Институт Золотого Сечения - Математика Гармонии

Stakhov А.P.

International Club of the Golden Section
6 McCreary Trail, Bolton, ON, L7E 2C8, Canada
goldenmuseum@rogers.com · www.goldenmuseum.com/

В статье рассматривается новый класс квадратных матриц, названных «золотыми» матрицами. Они являются обобщением классической «фибоначчиевой» Q-матрицы на непрерывную область. «Золотые» матрицы использованы для создания нового криптографического метода. Метод «золотой» криптографии является очень быстрым и простым для технической реализации и может быть использован для криптографической защиты различных телекоммуникационных систем (включая Интернет), работающих в реальном масштабе времени.

We consider a new class of square matrices called the «golden» matrices. They are a generalization of the classical Fibonacci Q-matrix for continuous domain. The «golden» matrices can be used for creation of a new kind of cryptography called the «golden» cryptography. The method is very fast and simple for technical realization and can be used for the cryptographic protection of different telecommunication systems (including Internet) functioning in real scale of time.

1. Introduction

In the last decades the theory of Fibonacci numbers [1-3] was reinforced and invigorated by new scientific results and applications [4-31]. Let us consider only two of them.

Fibonacci Q-matrix

In [2] the following square (2ґ 2)-matrix was introduced:

.

(1)

The following property of the n^th power of the Q-matrix was proved in [2]:

,

(2)

where n = 0, ± 1, ± 2, ± 3, …, F_n-1, F_n, F_n+1 are Fibonacci numbers given by the following recurrence relation:

Fn+1 = Fn + Fn-1

(3)

with the seed:

F1 = F2 = 1.

(4)

In the form (2) the Q-matrix shows its connection to the Fibonacci numbers given by (3), (4). It is easy to prove that the determinant of the matrix (2) coincides with the famous «Cassini formula»

(5)

that was named in honor of the 17-th century astronomer Giovanni Cassini (1625-1712).

Hyperbolic Fibonacci and Lucas functions

Alexey Stakhov, Ivan Tkachenko and Boris Rozin developed recently a theory of the hyperbolic Fibonacci and Lucas functions [8, 14, 15].

Let us consider so-called symmetrical hyperbolic Fibonacci functions introduced in [15].

Symmetrical hyperbolic Fibonacci sine:

(6)

Symmetrical hyperbolic Fibonacci cosine:

(7)

where (the Golden Proportion).

Note that the symmetrical hyperbolic Fibonacci functions are connected to the Fibonacci numbers (3), (4) by the following correlations:

(8)

It was proved in [15] that the following identities connect the symmetrical hyperbolic Fibonacci functions:

[sFs(x)]2 - cFs(x+1) сFs(x-1) = -1

(9)

[cFs(x)]2 - sFs(x+1) sFs(x-1) = 1.

(10)

Note that the identities (9), (10) are a generalization of the «Cassini formula» (5) for continuous domain.

The main purpose of the present paper is to develop a theory of the «golden» matrices that are a generalization of the matrix (2) for continuous domain. It is also considered to be a new kind of cryptography, which we call the «golden» cryptography, because it is based on the «golden» matrices.

2. The «golden» matrices

Some properties of the Q-matrix

Let us represent the Q-matrix (2) in the following form:

(11)

or

Qn = Qn-1 + Qn-2.

(12)

Let us write the expression (11) in the form:

Qn-2= Qn — Qn-1.

(13)

It was shown in [2] that the following property of the Q-matrix is valid:

QnQm = QmQn = Qn+m.

(14)

A representation of the matrices Qⁿ (n = 0, ± 1, ± 2, ± 3,...) in explicit form based on the recurrence relations (12), (13) are given in Table 1.

Table 1. The explicit forms of the matrices Qⁿ

Note that Table 1 gives the «direct» matrices Qⁿ and their «inverse» matrices Q^-n in explicit form. Comparing the «direct» and «inverse» Fibonacci matrices Qⁿ and Q^-n given in Table 1 it is easy to see that there is a very simple method to get the «inverse» matrix Q^-n from its «direct» matrix Qⁿ. In fact, if the power of the «direct» matrix Qⁿ given with (2) is odd number (n=2k) then for getting of its «inverse» matrix Q^-n it is necessary to rearrange in matrix (2) its diagonal elements F_n+1 and F_n-1 and to take its diagonal elements F_n with an opposite sign. It means that for the case n=2k the «inverse» matrix Q^-n has the following form:

(15)

For the case n=2k+1 for obtaining the «inverse» matrix Q^-n from the «direct» matrix Qⁿ it is necessary to rearrange in (2) the diagonal elements F_n+1 and F_n-1 and to take them with an opposite sign, so that one finds:

.

(16)

The direct «golden» matrices

Let us represent now the matrix (2) in the form of two matrices that are given for the odd (n=2k) and even (n=2k+1) values of n:

(17)

(18)

If we use the correlation (8), we can write the matrices (17), (18) in the terms of the symmetrical hyperbolic Fibonacci functions (6), (7):

(19)

(20)

where k is a discrete variable, k=0, ± 1, ± 2, ± 3, ….

If we replace now the discrete variable k in the matrices (19), (20) by the continuous variable x, then we will come to the two unusual matrices that are the functions of the continuous variable x:

(21)

(22)

It is clear that the matrices (21), (22) are a generalization of the Q-matrix (2) for continuous domain. They have a number of the unique mathematical properties. For example, for the matrix (14) takes the following form:

(23)

It is impossible to imagine a meaning for the «square root of the Q-matrix», but such «Fibonacci fantasy» follows from (23) amazing on it may be.

The inverse «golden» matrices

Let us represent the inverse matrices (15), (16) in the terms of the symmetrical hyperbolic Fibonacci functions (6), (7):

(24)

(25)

where k is a discrete variable, k=0, ± 1, ± 2, ± 3, ….

If we replace now the discrete variable k in the matrices (24), (25) by the continuous variable x, then we will come to the following matrices that are the functions of the continuous variable x:

(26)

(27)

Let us prove that the matrix (26) is the inverse of the matrix (21). To this end let us calculate a product of the matrices (21) and (26):

Q2x ґ Q-2x = ґ = =,

(28)

where

c11 = cFs(2x+1)ґ cFs(2x-1) – [sFs(2x)]2

(29)

c12 = — cFs(2x+1)ґ sFs(2x) + cFs(2x+1)ґ sFs(2x)

(30)

c21 = sFs(2x)ґ sFs(2x-1) — sFs(2x-1)ґ sFs(2x)

(31)

c22 = — [sFs(2x)]2 + cFs(2x+1)ґ cFs(2x-1)

(32)

It follows from (30), (31) that

c12 = c21 = 0

(33)

Using the identity (9) we can write the expressions (29), (32) as follows:

c11 = c22 = 1.

(34)

Then using (33), (34) we can write the expression (28) in the form:

Q2xґ Q-2x =

(35)

The identity (35), which is valid for any value of the variable x, is a proof that the matrix (26) is the inverse matrix.

In the same manner we can prove:

Q2x+1ґ Q-2x+1 =

(36)

This means that (27) is the inverse matrix to (25).

Determinants of the «golden» matrices

Let us calculate now the determinants of the matrices (21) and (22):

Det Q2x = cFs(2x+1)ґ cFs(2x-1) – [sFs(2x)]2

(37)

Det Q2x+1 = sFs(2x+2)ґ sFs(2x) – [cFs(2x+1)]2

(38)

Let us compare the expression (37) to the identity (8). Because the identity (8) is true for any values of the variable x, in particular, for the value 2x, it follows from this consideration the following identity:

Det Q2x = 1

(39)

By analogy, we can write:

Det Q2x+1 = - 1

(40)

If we now return back to the «Cassini formula» (5), we can conclude that the unusual identities (39), (40) are a generalization of the «Cassini formula» (5) for continuous domain.

3. The «golden» cryptography

The «golden» cryptographic method

What we he introduced above, namely the «golden» direct and inverse matrices (21), (22), (26), (27), allow us to develop the following application to cryptography. Let the initial message be a «digital signal», which is any sequence of real numbers:

a1, a2, a3, a4, a5, a6, a7, a8, …

(41)

Separate real numbers of the sequence (41) are called readings.

There are many examples of the «digital signals» (41): digital telephony, digital TV, digital measurement systems and so on.

The problem of protecting the «digital signal» (41) from the hackers is solved usually with application of cryptographic methods. Consider a new cryptographic method based on the «golden» matrices.

To this end let us choose the first four readings a₁, a₂, a₃, a₄ of (41) and form from them a square 2ґ 2-matrix M:

(42)

Note that the initial matrix M can be considered as plaintext [32].

Note that there are 4! = 4ґ 3ґ 2ґ 1 = 24 variants (permutations) to form the matrix (42) from the four readings a₁, a₂, a₃, a₄. Let us designate the i-th permutation by P_i (i=1, 2, …, 24). The first step of cryptographic protection of the four readings a₁, a₂, a₃, a₄ is a choice of the permutation P_i.

Then we choose the direct «golden» matrices (21) or (22) as enciphering matrices and their inverse matrices (26), (27) as deciphering matrices.

Let us consider now the following encryption/decryption algorithms based on matrix multiplication (see Table 2).

Table 2. Encryption/decryption algorithm

Here M is the plaintext (42) that is formed according to the permutation P_i; E₁(x), E₂(x) are ciphertexts; Q^2x, Q^2x+1 are the enciphering matrices (21), (21); Q^-2x and Q^-2x+1 are the deciphering matrices (26), (27). We can use the variable x as a cryptographic key or simply a key. This means that in dependence on the value of the key x there is an infinite number of transformation of the plaintext M into the ciphertext E(x).

In general the key K consists of two parts: permutation P_i and the variable x, that is,

K = {P, x}.

Let us prove that the cryptographic method given with Table 2 ensures one-valued transformation of the plaintext into the ciphertext E and then the ciphertext E into the plaintext M. Let us consider this transformation for the case when we choose the «golden» matrix (21) as the enciphering matrix. For the given value of the cryptographic key x = x₁ the «golden» encryption can be represented as follows:

Mґ Q2x = ґ = =E(x1)

(43)

where

e11 = a1cFs(2x1+1) + a2sFs(2x1)

(44)

e12 = a1sFs(2x1) + a2cFs(2x1-1)

(45)

e21 = a3cFs(2x1+1) + a4sFs(2x1)

(46)

e22 = a3sFs(2x1) + a4cFs(2x1-1)

(47)

Let us consider the «golden» decryption for this case:

E(x1)ґ Q-2x = ґ == D

(48)

where

d11 = e11cFs(2x1-1) – e12sFs(2x1)

(49)

d12 = — e11sFs(2x1) + e12cFs(2x1-1)

(50)

d21 = e21cFs(2x1-1) – e22sFs(2x1)

(51)

d22 = — e21sFs(2x1) + e22cFs(2x1-1)

(52)

For calculation of the matrix elements given by (49)-(52) we can use the expressions (44)-(47). Then we have:

d11 = [a1cFs(2x1+1) + a2sFs(2x1)] cFs(2x1-1) – [a1sFs(2x1) + a2cFs(2x1-1)]sFs(2 x1) = = a1cFs(2x1+1) cFs(2x1-1) + a2sFs(2x1) cFs(2x1-1) — a1[sFs(2x1)]2 – - a2cFs(2x1-1) sFs(2x1) = a1{cFs(2x1+1) cFs(2x1-1) – [sFs(2x1)]2}

(53)

Using the fundamental identity (9) we can write the expression (53) as follows:

d11 = a1.

(54)

In the same manner after corresponding transformations we can write:

d12 = a2

(55)

d21 = a3

(56)

d22 = a4

(57)

Using (54)-(55) we can write the matrix D (48) as follows:

D = = = M

(58)

This means that a cryptographic method given by Table 2 ensures one-valid transformation of the initial plaintext M at the entrance of the coder into the same plaintext M at the exit of the decoder.

Determinants of the «golden» matrices

Let us calculate now the determinant of the ciphertexts, that is, the matrices E₁(x), E₂(x):

Det E1(x) = Det Mґ Det Q2x

(59)

Det E2(x) = Det Mґ Det Q2x+1

(60)

If we use the identities (39), (40), we can write the expressions (59), (60) in the following form:

Det E1(x) = Det M

(61)

Det E2(x) = — Det M

(62)

This means that the determinants of the matrices E₁(x) and E₂(x) is determined identically by the determinant of the initial matrix M.

Encryption and decryption time

According to (43) the encryption consists in calculation of four elements e₁₁, e₁₁, e₁₁, e₁₁ of the matrix (43). According to (44)-(47) a calculation of every element include two multiplications and one addition. This means that a full encryption time T_e is equal:

Te = 8D tm + 4D tad

(63)

where D t_m is a time of one multiplication and D t_ad is a time of one addition.

By analogy, if we consider the expressions (48)-(52) we can write the expression for a full decryption time:

Td = 8D tm + 4D tad

(64)

Analysis of the expressions (63), (64) show that the «golden» cryptography is fast cryptography. This means that the «golden» cryptography can be used for cryptographic protection of digital signals in real scale of time.

Improvement of cryptographic protection

We can improve the cryptographic protection of the method if we use multiple encryption and decryption. This idea consists in the following. The first step of encryption is a use of the key

K₁={P_i, x₁}

that consists of any permutation P_i and any value x₁, taken in random manner. As a result of the encryption we got the matrix E(P_i, x₁) given by (43). The second step of encryption is to use the matrix E(x₁) as the initial matrix for encryption. To this end we can use the second cryptographic key

K₂={P_j, x₂}

where P_j is the next permutation and x₂ is the next value of x. After performing the «golden» encryption with the key x₂ we can get a new matrix E that is a function of the two permutation P_i and P_j and two cryptographic keys x₁ and x₂, that is, E = E (P_j, x₁; P_j, x₂). In general case we can repeat this procedure n times, that is, the cryptographic key K is a totality of n random permutations P_i, P_j, …, P_k and n cryptographic keys x₁, x₂,..., x_n, that is,

K={Pi, x1; Pj, x2;...; Pk, xn }

(65)

As a result of multiple encryption we can get the matrix

E = E (K).

For the decryption we have to apply the inverse cryptographic key K^-1 that is an inverse form of the initial cryptographic key, that is,

K^-1 ={ P_k, x_n; P_r, x_n-1; …; P_j, x₂; P_i, x₁}.

Transmission of the key

It is clear the «golden» cryptographic method relates to symmetrical cryptography [32]. It is considered now that the main deficiency of the symmetrical cryptography is a problem of key transmission. To this end in the recent decades so-called asymmetrical cryptography was developed [32]. In the asymmetric cryptosystems we use two keys: open or public key and secret key. The encryption of the message before transmission is realized by the public key and the decryption of ciphertext is realized by the secret key. However, the asymmetric cryptography has two deficiencies:

1. They use very complicated encryption and decryption algorithms. This means that sometimes they cannot be used for protection of digital signals in real scale of time.
2. Because encryption and decryption algorithms are very complicated this fact puts forward a problem to guarantee that encryption and decryption would be realized without errors (a problem of reliable computations).

The «golden» cryptographic method allows us to solve these two problems very elegantly for the following conditions:

1. We can use the existing asymmetrical cryptosystems for the transmission of the cryptographic key (65).
2. We can use the «golden» cryptographic system given by Table 2 for the fast transmission of the digital signal (41).
3. We can use the unique mathematical property of the «golden» cryptography given with (61), (62) for checking of the encryption and decryption algorithms.

This means that using the «golden» cryptography method (Table 2) we can design fast, simple for technical realization and reliable cryptosystems.

Note that for every session of transmission we can change the cryptographic key (65). This means that the analysis of the previous transmissions cannot be used for uncovering of the current cryptographic key (65). We can change the cryptographic key (65) using a generator of random numbers and so on.

4. Conclusion

As is shown in [4-31] the application field of the Fibonacci numbers theory applications is developing with such intensity that we can call that it is the beginning of an era of «global Fibonaccization» of modern science. «Harmony Mathematics» [10, 12, 13, 16, 20], which is modern development of the Fibonacci numbers theory, touches the fundamentals of mathematics and can lead to a new number theory [8, 16] and a new measurement theory [4, 5, 7]. The hyperbolic Fibonacci and Lucas functions [8, 14, 15] can lead to creation of Lobachevsky-Fibonacci and Minkovsky-Fibonacci geometry [15] that is of great importance for theoretical physics. However, the works of Mauldin and Willams, El Naschie, Vladimirov [22-31] and other physicists show that it is impossible to imagine a development of modern physical research without deep knowledge of the Golden Section.

The main result of the present article is a development of one more application of the Golden Proportion, that is, a creation of a new kind of matrices called the «golden» matrices. They are an original synthesis of two important discoveries of modern Fibonacci numbers theory, the Fibonacci Q-matrix (2) and hyperbolic Fibonacci functions (6), (7). A new kind of cryptography, the «golden» cryptography, follows from this approach. A common use of the existing asymmetric cryptosystems (for the transmission of the secret key) and the «golden» cryptosystems (for transmission of ciphertexts) can lead to the fast, simple for technical realization and reliable cryptosystems for protection of digital signals used in telecommunication and measurement systems.

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