The general idea of the Fibonacci cryptography is similar to the Fibonacci coding and based on the application of the generalized Fibonacci matrices, the Qp-matrices, for encryption and decryption of the initial message M. Table 1 demonstrates the general idea of the Fibonacci encryption/decryption algorithm.
Note that the encryption/decryption key is the pair of the numbers of p and n. Since p = 0, 1, 2, 3, ... and n = 1, 2, 3, ... this means that this method has theoretically unlimited number of the encryption/decryption keys.
Let's consider the Fibonacci encryption method:
and then the Fibonacci decryption method
It follows from (1) and (2) that the Fibonacci encryption algorithm (1) is reduced to the n-multiple multiplication of the initial matrix M by the matrix Qp and the Fibonacci decryption algorithm is reduced to the n-multiple multiplication of the secret message E by the inverse matrix .
Let's consider now the multiplication of the initial matrix M by the matrix Qp. Let's consider the concrete example when the initial message is represented in the form of the 4 ´ 4 matrix:
For this case the Qp-matrix of the 4-d order (p + 1 = 4) is used for encryption:
For calculation of the matrix of E = M ´ Q3 we can represent it in the following form:
After the execution of the matrix multiplication (5) the matrix E takes the following form:
Comparing the initial matrix (3) with its secret equivalent (6) we can formulate the following rule concerning the multiplication of the initial matrix M by the coding Qp-matrix (4):
For the multiplication of the initial matrix by the Qp-matrix it is necessary to shift all the matrix entries of the initial matrix (3) to the right by one column, and form the first entries of each row by means of the addition of the first entry of each row of the initial matrix with its last entry.
It is easy to show that this rule is true for the matrix multiplication by the Qp-matrix of the arbitrary order (p + 1).
Let's consider now the multiplication by the inverse matrix of . For that we can use the example considered above. It is clear that the initial matrix (3) can be represented as the product:
Comparing the matrix E given by (6) with the initial matrix M given by (3) we can formulate the following rule for the multiplication by the inverse matrix .
For the multiplication of the initial matrix (6) by the inverse matrix it is necessary to shift all the matrix entries of the initial matrix (6) to the left by one column, and form the last entries of each row by means of the subtraction of the second entry of each row of the initial matrix from its first entry.
It is easy to show that this rule is true for matrix multiplication by the inverse matrix of with arbitrary order (p + 1).
Thus, the Fibonacci encryption algorithm is reduced to the n-multiple application of Rule 1 to the initial matrix M message, but the Fibonacci decryption algorithm is reduced to the n-multiple application of Rule 2 to the secret matrix E.
Let's consider now the expression (1) for the Fibonacci encryption and calculate the determinants of the matrices in the left-hand and right-hand parts of the expression (1). Then according to the matrix theory we have:
But we know from the preceding pages of our Museum that the matrices possess the following wonderful property:
If we substitute the expression (8) into (7) we get the following property of the Fibonacci cryptography:
What practical importance has the property (9)? The property (9) plays the role of the main check property of the Fibonacci cryptography! This means that we cannot only secret our initial message M and do it inaccessible to "hacker" but we can protect our secret message E from "noise" in the "channel"! This means that using the Fibonacci encryption/decryption method we can design the super-reliable cryptography allowing protecting the information from "hackers" and "noise" simultaneously! And we hope that you can use our Fibonacci encryption/decryption method in your informational praxis.
Of course, for practical implementation of the new cryptography method the additional researches are demanded. Let's make it together! We are ready to cooperate with you! Address to us!