# A novel public key cryptography based on generalized Lucas matrices

Kalika Prasad1111E-mail<EMAIL_ADDRESS>, Hrishikesh Mahato2222E-mail:
<EMAIL_ADDRESS>and Munesh Kumari3333E-mail:
<EMAIL_ADDRESS>
1,2,3Department of Mathematics, Central University of Jharkhand, India, 835205

###### Abstract

In this article, we have proposed a generalized Lucas matrix (recursive matrix
of higher order) having relation with generalized Fibonacci sequences and
established many special properties in addition to that usual matrix algebra.
Further, we have proposed a modified public key cryptography using these
matrices as keys in Affine cipher and key agreement for encryption-decryption
with the combination of terms of generalized Lucas sequences under residue
operations. In this scheme, instead of exchanging the whole key matrix, only a
pair of numbers(parameters) need to be exchanged, which reduces the time
complexity as well as space complexity of the key transmission and has a large
key-space.

Keywords: Affine Hill Cipher; Cryptography; Elgamal; Lucas Sequence; Lucas
Matrices
Mathematics Subject Classifications: 11T71, 11B39, 94A60, 14G50, 68P30

##  1 Introduction

Matrix theory is rich in special properties, some of such properties are based
on its construction, interrelations in eigen values and structure of its
invertibility. Based on its construction, matrices are used in different
branches of science as well as in engineering & technologies. One of such area
is cryptography[20, 18, 15], where matrix theory plays a vital role in storing
data, efficiency of encryption-decryption and enlarging the key-spaces.

It is well known that recursive sequences are defined in the terms of sum or
difference or product(elementary operations) on preceding terms of
corresponding sequences. Nowadays a lot of research work is going on in the
direction of generalizing the existing sequences for higher order as well as
generalizing for arbitrary initial values. While some of the authors made
generalizations by considering the same relation but with different
multipliers(constant/arbitrary functions as coefficients), some of such recent
developments and their applications may be seen in [21, 7, 2, 10, 11].

Akbary & Wang[1], proposed a variant of generalized Lucas sequence and based
on it constructed all the permutation binomials $P(x)=x^{r}+x^{u}$ of the
finite Field $\mathbb{F}_{q}$, where $q=p^{m}$ with p is some odd prime and
further they investigated permutations properties of a binomial. Cerda-
Morales[3] defined a new generalized Lucas $V(p,q)$-matrix similar to the
generalized Fibonacci $U(1,-1)$-matrix which is different from the Fibonacci
$U(p,q)$-matrix and further they established some well-known equalities and a
Binet-like formula by matrix method for both generalized sequences. Munesh
et.al.[12] proposed k-Mersenne sequence and generalized k-Gaussian Mersenne
sequence which are equivalent to Fibonacci sequences and further they
constructed recursive polynomials and established many well-known identities
like Binet formula, Cassini’s identity, D’Ocagne’s identity, and generating
functions. Halici, et.al.[6], discussed the Fibonacci quaternion matrix by
considering entries as n-th Fibonacci quaternion number and derived some
identities like Cassini’s identity, Binet formula, …etc. In[19] Stanimirovic,
et.al defined a generalization of Fibonacci and Lucas matrices whose elements
are defined by the general second-order non-degenerated sequence and in some
cases, they also obtained inverse for those matrices.

E. Özkan, et.al[14] obtained the terms of n-step Lucas polynomials by using
matrices & generalizing the concept and then establishing the relationship
between Lucas polynomials and Fibonacci polynomials. In[17] Singh, et.al
presented a generalization of Fibonacci-Lucas sequences, constructed matrix of
order three and established some identities but they followed it for second
order sequences, which can be further extended to higher order sequences and
matrix construction.

We know that, well-known identities Fibonacci sequence & Lucas sequence[9] is
given by recurrence relation $f_{k+2}=f_{k}+f_{k+1},~{}~{}k\geq 0$ with
initial values $0,1$ and $2,1$ respectively. Similarly, Tribonacci sequence &
Lucas sequence of order three is given by recurrence relation
$f_{k+3}=f_{k}+f_{k+1}+f_{k+2},~{}~{}k\geq 0$ with initial values $0,0,1$
[A000073] and $3,1,3$ [A001644] respectively. And matrix representations[9]
corresponding to above recursive sequences of order two and three has been
obtained as follow, where $f_{k,n}$ represents $n^{th}$ term of the sequence
of order k:

$\displaystyle\begin{bmatrix}f_{2,n+1}&f_{2,n}\\\ f_{2,n}&f_{2,n-1}\\\
\end{bmatrix},~{}~{}\begin{bmatrix}f_{3,n+2}&f_{3,n+1}+f_{3,n}&f_{3,n+1}\\\
f_{3,n+1}&f_{3,n}+f_{3,n-1}&f_{3,n}\\\
f_{3,n}&f_{3,n-1}+f_{3,n-2}&f_{3,n-1}\\\ \end{bmatrix}$ (1.1)

These matrices are recursive in nature consisting of many properties based on
initial values of corresponding sequences. For example, let us assume that
$Q_{k}^{n}$ be the matrix of order $k$ denoting multiplication $Q_{k}$ to n
times. Now, if we consider initial value $0,1$ for order two or $0,0,1$ for
order three then $(Q_{k}^{1})^{n}=Q_{k}^{n}$ holds but in other cases, it does
not hold, some of such observations has been found in[16, 22, 23, 13, 8].

In this paper, we are working on generalizing the Lucas sequences to higher
order preserving the Fibonacci trace properties(the terms of Lucas sequence
are obtained by considering the trace of the corresponding Fibonacci
matrices). The construction of recursive matrices has been done with entries
from combinations of terms of proposed sequences and established a
relationship with generalized Fibonacci matrices(GFM)[16]. Further, we
implement these matrices in the Affine-Hill technique and examine its behavior
and strength.

This paper is organized as follows. Section 1 introduces the related work on
the construction of recursive matrices and their applications. In section 2,
preliminaries on the cryptographic scheme, signature scheme & their
mathematical formulation are studied. In section 3, we have established
generalized Lucas sequences & matrices from GFM and discussed some remarkable
properties. In section 4, we proposed a new algorithm for key exchange and
encryption-decryption scheme with an example. Finally, in section 5 we
discussed the implementation, complexity and strength of the proposed scheme
followed by a conclusion in section 6.

##  2 Encryption Scheme and Mathematical Flow

### Affine-Hill Cipher

Affine-Hill Cipher is one of polygraphic block cipher equivalent to Hill
Cipher[18, 20] which encrypts blocks of size $n\geq 1$ successive plaintext to
ciphertext and vice-versa. Let us assume that $P$ be the plaintext
$(p_{1},p_{2},...,p_{r})$, $K$ be key matrix (simply key) and
$C=(c_{1},c_{2},...,c_{r})$ be corresponding ciphertext of sizes $1\times rn$,
$r\times r$ and $1\times rn$ respectively, where $p_{i}~{}\&~{}c_{i}$ are
block vectors of size $1\times n$. i.e

$\displaystyle
P=\begin{bmatrix}p_{1}&p_{2}&...&p_{r}\end{bmatrix},K=\begin{bmatrix}k_{11}&k_{12}&...&k_{1n}\\\
k_{21}&k_{22}&...&k_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
k_{n1}&k_{n2}&...&k_{nn}\end{bmatrix}~{}~{}\text{and}~{}~{}C=\begin{bmatrix}c_{1}&c_{2}&...&c_{r}\end{bmatrix}$

The Affine-Hill Cipher’s technique is described as:

$\displaystyle Enc(P):\hskip 19.91684ptc_{i}$ $\displaystyle\equiv$
$\displaystyle(p_{i}K+B)\pmod{p}$ (2.1) $\displaystyle Dec(C):\hskip
19.91684ptp_{i}$ $\displaystyle\equiv$ $\displaystyle(c_{i}-B)K^{-1}\pmod{p}$
(2.2)

with $(|K|,p)=1$ where $B$ is a $1\times n$ row vector, $p$ is a prime number
and $Enc(P)$ represents encryption scheme & $Dec(C)$ represents decryption
scheme.

###  2.1 ElGamal and Signature Scheme

Elgamal cryptosystem[18, 20, 15], proposed by T. Elgamal[5] in 1984 is a
public-key scheme with digital signature whose strength is based on discrete
logarithms (closely related to Deffie-Hellman technique). One of such similar
digital signatures is ’Schnorr signature scheme’[18] which minimizes the
message based computation required to generate a signature. Usually, the
digital signature scheme involves the use of a private key for the generation
of the digital signature and a public key for its verification purpose. The
design of the Elgamal technique is as encryption is done by user’s public key
while decryption using user’s private key.

#### Primitive root

Primitive roots[18] play a crucial role in securing strength of cryptographic
schemes. Let $n$ be any positive integer, an integer $\alpha$ is called
primitive root of $n$ if $\alpha^{k}\equiv 1\pmod{n}$ where $k=\phi{(n)}$ is
least positive integer, i.e. there does not exists any $r,1\leq r<k$ such that
$\alpha^{r}\equiv 1\pmod{n}$. Further, when $\alpha$ is primitive root of $n$,
the powers of $\alpha$, $\alpha,\alpha^{2},\alpha^{3},...,\alpha^{\phi{(n)}}$
are distinct $\pmod{n}$ and also co-prime to n. Note that not all integers
have primitive roots, in fact integers of the form $2,4,p^{k}~{}or~{}2p^{k}$
where $p$ is odd prime & $k\in\mathbb{N}$ have primitive roots.

Similar to the Diffie-Hellman scheme, in the Elgamal technique, the global
elements are the prime $p$ and a primitive root of $p$. Elgamal scheme is
defined as:

####  2.1.1 Public key setup

Let us assume that $p$ be any odd prime number, $\alpha$ be a primitive root
of $p$ and an integer D is chosen such that $1<D<\phi(p)$. Now, assign
$E_{1}=\alpha$ and $E_{2}=E_{1}^{D}\pmod{p}$. Then $pk(p,E_{1},E_{2})$ will be
made as a public key and chosen $D$ is kept as a secret key. Elgamal technique
can be understood as follows. Suppose two parties Alice and Bob want to
communicate with each other, then they go through the following path:

####  2.1.2 Key exchanging

Alice(wish to send message to Bob) first choose a random integer $e$ such that
$1<e<\phi(p)$ and then generates a signature key (say $s$) using public key
$pk(p,E_{1},E_{2})$ where $s=E_{1}^{e}\pmod{p}$. Now she computes her secret
key $\lambda$ for encryption as $\lambda=E_{2}^{e}\pmod{p}$. Thus Alice
generates the parameters$(\lambda,s)$ of an encryption key using Bob’s public
key ($p,E_{1},E_{2}$), then encrypt the plaintext with the encryption key and
send $(\lambda,C)$ to Bob, where C is the corresponding ciphertext.

####  2.1.3 Key recover by Bob

On the other side Bob after receiving ($s,C)$ from Alice, recovers the same
parameters$(\lambda,s)$ of encryption key using his secret key $sk(D)$ as:

$\displaystyle s^{D}\pmod{p}$ $\displaystyle\equiv$
$\displaystyle(E_{1}^{e})^{D}\pmod{p}$ (2.3) $\displaystyle\equiv$
$\displaystyle(E_{1}^{D})^{e}\pmod{p}\equiv(E_{2})^{e}\pmod{p}=\lambda$

Thus, Bob and Alice agree on the same parameters$(\lambda,s)$ of encryption
key i.e. the parameters$(\lambda,s)$ of encryption key exchanged securely and
using these parameters, Bob can decrypt the ciphertext $C$, and recover the
original plaintext $P$.

##  3 Generalized Lucas Sequences and Matrix Construction

In this section, we discuss the construction of generalized Lucas sequences
and generalized Lucas matrices(GLM). Further, we obtain some new properties
for GLM and establish a relationship with GFM.

###  3.1 Generalized Fibonacci Matrices(GFM)

For $n\in\mathbb{Z}$, the generalized Fibonacci sequences$\\{f_{k,n}\\}$ of
order $k\geq 2$ is defined as follow:

$\displaystyle
f_{k,k+n}=f_{k,k+n-1}+f_{k,k+n-2}+f_{k,k+n-3}+...+f_{k,n+2}+f_{k,n+1}+f_{k,n}~{}~{}~{}~{}k(\geq
2)\in\mathbb{N}$ (3.1)

where

$\displaystyle
f_{k,0}=f_{k,1}=f_{k,2}=...=f_{k,k-2}=0~{}~{}\&~{}~{}f_{k,k-1}=1$ (3.2)

The corresponding generalized Fibonacci matrix($Q_{k}^{n}$) of order $k$
denoted as $GFM(k,n)$ is defined[16] as
$Q_{k}^{n}=\begin{bmatrix}f_{k,k+n-1}&f_{k,k+n-2}+f_{k,k+n-3}+...+f_{k,n}&f_{k,k+n-2}+...+f_{k,n+1}&...&f_{k,k+n-2}\\\
f_{k,k+n-2}&f_{k,k+n-3}+f_{k,k+n-4}+...+f_{k,n-1}&f_{k,k+n-3}+...+f_{k,n}&...&f_{k,k+n-3}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
f_{k,k+n-(k-1)}&f_{k,n}+f_{k,n-1}+...+f_{k,-k+n+2}&f_{k,n}+...+f_{k,-k+n+3}&...&f_{k,n}\\\
f_{k,n}&f_{k,n-1}+f_{k,n-2}+...+f_{k,-k+n+1}&f_{k,n-1}+...+f_{k,-k+n+2}&...&f_{k,n-1}\\\
\end{bmatrix}$ where initial matrices are given by

$\displaystyle Q_{k}=Q_{k}^{1}=\begin{bmatrix}1&1&1&...&1&1\\\
1&0&0&...&0&0\\\ 0&1&0&...&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&...&1&0\\\
\end{bmatrix}~{}~{}and~{}~{}Q_{k}^{-1}=\begin{bmatrix}0&1&0&...&0&0\\\
0&0&1&...&0&0\\\ 0&0&0&...&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&...&0&1\\\ 1&-1&-1&...&-1&-1\\\ \end{bmatrix}_{k\times k}$

######  Theorem 3.1.

[16] Let $k(\geq 2)\in\mathbb{N},n\in\mathbb{Z}$, $Q_{k}^{n}$ be generalized
Fibonacci matrices ($GFM(k,n)$) and $I_{k}$ is identity matrix of order k then
following properties holds for every $n$ and $k$:

  1. 1.

${(Q_{k}^{1})^{n}}=Q_{k}^{n}$ and ${(Q_{k}^{-1})^{n}}=Q_{k}^{-n}$.

  2. 2.

${Q_{k}^{n}Q_{k}^{l}}=Q_{k}^{n+l}={Q_{k}^{l}Q_{k}^{n}}$.

  3. 3.

$Q_{k}^{n}Q_{k}^{-n}=Q_{k}^{0}=I_{k}$.

  4. 4.

$det(Q_{k}^{n})=[(-1)^{k-1}]^{n}=(-1)^{(k-1)n}$.

######  Lemma 3.2.

Let $Q_{k}^{1}$ be first Fibonacci matrix of order k and $A=(a_{ij})$ be any
square matrix of same size, then on multiplication with $Q_{k}^{1}$ to A, the
first row of $Q_{k}^{1}A$ will be the sum of corresponding columns of A and
second row to $k^{th}$-row of $Q_{k}^{1}A$ becomes first to $(k-1)^{kth}$ rows
of A respectively.

###### Proof.

It can be easily proved by the usual matrix multiplication of $Q_{k}^{1}$ and
$A$. ∎

###  3.2 Generalized Lucas Matrices(GLM)

Now, we are establishing a new sequence$\\{l_{k,n}\\}$ analogous to
generalized Fibonacci sequence as follow:

######  Definition 3.1.

Consider the recurrence relation $\\{l_{k,n}\\}$ of order $k\geq 2$ defined as

$\displaystyle
l_{k,k+n}=l_{k,k+n-1}+l_{k,k+n-2}+l_{k,k+n-3}+...+l_{k,n+2}+l_{k,n+1}+l_{k,n}~{}~{}~{}~{}n\geq
0,k(\geq 2)\in\mathbb{N}$ (3.3) with ${l_{k,r}}=trace(Q_{k}^{r})$, $0\leq
r<k$.

To obtain initial values for above proposed sequence, we are considering trace
of first k generalized Fibonacci matrices. Using equation(3.3) and by the
definition of $Q_{K}^{n}$ it can be observed that

$\displaystyle l_{k,n}$ $\displaystyle=$ $\displaystyle
trace(Q_{k}^{n})~{}~{}~{}~{}~{}~{}~{}for~{}n\in\mathbb{Z},~{}k(\geq
2)\in\mathbb{N}$ (3.4) $\displaystyle i.e.~{}~{}~{}~{}l_{k,n}$
$\displaystyle=$ $\displaystyle
f_{k,k+n-1}+1f_{k,k+n-3}+2f_{k,k+n-4}+3f_{k,k+n-5}+...+(k-3)f_{k,n+1}$ (3.5)
$\displaystyle+(k-2)f_{k,n}+(k-1)f_{k,n-1}$ $\displaystyle=$ $\displaystyle
f_{k,k+n-1}+\sum_{i=n-1}^{k+n-3}f_{k,i}+\sum_{i=n-1}^{k+n-4}f_{k,i}+...+\sum_{i=n-1}^{n-2}f_{k,i}+\sum_{i=n-1}^{n-1}f_{k,i}$

which yields $k,1,3,7,15,31,63,127,255,511,...,2^{k-1}-1$ as initial values of
$\\{l_{k,n}\\}$.

Since the generalized Fibonacci sequences $\\{f_{k,n}\\}$ and GFM(k,n)
$\\{Q_{k}^{n}\\}$ are both two ended sequences, hence $\\{l_{k,n}\\}$ can also
be extended in negative direction. The new sequence $\\{l_{k,n}\\}$ is called
the generalized Lucas sequence of order k. In particular, $k=2$ gives the
standard Lucas sequence of order 2.

Here, the recursive matrix($L_{k}^{(n)}$) corresponding to the above sequence
is defined as

$\displaystyle
L_{k}^{(n)}=\begin{bmatrix}l_{k,k+n-1}&l_{k,k+n-2}+l_{k,k+n-3}+...+l_{k,n}&l_{k,k+n-2}+...+l_{k,n+1}&...&l_{k,k+n-2}\\\
l_{k,k+n-2}&l_{k,k+n-3}+l_{k,k+n-4}+...+l_{k,n-1}&l_{k,k+n-3}+...+l_{k,n}&...&l_{k,k+n-3}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
l_{k,k+n-(k-1)}&l_{k,n}+l_{k,n-1}+...+l_{k,-k+n+2}&l_{k,n}+...+l_{k,-k+n+3}&...&l_{k,n}\\\
l_{k,k+n-k}&l_{k,n-1}+l_{k,n-2}+...+l_{k,-k+n+1}&l_{k,n-1}+...+l_{k,-k+n+2}&...&l_{k,n-1}\\\
\end{bmatrix}.$ (3.6)

And the matrix($L_{k}^{(n)}$) is known as generalized Lucas Matrix of order
$k$ and we denote it by GLM(k,n). Further, the initial Lucas matrix
$L_{k}^{(0)}$ is

$\displaystyle L_{k}^{(0)}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}2^{k-1}-1&2^{k-1}&2^{k-1}-k&...&7.2^{k-4}&3.2^{k-3}&2^{k-2}-1\\\
2^{k-2}-1&2^{k-2}&2^{k-2}+1&...&7.2^{k-5}&3.2^{k-4}&2^{k-3}-1\\\
2^{k-3}-1&2^{k-3}&2^{k-3}+1&...&7.2^{k-6}&3.2^{k-5}&2^{k-4}-1\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 1&2&3&...&k-2&k-1&k\\\
k&1-k&2-k&...&-3&-2&-1\\\ \end{bmatrix}$ (3.7)

######  Example 1.

Initial Lucas matrix of order two, three, four and five are respectively

$\displaystyle\begin{bmatrix}1&2\\\ 2&-1\\\
\end{bmatrix},\begin{bmatrix}3&4&1\\\ 1&2&3\\\ 3&-2&-1\\\
\end{bmatrix},\begin{bmatrix}7&8&4&3\\\ 3&4&5&1\\\ 1&2&3&4\\\ 4&-3&-2&-1\\\
\end{bmatrix}~{}and~{}\begin{bmatrix}15&16&11&10&7\\\ 7&8&9&4&3\\\
3&4&5&6&1\\\ 1&2&3&4&5\\\ 5&-4&-3&-2&-1\\\ \end{bmatrix}$

######  Theorem 3.3.

Let $n\in\mathbb{Z}$, $Q_{k}^{n}$ be generalized Fibonacci matrix,
$L_{k}^{(n)}$ be $GLM(k,n)$. Suppose $L_{k}^{(0)}$ be the initial matrix as
defined by equation(3.7), then we have

$\displaystyle{L_{k}^{(n)}}=Q_{k}^{n}L_{k}^{(0)}=L_{k}^{(0)}Q_{k}^{n}$ (3.8)

###### Proof.

We prove it using mathematical induction on n, for $n\geq 1$. Since for $n=0$,
result holds obviously. For $n=1$, we have

$\displaystyle Q_{k}^{1}L_{k}^{(0)}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}1&1&1&...&1&1\\\ 1&0&0&...&0&0\\\
0&1&0&...&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&...&0&0\\\
0&0&0&...&1&0\\\
\end{bmatrix}\begin{bmatrix}2^{k-1}-1&2^{k-1}&2^{k-1}-k&...&7.2^{k-4}&3.2^{k-3}&2^{k-2}-1\\\
2^{k-2}-1&2^{k-2}&2^{k-2}+1&...&7.2^{k-5}&3.2^{k-4}&2^{k-3}-1\\\
2^{k-3}-1&2^{k-3}&2^{k-3}+1&...&7.2^{k-6}&3.2^{k-5}&2^{k-4}-1\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 1&2&3&...&k-2&k-1&k\\\
k&1-k&2-k&...&-3&-2&-1\\\ \end{bmatrix}$ using lemma(3.2) $\displaystyle=$
$\displaystyle\begin{bmatrix}2^{k}-1&2^{k}-k-1&2^{k}-2-k&...&7.2^{k-3}&3.2^{k-2}&2^{k-1}-1\\\
2^{k-1}-1&2^{k-1}&2^{k-1}-k&...&7.2^{k-4}&3.2^{k-3}&2^{k-2}-1\\\
2^{k-2}-1&2^{k-2}&2^{k-2}+1&...&7.2^{k-5}&3.2^{k-4}&2^{k-3}-1\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 1&2&3&...&k-2&k-1&k\\\
\end{bmatrix}=L_{k}^{(1)}$

Now, assuming the result is true for $n=r$ i.e.
$Q_{k}^{r}L_{k}^{(0)}=L_{k}^{(r)}$. Further we show that the statement is also
true for $n=r+1$. We have,

$\displaystyle Q_{k}^{r+1}L_{k}^{(0)}$ $\displaystyle=$ $\displaystyle
Q_{k}^{1}Q_{k}^{r}L_{k}^{(0)}=Q_{k}^{1}L_{k}^{(r)}$ $\displaystyle=$
$\displaystyle
Q_{k}^{1}\begin{bmatrix}l_{k,k+r-1}&l_{k,k+r-2}+l_{k,k+r-3}+...+l_{k,r}&l_{k,k+r-2}+...+l_{k,r+1}&...&l_{k,k+r-2}\\\
l_{k,k+r-2}&l_{k,k+r-3}+l_{k,k+r-4}+...+l_{k,r-1}&l_{k,k+r-3}+...+l_{k,r}&...&l_{k,k+r-3}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
l_{k,k+r-(k-1)}&l_{k,r}+l_{k,r-1}+...+l_{k,-k+r+2}&l_{k,r}+...+l_{k,-k+r+3}&...&l_{k,r}\\\
l_{k,k+r-k}&l_{k,r-1}+l_{k,r-2}+...+l_{k,-k+r+1}&l_{k,r-1}+...+l_{k,-k+r+2}&...&l_{k,r-1}\\\
\end{bmatrix}$ using lemma(3.2) $\displaystyle=$
$\displaystyle~{}~{}~{}~{}~{}\begin{bmatrix}l_{k,k+r}&l_{k,k+r-1}+l_{k,k+r-2}+...+l_{k,r+1}&l_{k,k+r-1}+...+l_{k,r+2}&...&l_{k,k+r-1}\\\
l_{k,k+r-1}&l_{k,k+r-2}+l_{k,k+r-3}+...+l_{k,r}&l_{k,k+r-2}+...+l_{k,r+1}&...&l_{k,k+r-2}\\\
l_{k,k+r-2}&l_{k,k+r-3}+l_{k,k+r-4}+...+l_{k,r-1}&l_{k,k+r-3}+...+l_{k,r}&...&l_{k,k+r-3}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
l_{k,k+r-(k-1)}&l_{k,r}+l_{k,r-1}+...+l_{k,-k+r+2}&l_{k,r}+...+l_{k,-k+r+3}&...&l_{k,r}\\\
\end{bmatrix}$ $\displaystyle=$ $\displaystyle~{}~{}~{}~{}~{}L_{k}^{(r+1)}.$

For negative direction, consider $n=-r$, where $r\geq 0$. Then by mathematical
induction on $r$, it can be proved for negative direction with a similar
argument. Combining both the cases, the result holds for all $n\in\mathbb{Z}$.
The second equality can be proved similarly as first. ∎

######  Theorem 3.4.

Let $~{}n\in\mathbb{N}$, then we have
$(L_{k}^{(1)})^{n}=(L_{k}^{(n)})(L_{k}^{(0)})^{n-1}$.

###### Proof.

$\displaystyle(L_{k}^{(1)})^{n}=(Q_{k}^{1}L_{k}^{(0)})^{n}$ $\displaystyle=$
$\displaystyle
Q_{k}^{1}L_{k}^{(0)}Q_{k}^{1}L_{k}^{(0)}...n-times...Q_{k}^{1}L_{k}^{(0)}=(Q_{k}^{1})^{n}(L_{k}^{(0)})^{n}$
(3.9) $\displaystyle=$
$\displaystyle(Q_{k}^{n})(L_{k}^{(0)})^{n}=(Q_{k}^{n}L_{k}^{(0)})(L_{k}^{(0)})^{n-1}=(L_{k}^{(n)})(L_{k}^{(0)})^{n-1}$

∎

######  Theorem 3.5.

Determinant of generalized Lucas matrices are given by

$\displaystyle
det(L_{k}^{(n)})=\left\\{\begin{array}[]{lr}det(L_{k}^{(0)})&:if~{}k=odd{}{}\\\
(-1)^{n}det(L_{k}^{(0)})&:if~{}k=even\end{array}\right.$ (3.12)

###### Proof.

We have

$\displaystyle det(L_{k}^{(n)})=det(Q_{k}^{(n)}L_{k}^{(0)})$ $\displaystyle=$
$\displaystyle det(Q_{k}^{n})det(L_{k}^{(0)})=(-1)^{(k-1)n}det(L_{k}^{(0)})$
(3.15) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lr}det(L_{k}^{(0)})&:if~{}k=odd{}{}\\\
(-1)^{n}det(L_{k}^{(0)})&:if~{}k=even\end{array}\right.$

∎

######  Theorem 3.6.

For $m,n\in\mathbb{Z},~{}k(\geq 2)\in\mathbb{N}$, we have

$\displaystyle{L_{k}^{(m)}L_{k}^{(n)}}=L_{k}^{(m+n)}L_{k}^{(0)}={L_{k}^{(n)}L_{k}^{(m)}}$
(3.16)

###### Proof.

From theorem(3.3) and theorem(3.1) we have,

$\displaystyle
L_{k}^{(m)}L_{k}^{(n)}=Q_{k}^{(m)}L_{k}^{(0)}Q_{k}^{(n)}L_{k}^{(0)}=Q_{k}^{(m)}Q_{k}^{(n)}L_{k}^{(0)}L_{k}^{(0)}=Q_{k}^{(m+n)}L_{k}^{(0)}L_{k}^{(0)}=L_{k}^{(m+n)}L_{k}^{(0)}$
$\displaystyle
L_{k}^{(n)}L_{k}^{(m)}=Q_{k}^{(n)}L_{k}^{(0)}Q_{k}^{(m)}L_{k}^{(0)}=Q_{k}^{(n)}Q_{k}^{(m)}L_{k}^{(0)}L_{k}^{(0)}=Q_{k}^{(m+n)}L_{k}^{(0)}L_{k}^{(0)}=L_{k}^{(m+n)}L_{k}^{(0)}$

∎

######  Corollary 3.6.1.

For $k(\geq 2)\in\mathbb{N},~{}n\in\mathbb{Z}$, we have
$L_{k}^{(n)}L_{k}^{(0)}=L_{k}^{(0)}L_{k}^{(n)}$.

######  Theorem 3.7.

Let $L_{k}^{(n)}$ be the $k^{th}$ order generalized Lucas matrix and
$Q_{k}^{n}$ be GFM(k,n), then

$\displaystyle L_{k}^{(n)}L_{k}^{(-n)}=H\hskip
11.38092pt~{}~{}\forall~{}~{}n\in\mathbb{Z},~{}~{}where~{}H=(L_{k}^{(0)})^{2}$
(3.17)

###### Proof.

For $n\in\mathbb{N}$, we have

$\displaystyle L_{k}^{(n)}L_{k}^{(-n)}$ $\displaystyle=$ $\displaystyle
Q_{k}^{n}L_{k}^{(0)}Q_{k}^{-n}L_{k}^{(0)}=Q_{k}^{n}Q_{k}^{-n}L_{k}^{(0)}L_{k}^{(0)}=I_{k}H=H$

∎

######  Theorem 3.8.

Suppose $L_{k}^{(n)}$ be a GLM(k,n) and $H=(L_{k}^{(0)})^{2}$ is invertible,
then inverse is given by

$Inv(L_{k}^{(n)})=L_{k}^{(-n)}H^{-1}$

###### Proof.

Since, we have $L_{k}^{(n)}L_{k}^{(-n)}=H$ and $(L_{k}^{(0)})$ is invertible
matrix hence does H. Now, on post multiplication with $H^{-1}$, we get

$L_{k}^{(n)}(L_{k}^{(-n)}H^{-1})=I_{k};~{}~{}~{}~{}~{}I_{k}~{}\text{is
identity matrix.}$

Thus, for every GLM(k) $L_{k}^{(n)}$ there is a unique matrix
$(L_{k}^{(-n)}H^{-1})$ such that their product is an identity matrix. Hence

$Inv(L_{k}^{(n)})=L_{k}^{(-n)}H^{-1}$

∎

######  Lemma 3.9.

Let p be a prime number and L is a generalized Lucas matrix, then we have

$\displaystyle det(L)\pmod{p}=det(L\pmod{p})$ (3.18)

######  Theorem 3.10.

[18] Let $A=(a_{ij})$ be any matrix, then we have

$A\pmod{p}=[a_{ij}\pmod{p}]$

##  4 Encryption Scheme & Algorithm

Let us assume that receiver’s public key is $pk(p,E_{1},E_{2})$ whose
components are constructed by receiver(Bob) as discussed in subsection(2.1.1).
Now, a sender(Alice) constructs a secret key (say $\lambda$) with this public
key by choosing a restricted integer and form an encryption matrix with their
signature. Further, after receiving the encrypted message with the signature
from Alice, Bob retrieve the secret key($\lambda$) and after some calculation
recover the plain text (see section 2.1). The following algorithm summarizes
the methodology.

###  4.1 Algorithm

#### Encryption Algorithm(sender have access to $pk(p,E_{1},E_{2})$):

  1. 1.

Sender(Alice) first chooses a secret number $e$, such that $1<e<\phi(p)$.

  2. 2.

Signature: $s\leftarrow E_{1}^{e}\pmod{p}$.

  3. 3.

Secret key: $\lambda\leftarrow E_{2}^{e}\pmod{p}$.

  4. 4.

Initiate Lucas sequence$\\{l_{\lambda,s}\\}$ of order $\lambda$.

  5. 5.

Key Matrix: $K\leftarrow L_{\lambda}^{(s)}\pmod{p}$, where $L_{\lambda}^{(s)}$
may be obtained from step-4 and equ.(3.6).

  6. 6.

Shift vector:
$B\leftarrow[l_{\lambda,\lambda},l_{\lambda,\lambda+1},...,l_{\lambda,2\lambda-1}]$
of size $1\times\lambda$ (using step-4).

  7. 7.

Encryption: $C=Enc(P):~{}~{}~{}~{}c_{i}\leftarrow(p_{i}K+B)\pmod{p}$.

  8. 8.

transmit $(C,s)$ to Bob.

Decryption Algorithm: After receiving $(C,s)$ from sender.

  1. 1.

Secret key: $\lambda\leftarrow s^{D}\pmod{p}$, where $D$ is Bob’s chosen
secret key.

  2. 2.

Initiate Lucas sequence$\\{l_{\lambda,s}\\}$.

  3. 3.

Key Matrix: $K^{*}\leftarrow L_{\lambda}^{(-s)}H^{-1}\pmod{p}$, where both
$L_{\lambda}^{(-s)}$ & $H=(L_{\lambda}^{(0)})^{2}$ may be obtained from step-2
using eqn.(3.6) & eqn.(3.7 respectively.

  4. 4.

Shift vector:
$B\leftarrow[l_{\lambda,\lambda},l_{\lambda,\lambda+1},...,l_{\lambda,2\lambda-1}]$
of size $1\times\lambda$ (using step-2).

  5. 5.

Decryption: $P=Dec(C):~{}~{}p_{i}\leftarrow(c_{i}-B)K^{*}\pmod{p}$

  6. 6.

Plaintext(P) recovered.

###  4.2 Example

######  Example 2.

Let us assume that Alice(sender) wish to send message to Bob(receiver).
Consider the prime number $p=37$. Establish the public key and secret key of
communication for Bob.

###### Solution.

Bob first choose an integer D is such that $1<D<\phi(37)=36$, say $D=10$. The
set of primitive roots of 37 is given by X= {2, 5, 13, 15, 17, 18, 19, 20, 22,
24, 32, 35 }. Now Bob selects a primitive root $\alpha=17$ of $p$ from X. Now,
according to public key setup(2.1.1), Bob assigns
$E_{1}=17,E_{2}=E_{1}^{D}\pmod{p}\equiv 17^{10}\pmod{37}\equiv 28$. Thus
public key $pk(p,E_{1},E_{2})$ for Bob is $pk(37,17,28)$ and secret key is
$sk(10)$. Now using $pk(37,17,28)$ anyone can send message to Bob (explained
in next example). ∎

######  Example 3 (Encryption-Decryption).

Using $pk(37,17,28)$, construct the key matrix & shift vector and encrypt the
plaintext NOBLE2022.

###### Solution.

Here the numerical values equivalent to NOBLE2022 is
$[13,14,01,11,04,28,26,28,28]$. Let us consider the alphabets
$\Sigma=\mathbb{Z}_{37}$ defined as for letters from A-Z equivalent to 00-25,
digits 0-9 are that to 26-35 and 36 for blank/white space. Now, according the
algorithm(4.1),
Alice first choose an integer $e$ such that $1<e<\phi(37)$, say $e=23$.
Now, Alice make signature(s) as $s=E_{1}^{e}=17^{23}\pmod{37}\equiv 18$.
Thus secret key for Alice is $\lambda=E_{2}^{e}=28^{23}\pmod{37}\equiv 3$ and
the key matrix $K=L_{\lambda}^{(s)}\pmod{p}$ for encryption is given by

$\displaystyle K=L_{3}^{(18)}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}l_{3,20}&l_{3,19}+l_{3,18}&l_{3,19}\\\
l_{3,19}&l_{3,18}+l_{3,17}&l_{3,18}\\\ l_{3,18}&l_{3,17}+l_{3,16}&l_{3,17}\\\
\end{bmatrix}\pmod{37}$ (4.1) $\displaystyle=$
$\displaystyle\begin{bmatrix}196331&164778&106743\\\ 106743&89588&58035\\\
58035&48708&31553\\\ \end{bmatrix}\pmod{37}~{}=\begin{bmatrix}9&17&35\\\
35&11&19\\\ 19&16&29\\\ \end{bmatrix}$

which is obtained by substituting the values of corresponding terms of
generalized Lucas sequence for $\lambda=3$ as given in the table

index (n) | … | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | … | 15 | 16 | 17 | 18 | 19 | 20 | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Lucas Seq.($l_{3,n}$) | … | -1 | 3 | 1 | 3 | 7 | 11 | 21 | 39 | … | 9327 | 17155 | 31553 | 58035 | 106743 | 196331 | …

Table 1: Lucas sequence of order 3

And shift vector(B) is, $B=[l_{3,3},l_{3,4},l_{3,5}]=[07,11,21]$.
Now divide the plaintext NOBLE2022 in blocks of size $1\times\lambda$ as
follow:
$P_{1}=[~{}N~{}O~{}B]=[13~{}~{}14~{}~{}01],P_{2}=[L~{}E~{}2]=[11~{}04~{}28]~{}~{}\text{and}~{}~{}P_{3}=[0~{}2~{}2]=[26~{}28~{}28]$.
Encryption takes places as: $C_{i}\leftarrow(P_{i}K+B)\pmod{37}$.
$C_{1}=(P_{1}K+B)\equiv$ $\left(\begin{bmatrix}13&14&01\\\ \end{bmatrix}\\\
\begin{bmatrix}9&17&35\\\ 35&11&19\\\ 19&16&29\\\
\end{bmatrix}+\begin{bmatrix}07&11&21\\\ \end{bmatrix}\right)\pmod{37}$
$\equiv(04~{}~{}32~{}~{}31)\sim$ (E 7 6)
$C_{2}=(P_{2}K+B)\equiv$ $\left(\begin{bmatrix}11&04&28\\\ \end{bmatrix}\\\
\begin{bmatrix}9&17&35\\\ 35&11&19\\\ 19&16&29\\\
\end{bmatrix}+\begin{bmatrix}07&11&21\\\ \end{bmatrix}\right)\pmod{37}$
$\equiv(01~{}~{}24~{}~{}36)\sim$ (B Y $\square$)
$C_{3}=(P_{3}K+B)\equiv$ $\left(\begin{bmatrix}26&28&28\\\ \end{bmatrix}\\\
\begin{bmatrix}9&17&35\\\ 35&11&19\\\ 19&16&29\\\
\end{bmatrix}+\begin{bmatrix}07&11&21\\\ \end{bmatrix}\right)\pmod{37}$
$\equiv(14~{}~{}25~{}~{}18)\sim$ (O Z S)
Thus, Alice encrypted the plaintext NOBLE2022 to E76BY$\square$OZS, and send
it to Bob along with her signature i.e Alice sends
$\\{s=18,C=C_{1}C_{2}C_{3}\\}$ to Bob.
Decryption: On the other side ’Bob’ receives $(C,s)$ from Alice. To construct
the decryption key $K^{*}$ Bob first recovers $\lambda$. It can be obtained
using their secret key $D$ as follow:

$\displaystyle\lambda=s^{D}\pmod{37}=18^{10}\pmod{37}\equiv 3.$

Thus $K^{*}=L_{3}^{(-s)}H^{-1}\pmod{p}$, where
$H=(L_{3}^{(0)})^{2}=\begin{bmatrix}16&18&14\\\ 14&2&4\\\ 4&10&-2\\\
\end{bmatrix}$ is given as

$\displaystyle K^{*}$ $\displaystyle=$ $\displaystyle
L_{3}^{(-18)}H^{-1}\pmod{37}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}l_{3,-16}&l_{3,-17}+l_{3,-18}&l_{3,-17}\\\
l_{3,-17}&l_{3,-18}+l_{3,-19}&l_{3,-18}\\\
l_{3,-18}&l_{3,-19}+l_{3,-20}&l_{3,-19}\\\
\end{bmatrix}\dfrac{1}{44}\begin{bmatrix}-1&4&1\\\ 1&-2&3\\\ 3&-2&-5\\\
\end{bmatrix}\pmod{37}$ $\displaystyle=$
$\displaystyle\dfrac{1}{44}\begin{bmatrix}-253&318&271\\\ 271&-524&47\\\
47&-224&-571\\\ \end{bmatrix}\begin{bmatrix}-1&4&1\\\ 1&-2&3\\\ 3&-2&-5\\\
\end{bmatrix}\pmod{37}=\begin{bmatrix}18&36&7\\\ 7&11&29\\\ 29&15&19\\\
\end{bmatrix}$

Clearly, $K.K^{*}=I\pmod{37}$ (by the theorem (3.8)).
And shift vector is recovered by $\lambda$ as
$B=[l_{3,3},l_{3,4},l_{3,5}]=[07,11,21]$. Note that, entries of both
$L_{3}^{-18}$ and H may be obtained from the table(1)
Thus plaintext may be obtained by $P_{i}\leftarrow(C_{i}-B).K^{*}\pmod{37}$.
$P_{1}=(C_{1}-B)K^{*}\equiv$ $\left(\begin{bmatrix}04&32&31\\\
\end{bmatrix}-\begin{bmatrix}07&11&21\\\
\end{bmatrix}\right)\begin{bmatrix}18&36&7\\\ 7&11&29\\\ 29&15&19\\\
\end{bmatrix}\pmod{37}\\\ \indent\equiv(13~{}~{}14~{}~{}01)\sim$ (N O B)
$P_{2}=(C_{2}-B)K^{*}\equiv$ $\left(\begin{bmatrix}01&24&36\\\
\end{bmatrix}-\begin{bmatrix}07&11&21\\\
\end{bmatrix}\right)\begin{bmatrix}18&36&7\\\ 7&11&29\\\ 29&15&19\\\
\end{bmatrix}\pmod{37}\\\ \indent\equiv(11~{}~{}04~{}~{}28)\sim$ (L E 2)
$P_{3}=(C_{3}-B)K^{*}\equiv$ $\left(\begin{bmatrix}14&25&18\\\
\end{bmatrix}-\begin{bmatrix}07&11&21\\\
\end{bmatrix}\right)\begin{bmatrix}18&36&7\\\ 7&11&29\\\ 29&15&19\\\
\end{bmatrix}\pmod{37}\\\ \indent\equiv(26~{}~{}28~{}~{}28)\sim$ (0 2 2)
Thus, the plaintext NOBLE2022 successfully received by Bob. ∎

##  5 Strength-Analysis

In our proposed scheme, generalized Lucas matrix and Elgamal technique have
been considered as a key element of system and decryption matrix is set up as
$L_{\lambda}^{(-s)}H^{-1}$ constructed with combinations of terms of
$GLS(\lambda,s)$. The construction of key matrices is quite easy for
authorized parties as $\lambda$ are known for both of them but it is very
difficult for an adversary to obtain $\lambda$, as the adversary needs to
solve a discrete logarithm problem[5]. Further matrix construction is based on
only two elements $(\lambda,s)$, so it reduces time complexity as well as
space complexity of key formation and calculation of its inverse. In the
context of attacks based on public data, one of the popular attacks is brute
force attack [18, 20, 15] which have been discussed here. In case of brute
force attack, the adversary needs to calculate $\lambda$ which is almost
impossible (discrete logarithm problem), and the next challenge for the
adversary is to identify the correct key matrix out of $|GL(\lambda)|$
matrices, where $GL(\lambda)$ represents general linear group[4] of order
$\lambda$. Since $|GL(\lambda)|$ given by

$\displaystyle|GL_{\lambda}(F_{p})|=(p^{\lambda}-p^{\lambda-1})(p^{\lambda}-p^{\lambda-2})\cdots(p^{\lambda}-p^{1})(p^{\lambda}-1)$
(5.1)

From equation(5.1), it is clear that security for generalized Lucas matrices
$L_{\lambda}^{(s)}$ depends on $\lambda$ only, not on signature $s$. So $s$
does not compromise the security even though it is known to the adversary. For
example, consider $p=37$ and $\lambda=50$, then by equation(5.1) total number
of possible key space over $\mathbb{F}_{37}$ is approximately $3.105\times
10^{3920}$ which is too large. And in case of $\lambda$ and/or prime p
increasing, then the key space grows exponentially.

##  6 Conclusion

We know that key component in a cryptosystem plays an important role to keep
cryptography secure and strong. Many researchers are working in the direction
of developments of new cryptography and modification in existing cryptography
based on properties of number theory and linear algebra.

Here, in this article, we have first proposed a generalized Lucas sequence
which is constructed with the trace of $GFM(\lambda,s)$ and its initial values
are also obtained. Further, we have developed a recursive matrix whose entries
are constructed from linear combinations of generalized Lucas sequences and
investigated some properties like direct calculation of its inverse, product
of two matrices, …etc irrespective of matrix algebra. The purpose behind
considering the field for a key element is the necessity of the existence of
inverse. In our case, we have considered generalized Lucas matrices which does
not form a multiplicative group but it assure the existence of inverse matrix
for each $GLM(\lambda,s)$ for every $\lambda(\geq 2)\in\mathbb{N}$, i.e. we
have shown that for every integer $s$, we have matrix
$K^{*}=L_{\lambda}^{(-s)}H^{-1}$ such that
$L_{\lambda}^{s}K^{*}=K^{*}L_{\lambda}^{s}=I_{\lambda}$.

Also, we have proposed a modified public key cryptography using Affine-Hill
cipher & Elgamal signature schemes with generalized Lucas matrices and show
the implementation of Lucas matrices as a key matrix.

Thus generalized Lucas matrices as key components in cryptosystem play a
crucial role. It enlarges the keyspace, reduces the time complexity as well as
the space complexity of key formation in cryptography. Our proposed method is
based on construction with two parameters and has three digital signatures
($\lambda,s~{}\&~{}\text{shift vector(B)}$) which strengthen the security of
the modified cryptography. Since $\lambda$ is known only to both end
parties(Alice and Bob) so shift vector B constructed with $\lambda$ is also
known only to Alice and Bob, thus it is practically impossible to recover
$\lambda$ by anyone else as it is based on discrete logarithm problem. Hence,
our proposed method is mathematically simple for authorized party and tedious
for an intruder, mathematically strong and have a large keyspace.

### Acknowledgment

The first and third authors acknowledge the University Grant Commission, India
for providing fellowship for this research work.

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