Determinants and Permanents of Hessenberg Matrices with Perrin’s
Bivariate Complex Polynomials and Its Application
JIRAWAT KANTALO
Department of Mathematics and Statistics,
Sakon Nakhon Rajabhat University,
Sakon Nakhon 47000,
THAILAND
Abstract: - In this paper, we define some
nn
Hessenberg matrices and then we obtain determinants and
permanents of their matrices that give the odd and even terms of bivariate complex Perrin polynomials.
Moreover, we use our results to apply the application cryptology area. We discuss the Affine-Hill method over
complex numbers by improving our matrix as the key matrix and present an experimental example to show that
our method can work for cryptography.
Key-Words: Perrin Complex Bivariate Polynomials, Determinant, Permanent, Hessenberg Matrix,
Cryptography
Received: September 19, 2022. Revised: April 9, 2023. Accepted: May 2, 2023. Published: May 17, 2023.
1 Introduction
Perrin’s complex bivariate polynomials
,
nxyP
have been introduced by [1], and are defined by the
recurrence relation, for
3n
,
22
23
, , , ,
n n n
x y ix x y y x y

P P P
(1)
where initial conditions
0, 3,xy P
1, 0,xy P
and
21i
. The first terms of the
above sequences are presented in Table 1.
In recent years, the determinants and permanents
of one type of Hessenberg matrices representation
of many sequences. For example, [2], introduced
determinants and permanents of Hessenberg
matrices as the generalized Fibonacci and Pell
sequences. In 2014, [3], presented some
determinantal and permanental representations of
associated polynomials of Perrin and Cordonnier
numbers. In 2020, [4], defined tridiagonal matrices
whose permanent is equal to the 𝑘-Jacobsthal
sequence. See more examples in [5], [6], [7], [8].
In addition, the applications of number theory
have been widely studied. One of the most
interesting applications is cryptography. Several
authors used the methods for encryption using their
obtained results as a key such as in 2017, [9],
presented a coding and decoding method using the
generalized Pell numbers. In 2019, [10], proposed a
new coding and decoding algorithm using Padovan
𝑄-matrices and Maxrizal, [11], showing the Hill
Cipher method can be generalized to key matrices
over complex numbers.
Table 1. The first terms of Perrin’s complex
bivariate polynomials.
n
,
nxyP
1
0
2
2
3
2
3y
4
2
2ix
5
2 2 2
23y ix y
6
44
32yx
7
4 2 2 2
34x y ix y
8
2 4 6 4
6 2 2ix y ix y
9
6 2 4 2 6
3 6 3ix y x y y
10
4 4 2 4 8
9 6 2x y ix y x
In 2021, [12], defined some third-order Bronze
Fibonacci sequences and developed the obtained
results in encryption theory. Moreover, the anti-
orthogonal and 𝐻-anti-orthogonal of type 𝐼 matrices
were firstly defined by [13], and they applied these
matrices in cryptology.
In this paper, we consider the bivariate Perrin’s
complex polynomials and then define new
nn
Hessenberg matrices which have determinants and
permanents related to these polynomials. In
addition, we consider an application in cryptology
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based on the Affine-Hill chipher which was
introduced by [14]. We improve and modify the
public key over complex numbers by using our
obtained matrix which is a non-singular matrix.
Finally, a numerical example of an encryption and
decryption algorithm is given.
2 Preliminaries
In this section, the following definitions and lemmas
for determinants and permanents of the Hessenberg
matrix are given.
Definition 2.1 [15], An
nn
matrix
,n r s
Aa


is
called a lower Hessenberg matrix if
,0
rs
a
when
1sr
, i.e.,
1,1 1,2
2,1 2, 2 2,3
3,1 3,2 3,3
1,1 1,2 1,3 1,
,1 ,2 ,3 ,
00
0
0
n
n n n n n
n n n n n
aa
a a a
a a a
A
a a a a
a a a a









.
(2)
Lemma 2.2 [16], Let
n
A
be a lower Hessenberg
matrix. The following determinant formula for
n
A
is
given by
,,1
1
1
, 1 1
1
d,tet de et 1d
n
nnt
n t j j t
tj
n n n n
t
a a AA a A








for
2n
, where
0
det 1A
, and
1 1,1
det Aa
.
Definition 2.3 Let
n
A
be
nn
a matrix, the
permanent of
n
A
is defined by
,
1
per ,
n
n
nii
i
Aa


S
(3)
where
n
S
denotes the set of permutations of
1,2, ,n
.
Lemma 2.4 [17], Let
n
A
be a lower Hessenberg
matrix. The following permanent formula for
n
A
is
given by
1
1
, 1 , , 1 1
1
per per per ,
n
n
n n n n n t j j t
tjt
A a A a a A








for
2n
, where
0
per 1A
, and
1 1,1
per Aa
.
3 Main Results
In this section, we will define new
nn
lower
Hessenberg matrices and present the determinants
and permanents of their matrices which are bivariate
Perrin’s complex polynomials, respectively.
Theorem 3.1 Let
,n r s
Bb


be a
nn
lower
Hessenberg matrix, is defined by
2
2
2
2
2
2
2
2
2
,
3 if 1
if for , 2
2 if 1
( ) if 2, 1
2
1if 2 for 4, 2
42
0 otherwise
.
r
r
rs
rs
y r s
ix r s r s
y s r
x
i r s
y
xi r s r s
y
b










(4)
Then
21
det , ,
nn
B x y
P
for
1.n
(5)
Proof. We proved this by mathematical induction on
n
. By hypothesis, the result holds for all
4n
.
Then, we suppose that the result is true for all
positive integer
k
such that
5k
. We will prove it
for
1.k
Firstly, we use elementary row operations on the
matrix
1k
B
. We multiply the
th
( 1)k
row by
4
4
4
x
y
then add to
th
( 1)k
row. So, we get the
th
( 1)k
row as
th
2 6 4 4
2
242
( 3) 8 4 2
0 0 0
k
ix ix y x ix
yyy




.
That is
22
2
2
2
4
4
62
1
62
2
2 6 4 4
2
242
3 2 0 0
1 0 0
0
2
1
4
4
88
0
102
4
0 0 0 8 4 2
.
k
yy
ix
ix
y
x
y
Bix ix
yy
y
ix ix y x ix
yyy





















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Now, using Lemma 2.2, we have
1
2
1 1, , 1 1
1
det det 1 det
k
kkt
k k k t j j t
tjt
B ix B b b B








31
2
1, , 1 1
1
det 1 det
k
kkt
k k t j j t
tjt
ix B b b B








1
1, , 1 1
2
1 det .
k
kkt
k t j j t
tk jt
b b B










Since
1, 0
kt
b
for
13tk
, then
2
1 2 1
det ,
kk
B i x x y

P
1
1, , 1 1
2
1 det
k
kkt
k t j j t
tk jt
b b B










2 2 4
2 1 2 5
, ,
kk
ix x y ix y x y

PP
6 4 4
2 3 2 1
, ,
kk
ix y x y x x y

PP
2 2 2 2
2 1 2 2 2 4
, , ,
k k k
ix x y ix y x y ix x y
P P P
6 4 4
2 3 2 3 2 1
, , ,
k k k
ix x y y x y x x y
P P P
2 4 2 2
2 1 2 3 2 4
, , ,
k k k
ix x y x ix x y y x y
P P P
2 2 2 4
2 2 2 3 2 1
, , ,
k k k
y ix x y y x y x x y
P P P
22
2 1 2
, ,
kk
ix x y y x y
PP
23 ,
kxy
P
2 1 1 ,.
kxy

P
Then,
21
det ,
nn
B x y
P
for all
1n
.
Example 3.2 Let
6
B
is defined by
4
. So, the
determinant of
6
B
which is as follows:
22
22
2
22
2
4
22
64
62
22
62
8 4 2
2
8 4 2
3 2 0 0 0 0
1 2 0 0 0
0 2 0 0
2
1
det 0 2 0
4
4
102
4
88
10
4
16 16 8
yy
ix y
ix ix y
y
x
Bix y
y
ix ix ix y
yy
x x ix ix
y y y

10 2 8 2 4 6 2 6
3 10 18 8ix y x y x y ix y
13 ,xyP
2 6 1 ,.xy
P
Theorem 3.3 Let
,n r s
Dd


be a
nn
lower
Hessenberg matrix, is defined by
2
1,1 ,dy
2
2 2 2
1,2 2,1 2,2 3,1
3
, 2 , 3 , ,
2
y
d ix d ix d y d
3,2 1d
and
2
2
3
2
1
2
2
2
,2
4
22
2
2
if for , 3
2 if 1 for 3
( ) if 3, 2
2
1.
if 2
42
for 5, 3
() 2 3 if 4, 1
42
0 otherwise
r
r
rs
rs
r
r
ix r s r s
y s r s
x
i r s
y
ix
drs
y
rs
ix ix r s
y













(6)
Then
22
det , ,
nn
D x y
P
for
2.n
(7)
Proof. We proved this by mathematical induction on
n
. By hypothesis, the result holds for all
25n
.
Then, we suppose that the result is true for all
positive integer
k
such that
6k
. We will prove it
for
1.k
We use elementary row operations on the matrix
1k
D
. We multiply the
th
( 1)k
row by
4
4
4
x
y
then
add to
th
( 1)k
th row. So, we get the
th
( 1)k
row
as
th
2 4 6 4
2
242
( 3) 8 4 2
0 0 0
k
ix y ix x ix
yyy




.
That is
22
2 2 2
2
2
22
2
4 2 4
1
24
2
2 4 6 4
2
242
00
2 3 2 0 0
31
2
23 0
42
3 2 1
4
84
0
102
4
0 0 0 8 4 2
k
y ix
ix y y
yix
ix ix
y
Dx x x
yy
y
ix y ix x ix
yyy





















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Now, using Lemma 2.2, we have
21
1, , 11 1
1
d 1det d eet t
k
kkt
k t j j t
t
kjt
k
D i D dx dD








31
2
1, , 1 1
1
det 1 det
k
kkt
k k t j j t
tjt
ix D d d D








1
1, , 1 1
2
1 det .
k
kkt
k t j j t
tk jt
d d D










Since
1, 0
kt
d
for
13tk
, then
2
1 2 2
det ,
kk
D i x x y

P
1
1, , 1 1
2
1 det
k
kkt
k t j j t
tk jt
d d D










2 2 4
2 2 2 4
, ,
kk
ix x y ix y x y

PP
6 4 4
2 2 2
, ,
kk
ix y x y x x y
PP
2 2 2 2
2 2 2 1 2 3
, , ,
k k k
ix x y ix y x y ix x y
P P P
6 4 4
2 2 2 2 2
, , ,
k k k
ix x y y x y x x y

P P P
2 4 2 2
2 2 2 2 2 3
, , ,
k k k
ix x y x ix x y y x y
P P P
2 2 2 4
2 1 2 2 2
, , ,
k k k
y ix x y y x y x x y

P P P
22
2 2 2 1
, ,
kk
ix x y y x y

PP
24 ,
kxy
P
2 1 2 ,.
kxy

P
Therefore,
22
det ,
nn
D x y
P
for all
2n
.
Example 3.4 Let
6
D
is defined by
6
. So, the
determinant of
6
D
which is as follows:
22
2 2 2
2
22
22
22
62
4 2 4
22
24
4 6 6 2
2
4 6 2
0 0 0 0
2 3 2 0 0 0
31 2 0 0
2
23 0 2 0
det 42
3 2 1 02
8 4 4
2 3 1 0
16 8 8 4
y ix
ix y y
yix y
ix ix ix y
Dy
x ix x ix y
yy
x ix ix ix ix
y y y



12 4 8 4 6 8 2 8
2 15 20 12 2x y x iy x iy x y
14 ,xyP
2 6 2 ,.xy
P
Now, we will present the permanents of matrices
that are bivariate complex Perrin polynomials. In
[18], this study gave the relationship between the
determinant and the permanent of a Hessenberg
matrix by using Lemmas 2.2 and 2.4.
Then, let
n
A
be
nn
lower Hessenberg matrix
,rs
Aa


is given in
2
and also
n
E
be
nn
a
lower Hessenberg matrix which is defined by
, 1 , 1r r r r
ea


for all
r
,
,,r s r s
ea
for
rs
and
0
otherwise. So, we have
det nn
EA
or
det per
nn
AE
. Then, we have the following
Corollary without proof.
Let
n
H
be
nn
matrix, is defined by
1 1 1 1
1 1 1 1
.
1 1 1 1
1
1 1 1 1
n
H






(8)
Corollary 3.5 Let
n
V
and
n
W
be
nn
matrices
and define
n n n
V H B
and
n n n
W H C
where
denotes the operator of Hadamard product of
matrix. Then,
21
per , ,
nn
V x y
P
(9)
22
per , .
nn
W x y
P
(10)
4 Applications in Cryptography
In this section, we present new encoding and
decoding algorithms over complex numbers based
on the Affine-Hill cipher method for encryption.
We give some obtained results as a key matrix.
Let
1 2 3
, , , , n
p p p p
be the plain text with
numerical characters. We consider the plain text
with complex number form, i.e.,
1 2 3 4 1
, , , .
nn
p p i p p i p p i
(11)
Define
j
P
as the
th
j
plain text in
22
matrix
form, for
1jl
where
8
n
l

, is the smallest
integer which is greater than or equal to the length
of plain text divided by 8. If the plain text matrix
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j
P
is not suitable, a zero will be added to complete
the matrix
j
P
.
Let us consider 37-characters with the numerical
values in Table 2.
Table 2. The 37-characters with
the numerical values
A
B
C
D
E
F
G
H
I
J
1
2
3
4
5
6
7
8
9
10
K
L
M
N
O
P
Q
R
S
T
11
12
13
14
15
16
17
18
19
20
U
V
W
X
Y
Z
0
1
2
3
21
22
23
24
25
26
27
28
29
30
4
5
6
7
8
9
blank
31
32
33
34
35
36
37
Example 4.1 Suppose the plain text with the
characters “CHOOSE HAPPY” . In Table 2, we
have the corresponding numerical characters as 3,
8, 15, 15, 19, 5, 37, 8, 1, 16, 16, and 25. Then, the
length of plain text is 12. So, we have
12
82.l


Finally, plain text with complex number forms
become
3 8 ,15 15 ,19 5 ,37 8 ,1 16 ,16 25 ,i i i i i i
and then
1
3 8 15 15 ,
19 5 37 8
ii
Pii





and
2
1 16 16 25 .
0 0 0 0
ii
Pii





4.1 Encryption and Decryption Algorithms
We will explain the following new coding and
decoding algorithms.
Firstly, we let
be a prime number and choose
a private key
G
such that
1G


where

is the Euler’s phi function. Then, we select
1
that is the primitive root of
and calculate
2
that
21
mod
G
. Finally, we have a public
key, denoted
12
,,
and
G
as the private key.
Encryption Algorithm
Step 1: The sender chooses a secret number
ς
such that
1.

ς
Step 2: The sender calculates the signature
such that
1 mod
ς
.
Step 3: The sender calculates the secret key
such that
2 mod
ς
.
Step 4: The sender constructs
K
as the key
matrix of size
22
which is obtained in our results
for
x
and
y
.
Step 5: The sender constructs
S
as the shifting
matrix of size
22
.
Step 6: The sender calculates
mod ,
jj
C P K S

where
j
P
and
j
C
are
th
j
of
22
matrix of
plain text and cipher text, respectively, for
1jl
.
Finally, the sender will send
,C
to the
recipient for decoding the cipher text.
Decryption Algorithm
After receiving
,C
, the recipient decrypts
the cipher text with the following steps.
Step 1: The recipient calculates the secret key
such that
mod
G
.
Step 2: The recipient receives
K
as the key
matrix with
,x
y
and calculates
1
K
.
Step 3: The recipient receives
S
as the shifting
matrix.
Step 4: The recipient calculates
1 mod .
jj
P C S K

Note that: The prime number
shall be at least
the number of different characters used in plain text
and
gcd det , 1.K
4.2 Numerical Example
We suppose the key matrix
K
is defined by
2
B
that
given in
4
and the shifting matrix
S
is defined
by
2
D
that given in
6
, respectively. So, we have
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2 2 2 2
2 2 2
32
and .
1 2 3
y y y ix
KS
ix ix y


Example 4.2 Assume that
37,
the key matrix
,
n
KB
private key
G
is
11
and primitive
the root of
,
15.
Then we calculate
2
such
that
11
25
2 37 .mod
So, the public key is
12
, , 37,5,2 .
We consider the plain text to be “STAY AT
HOME” in encryption and decryption algorithms.
Therefore, we obtain the plain text with numerical
characters 19, 20, 1, 25, 0, 1, 20, 0, 8, 15, 13, 5 and
12
82l


.
Then, the plain text matrix
j
P
for
12j
become
1
19 20 1 25 mod37 ,
0 20 0
ii
Pii





and
2
8 15 13 5 mod37 .
0 0 0 0
ii
Pii





Encryption Algorithm:
Step 1: Choosing a secret number
32ς
.
Step 2: Calculating the signature :
32
5 9 mod37 .

Step 3: Calculating the secret key :
32
2 7 mod37 .

Step 4: We have
K
as the key matrix for
9,x
7,y
is defined by
147 98 36 24 mod37 .
1 81 36 7
Kii

Step 5: We have
S
as shifting matrix for
9x
,
7y
, is defined by
49 81 12 7 mod37 .
162 147 23 36
ii
Sii

Step 6: So, we have
j
C
cipher text for
1,2j
, as
follows:
1
19 20 1 25 36 24 12 7
0 20 0 36 7 23 36
i i i
Ci i i i



29 29 22 13 mod37 .
17 22 36 16
ii
ii





2
8 15 13 5 36 24 12 7
0 0 0 0 36 7 23 36
i i i
Ci i i i



28 17 9 14 mod37 .
0 23 36 0
ii
ii





So, we have cipher text with numerical numbers
as 29, 29, 22, 13, 17, 22, 36, 16, 28, 17, 9, 14, 0, 23,
36, 0 and sent the cipher text “22VMQV9P1QIN
W9 ” and signature
9
to the recipient.
Decryption Algorithm :
Step 1: Firstly, calculating the secret key from
11
9 mod37 .
So, we have
7.
Step 2: Calculating
1
K
. By Theorem 3.1, we
obtain
11
5
det 9,7Kp

1
(24 30 ) mod37i

124 30 mod37
1476 i
9 24 7 mod37i
31 26 mod37 .i
Then, we have
17 13
31 26 mod37
1 36
i
Ki

 

3 32 33 5 mod37 .
31 26 6 11
ii
ii





Step 3: Calculating the shifting matrix for
7,
then
12 7 mod37 .
23 36
i
Si



Step 4: Finally, we decrypt the cipher text as
follows.
1
29 29 22 13 12 7
17 22 36 16 23 36
i i i
Pi i i






3 32 33 5
31 26 6 11
ii
ii





19 20 1 25 mod37 .
0 20 0
ii
ii





2
28 17 9 14 12 7
0 23 36 0 23 36
i i i
Pi i i






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3 32 33 5
31 26 6 11
ii
ii





8 15 13 5 mod37 .
0 0 0 0
ii
ii





First, we receive the plain text with numerical
characters after decrypting the cipher text, and then
we decrypt it again to obtain “STAY AT HOME”.
5 Discussion and Conclusion
In this paper, we have obtained the
nn
Hessenberg matrices whose determinants and
permanents are the odd and even terms of bivariate
Perrin’s complex polynomial. Moreover, we
demonstrate the significance of these in the field of
mathematics and cryptography and provide
experimental evidence of their usefulness in
cryptography applications. We have developed a
method over complex numbers based on the Affine-
Hill cipher method that requires an invertible key
matrix. We have shown that our matrices can be
used as the key matrix for encryption and decryption
algorithms. In future work, these matrices may be
applied in steganography. Finally, we hope that this
will inspire further research in this area and provide
a new algorithm for more secure encryption in the
future.
Acknowledgment:
The author would like to sincerely appreciate the
referees for their valuable comments and
suggestions on the manuscript. This work is
supported by Research and Development Institute,
Sakon Nakhon Rajabhat University under Grant
No.1/2022.
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Jirawat Kantalo
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Volume 22, 2023
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The research was financially supported by the
Research and Development Institute, Sakon Nakhon
Rajabhat University under Grant No.1/2022.
Conflict of Interest
The author has no conflict of interest to declare.
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DOI: 10.37394/23206.2023.22.40
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Volume 22, 2023
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