A NEW KIND OF DIGITAL SIGNATURE SCHEME USING GOLDEN MATRICES BASED ON FACTORING PROBLEM

A.P. Stakhov in [12] proposed the concepts golden matrices and new kind of cryptography. In this paper, we propose a new kind of digital signature scheme based on factoring problem and the golden matrices, called the golden digital signature. The method is very fast and simple for technical realization and can be used for signature protection of digital signals (telecommunication and measurement system).


Introduction
In the last decades the theory of Fibonacci numbers [1], [3] was complemented by the theory of the so-called Fibonacci Q−matrix [2], [3].The latter is a square 2 × 2 matrix of the following form In [3] the following property of the n th power of the Q − matrix was proved Where n = 0, ∓1, ∓2, ∓3, F n−1 , F n , F n+1 are Fibonacci numbers given by the following recurrence relation: with the initial terms F 1 = F 2 = 1 Rule (4), can be used to extend the sequence backwards, thus F −n = (−1) n+1 F n In [7] , represent matrix (2) is showed in the following form It is proved in [8] the following property of the matrix, Alexey Stakhov, Ivan Tkachenko and Boris Rozin developed recently a theory of the hyperbolic Fibonacci and Lucas functions [9,10,11] Let us consider so-called symmetrical hyperbolic Fibonacci functions introduced in [11] Symmetrical hyperbolic Fibonacci sine: Symmetrical hyperbolic cosine: Where sF s = τ x −τ −x √ 5 (the Golden Proportion).
Note that the symmetrical hyperbolic Fibonacci functions are connected to the Fibonacci numbers by the following correlations: It was proved in [11] that the following identities connect the symmetrical hyperbolic Fibonacci functions. [ Note that the identities (8) and ( 9) are a generalization of the Cassini formula (3) for continuous domain.Stakhov [12] developed a theory of the golden matrices that are a generalization of the matrix (2) for continuous domain.
He defined the golden matrices in the terms of the symmetrical hyperbolic Fibonacci function (5) and (6).The golden matrices that are the functions of the continuous variable x are the following form.
Stakhov [12] obtained inverse matrices of (10) and (11).The inverse golden matrices that are the functions of the continuous variable x are the following form.
Furthermore, in [12] the golden matrices were used for creation of a new kind of cryptography called the golden cryptography.In this paper, we propose a new kind of digital signature based on factoring and golden matrices.

The New Digital Signature
In this section, we propose a new signature scheme based on factoring problem and golden matrices.

Algorithm for Signing and Verification Message
Select two large primes p, q and n = pq.The signer picks randomly an integer e from 0, 1, 2, ..., n − 1 such that gcd(e, φ(n)) = 1.Calculate d such that ed ≡ 1 (modφ(n)) .Thus the public key and secret keys respectively given by e and d.
What we have introduced above, namely the "golden" direct and inverse matrices and allow us to develop the following application to digital signature.Let the initial message be a digital signal, which is any sequence of real numbers: The problem of protecting the "digital signal" (14) from the hackers is solved usually with application of digital signature methods.Consider a new signature method based on the "golden" matrices.To this end let us choose the first four readings a 1 , a 2 , a 3 , a 4 of ( 14) and form them a square 2 × 2-matrix M : Note that the initial matrix M can be considered as message Note that there are 24 variants (permutations) to form the matrix (15) from the four readings .Let us designate the i th permutation by (i = 1, 2, ..., 24).The first step of signature protection of the four readings a 1 , a 2 , a 3 , a 4 is a choice of the permutation P i .Let us consider now the following signing /verification algorithms based on matrix multiplication (see table 1).Here is the plaintext (15) that is formed according to the permutation P i ;s 1 (x), s 2 (x) are signatures; A = Q 2x , B = Q 2x+1 are the signature matrices (10) and (11) are the verification matrices ( 12) and (13).We can use the variable x as a signature key or private key.This means that in dependence on the value of the key x there is an infinite number of transformation of the message M into signature .In general the key consists of two parts: public key e , private keys (x, d) The signature on message M is (A, B, s 1 (x), s 2 (x)).where M d is matrix to the power d , also [s 1 (x)A −1 ] e = M (mod n) is matrix to the power e The above equations are true for valid signature since: Let us prove that the signature method given with table 1 ensures one -valued transformation of the message into the signature s and then the signature s into the message M .Let us consider this transformation for the case when we choose the" golden" matrix (10) as the digital matrix .For the given value of the digital signature key x = x 1 the "golden" digital can be represented as follows: suppose where Let us consider the "golden" verification for case: where For calculation of the matrix elements given by ( 22)-( 25) we can use the expression (17)-(20).Then we have: Using the fundamental identity (8) we can write the expression (26) as follows: In the same manner after corresponding transformations we can write:

Performance Evaluation
We use the following notation to analyze the performance of the scheme : • T s this means that a full signature time By analogy, if we consider the expressions (21) − (25) we can write the expressions for a full verification time: Analysis of the expressions ( 27) and (28) show that the golden signature is fast signature .This means that the golden signature can be used for signature protection of digital in scale of time.

Conclusion
In this paper, we presented a new signature scheme based on factor-ing and golden matrices, the main result of the present article is a develop of one more application of the golden proportion, that is, a creation of a new kind of digital signature called the golden digital signature.The proposed scheme is fast signature.This means that the golden signature can be used for signature protection of digital in scale of time

Table 1 :
Signing / verification algorithm • T v this means that a full verification time • T ad is the time of modular addition • T m is the time modular multiplication • T exp is the time for a modular exponentiation According to (16) the signature consists in calculation of four elements s 11 , s 12 , s 21 , s 22 of the matrix (16) .According to (17) − (20) a calculation of every element include eight multiplication and four additions.This means that a full signature time T s is equal: