ST) of equivalent number is mapped to the KT to build M1—A cipher template and the rest of its modular multiples. We believe that this discovery mixes up the deterministic and non-deterministic properties of a solution to decision making following a set of rules and defined variables.- This is a game changer. We have used these cipher templates to encrypt messages as well as finger prints or other grooved entities. The strings produced by each modes are encrypts, none can decipher this without the knowledge of the keys used e.g passwords, username, pin, crypto-keys etc. However, all these cryptographically secure keys and templates are not saved anywhere on the disk. They are generated on the fly. We believe that this system has all that is needed to take us to the new era of internet of things of values (IoToV).
We hope that many could share our views on the status quo. We hope that we can push what we know to the limits. We believe there is a way: We want to share this so that together we can secure and stabilize the technology which requires a great improvement. Blockchain must embrace a new set of encryption mechanism to reach the desired potentials.
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