Fibonacci cryptography The general idea of the Fibonacci cryptography is similar to the Fibonacci coding and based on the application of the generalized Fibonacci matrices, the
Note that the encryption/decryption key is the pair of the numbers of 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 Let's consider now the multiplication of the initial matrix
For this case the
For calculation of the matrix of
After the execution of the matrix multiplication (5) the matrix
Comparing the initial matrix (3) with its secret equivalent (6) we can formulate the following rule concerning the multiplication of the initial matrix
For the multiplication of the initial matrix by the It is easy to show that this rule is true for the matrix multiplication by the 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
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 ( Thus, the Fibonacci encryption algorithm is reduced to the 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 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! |