1 Introduction
Information is one of the most valuable assets since the dawn of civilizations. The secured transmission of information is of prime importance. Cryptography is the science of study about the security, privacy, and confidentiality of information transmitted over a secured channel. Although many encryptiondecryption schemes exist but the need for new nonstandard encryption algorithms raises to prevent any traditional opportunities to steal or change the information. Also reducing the time and space complexity of key generation with enhancement in security so that it is difficult for an intruder to steal or change the information becomes a need for today’s time..
The problem concerned to the topic is to develop a model of public key cryptography with the following parameters:

Reduction in complexity for key generation.

Enhancement of security in the transmission of information.
As an example, for the same level of security, the RSA cryptosystem uses a bigger key size than the key size of Elliptic curve cryptography.
There is a need for finding one model which has the least complexity but maximum possible security.
In 1976, Diffie and Hellman [2] provided a solution to the longstanding problem of key exchange and pointed the way to digital signature. In 1978 Rivest, Shamir & Adleman[9] proposed a public key cryptosystem which famed as RSA cryptosystem. The security of RSA cryptosystem depends on the difficulty level of factoring large integers.
R.ALVAREZA, et.al[1] proposed a public key cryptosystem (PKC) based on the generalization of the discrete logarithm problem for block matrices over the field with reduced key length for a given level of security.
Kuppuswamy, et.al [8] has given two different encryption algorithms; One is public key cryptography based on a linear block cipher and the second one is private key cryptography based on a simple symmetric algorithm . Viswanath and Kumar[12] proposed a public key cryptosystem using Hill’s cipher, in which security of the system depends on the involvement of two digital signatures.
M. Zeriouh, et.al [13], proposed the concept of key exchange between Alice and Bob using specially designed matrices. In this key exchange scheme, each of the sender and receiver first chooses a square matrix of suitable order and then both publish their corresponding set of matrices which commute with their corresponding chosen matrices.
In this paper, we propose a modified key exchange scheme and public key cryptography based on generalized Fibonacci matrices and affinehill cipher. In this proposed scheme, the time complexity of key matrix generation is reduced to . The security of our scheme depends on the discrete logarithm problem. So our proposed scheme is secure against various known attacks.
This paper is organized as follows. In section 2, the basic concepts of Affine Cipher, generalized matrices and block matrices are outlined. In section 3, we have shown commutativity of the generalized Fibonacci matrices and developed public key cryptography. In section 4, we have described an example of proposed scheme. Finally in section 5, we present the cryptanalytic strength of model.
2 Preliminaries
Definition 2.1 (AffineHill Cipher).
AffineHill Cipher is defined as:
(2.1)  
(2.2) 
where P, C & K represents plaintext, ciphertext & key matrix respectively and is a prime with .
2.1 Generalization of Fibonacci sequence and Fibonacci matrix
Definition 2.2 (Fibonacci Sequence).
Fibonacci Sequence is defined by the recurrence relation
(2.3) 
and the terms of this sequence are called Fibonacci numbers.
Note that, the Fibonacci sequence can also be extended in negative direction [5, 7] .
Fibonacci Matrix:
Fibonacci Matrix was first used by Brenner and it’s basic properties were enumerated by King [6]. In 1985, Honsberger [4] showed that the Fibonacci matrix F is a square matrix of order 22 of the form
and it’s power is defined as
(2.4) 
In notation, here we use instead of and in general instead of for .
Generalization of Fibonacci matrix
Definition 2.3.
The generalized Fibonacci sequence of order is defined by the recurrence relation
(2.5) 
with and .
It can also be stated as multinacci sequence [5].
In particular, for , relation (2.5) is Fibonacci sequence[A000045] and for , (2.5) is called tribonacci sequence [5, 7][A000073].
The generalized Fibonacci matrix of order n is given by
and power of is given by
,
for
where ,
is identity matrix of order of n.
2.2 Block Matrices
Definition 2.4 (Block Matrix).
Let and are square matrix of order , then we define block matrix as,
Properties of the block matrices
Let and are square matrix of order and . Then some properties of block matrices discussed by M. Zeriouh, et.al [13] are given as follow,

, where
(2.6) which can also be seen as .

.

.
Lemma 2.1.
Matrix if and .
3 Main Work
Lemma 3.1.
Generalized Fibonacci matrices commute with each other i.e. if and are two generalized matrices for some then =.
Proof.
We prove this lemma by mathematical induction on and fixing .
For, ,
(3.1)  
And,
(3.2)  
Thus,from equation(3.1) and (3.2),
(3.3) 
Therefore result holds for .
Assume that the statement is true for ,
(3.4) 
Then for ,
So, result is true for .
Hence, by the principal of mathematical induction,result is true for every .
Here, we see that generalized Fibonacci matrix commute with each other.
∎
Hence satisfy lemma(2.1).
3.1 Key Generation Algorithm
Let the set ={ is a generalized Fibonacci matrix, }.
Construction of public key:

[label=.]

Alice chooses a prime number , and matrices .

Calculate
Here, is Alice’s public key and is her secret key.
Key Generation and Encryption:
Using Alice’s public key , Bob will generate first encryption key and then encrypts his plaintext as follows:

[label=.]

Bob choose his secret key and .

Calculate as

Further,
Thus, encryption key (say .

Encryption takes place as,

Bob sends to Alice.
Decryption:
After receiving from Bob, Alice perform following operations to recover plaintext:

[label=.]

Alice first calculate key matrix as,

Thus, decryption key (say D) = .

Decryption of ciphertext
where B is a row vector whose column is the sum of elements of coloumn of in .
4 Example
Example 1.
Let prime p=47 and is a generalized Fibonacci matrix of order 3, . Encrypt the plaintext HEY.
Construction of Public Key:
(i). Suppose Alice’s secret key , secret matrices and public matrix are
(4.1) 
(ii). Construction of public key,
Thus, here Alice’s public Key is
and
Alice’s secret key
is .
Key Generation and Encryption:
Bob generate encryption key using Alice’s public key , and encrypt his plaintext as follows:

.

Bob chooses his secret number, say m=3 and secret matrices , which are
(4.2) 
Calculation of ,

Calculation of encryption key,
So, encryption key .

Row vector B over , .

Encryption: .
[ Z P].
Here, plaintext encrypted as . 
Bob sends = to Alice.
Decryption:
After receiving , Alice recover plaintext as

First calculate key matrix as

Decryption key, D = .

Decryption:
(4.3) 
[36 25 15] and row vector .
Now recovering the plaintext according to (4.3), we haveThus, Alice recovers from successfully.
5 Complexity of Key matrix
In our work, time complexity [11] of generating key matrix using equation(2.6) is reduced to
(5.1)  
where, , and .
So time complexity of construction is reduced to .
Security strength of proposed method depends upon the private keys of sender and receiver .
Since, it is almost impossible to recover from because there is no any determinastic algorithm (Discrete logarithm problem [3, 10]) even intruder knows the matrix .
6 Conclusion
Here we have proposed public key cryptography and modification in key exchange method discussed in paper [zeriouh2019cryptography] using generalized Fibonacci matrices. We have used commutative property of generalized Fibonacci matrices to reduce complexity of construction. In above proposed scheme, neither sender nor receiver need to publish their corresponding set of matrices which commute with chosen matrix of sender and receiver respectively. Also our proposed scheme is secured against various known attacks. So, Our method is quite robust and can be implemented easily.
7 Acknowledgment
First author would like to thank University Grant Commission, India for financial support. The authors are grateful to Central University of Jharkhand, India for kind support.
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