Ratio Mathematica Volume 46, 2023

Algebraic coding theory using Pell equation
x2 − 8y2 = 1

Janaki G*
Gowri Shankari A†

Abstract

An interdisciplinary field with significant practical use is cryptogra-
phy. The difficulty of specific mathematical computing tasks affects
a public key cryptosystem’s security. The technique for coding and
decoding the messages was described in this work, utilising the so-
lutions of Pell equation x2 − 8y2 = 1 and matrix Q8∗ . It was noted
that messages can be turned into an even size that is then divided
into slabs. Considering the information security has become a more
serious issue in recent years, coding and decoding algorithms are es-
sential in order to improve information security. In this article, a new
matrix Q8* and a decryption system based on the solutions of the Pell
Equation x2 −8y2 = 1 are devised. The messages can be divided into
even-sized slabs during encryption. This algorithm will not only im-
prove information security but also has a high degree of accuracy.
Keywords: Pell equation; Encryption-decryption algorithm; Q8

∗
ma-

trix; Cryptography
2020 AMS subject classifications: 11B37, 11C20, 11D09, 11T71.
1

*Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University,
Tiruchirappalli-18, Tamil Nadu, India. janakikarun@rediffmail.com.

†Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University,
Tiruchirappalli-18, Tamil Nadu, India. gowrirajinikanth@gmail.com.
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1062. ISSN: 1592-7415. eISSN: 2282-8214. ©Janaki G et al.
This paper is published under the CC-BY licence agreement.

101



Janaki and Gowri Shankari A

1 Introduction

Currently, almost all current coding methods can be supported by number the-
ory Carmichael [1950], Disckson [1952], Trappe and Washington [2006]. Samuel
F.B. Morse, an American inventor, developed the first coding technique that used
just two symbols—a dot and a comma—to transmit the first cypher message in
1844 through an electric telegraph. He called it Morse code. Later, binary code
encoding improved as a more secure method for encrypting messages, and it is
still in use today Saranya and Janaki [2019], Janaki and Saranya [2020]. Binary
code encoding divides each coded word into blocks of ones and zeros. The twen-
tieth century saw important advancements in coding. It is common knowledge
that the Fibonacci sequences are defined as Fn + 1 = Fn + Fn − 1 with the
initial parameters F0=0and F1=1. The definition of the Fibonacci Q-matrix is
Q = [(11@10)], and its nth power is of the form Carmichael [1950], Disckson
[1952], Trappe and Washington [2006]. Several approaches were used to study
the Fibonacci coding principle. For instance, utilising the key variability notion
in symmetric key algorithm and the Fibonacci Q-matrix, a novel method for safe
information transmission through communication channels was developed. By
using blocking matrices and Fibonacci numbers, a new cryptography algorithm
was recently described in Berges [1981], Gould [1981]. Using the help of the Pell
equation’s solutions x28y2 = 1 that were found in Tas et al. [2018], Sumeyra et al.
[2019] and the new matrix Q8*, we present new coding and decoding algorithms
in this work. Also, this method’s fundamental principle hinges on subdividing the
message into a 2x2 slab matrix. The security of information will be improved by
using this technology, which also has a high degree of accuracy in data transmis-
sion via communication channels Saranya and Janaki [2019], Janaki and Saranya
[2020].
The connection of recurrence for the Pell equation x2 − 8y2 = 1 is

xn+1 = 3xn + 8yn

yn+1 = xn + 3yn

For n ≥ 1, where x1 = 3, y1 = 1.

It is noted that Q8
∗
=

(
xj 8yj
yj xj

)
The primary concept is to turn the messages into order 2 × 2 block matrices. The
number of blocks and letter positions must be determined based on this. Then
encrypted matrix P is obtained and Q8

∗
utilized for decryption.

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Algebraic coding theory using Pell equation x2 − 8y2 = 1

2 Main results

2.1 Representations:
1. G− a message-to-be-sent matrix constituted by an even order.

2. Gj − jth block of G whose size is 2.

3. g− Number of slabs Gj of G.

4. m =

{
3 if g ≥ 3
g if g > 3

5. Dj = |Gj|.

6. Gj =
(
gj1 gj2
gj3 gj4

)
7. P− encrypted matrix defined by P =

(
dj gjk

)
k∈{1,2,4}

8.
(
Q8

∗)m
=

(
q11 q12
q21 q22

)
9. ∗− denotes the space between the words.

2.2 Assignment of alphabets:

A B C D E F G H I
m m+1 m+2 m+3 m+4 m+5 m+6 m+7 m+8
J K L M N O P Q R

m+9 m+10 m+11 m+12 m+13 m+14 m+15 m+16 m+17
S T U V W X Y Z *

m+18 m+19 m+20 m+21 m+22 m+23 m+24 m+25 m-1

2.3 Algorithm for encryption:
1. Create an even order matrix G for the provided message.

2. Separate Gj into slabs of size 2 and calculate g.

3. Selecting m using g.

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Janaki and Gowri Shankari A

4. Replace the alphabets with the provided numbers to get the elements of Gj.

5. Find Dj.

6. Create P .

2.4 Algorithm for decryption:
Using P , one must find G. The goal is to locate gj3 as P contains gj1, gj2, gj4.

1. Identify
(
Q8

∗)m
2. Identify its constituents as qij’s.

3. Explore pj1 = q11gj1 + q21gj2

4. Explore pj2 = q12gj1 + q22gj2

5. Solve Dj = pj1(q12sj + q22gj4) − pj2(q11sj + q21gj4) for sj.

6. Replace sj = gj3

7. Establish Gj

8. Establish G

2.5 Encryption decryption algorithm using (x, y) such that
x2 − 8y2 = 1

The following are a few Pell equation x2 − 8y2 = 1 solutions:

j 1 2 3 4 5 6 7 8
xj 3 17 99 577 3363 19601 114243 665857
yj 1 6 35 204 1189 6930 40391 235416

Case (i): g=1

Example: 1 Take “ARMY” as the encrypted message.

Encryption:

G =

(
A R
M Y

)

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Algebraic coding theory using Pell equation x2 − 8y2 = 1

Here there is only one block so g = 1.

This implies m = 3.

Thus G1 =
(
3 20
15 27

)
and so g11 = 3, g12 = 20, g13 = 15, g14 = 27.

∴ D1 = −219

P =
(
−219 3 20 27

)
Decryption: (

Q8
∗)3

=

(
x3 8y3
y3 x3

)
=

(
99 280
35 99

)
⇒ q11 = 99, q12 = 280, q21 = 35, q22 = 99.

Apply pj1 and pj2, one can get p11 = 997 and p12 = 2820.

D1 = p11(q12s1 + q22g14) − p12(q11s1 + q21g14)
−219 = 997(280s1 + 2673) − 2820(99s1 + 945)
⇒ s1 = 15

Thus g13 = s1 = 15.

∴ G1 =

(
3 20
15 27

)
from P

Hence G =
(
A R
M Y

)
.

Case (ii): g=4

Example: 2 Take “AMOUNT RECEIVED” as the encrypted message.

Encryption:

G =



A M O U
N T ∗ R
E C E I
V E D ∗




Here there are four blocks so g = 4.

∴ G1 =

(
A M
N T

)
, G2 =

(
O U
∗ R

)
, G3 =

(
E C
V E

)
, G4 =

(
E I
D ∗

)

105



Janaki and Gowri Shankari A

Choose m = 4

Thus G1 =
(
4 16
17 23

)
, G2 =

(
18 24
3 21

)
, G3 =

(
8 6
25 8

)
, G4 =

(
8 12
7 3

)
Hence D1 = −180, D2 = 306, D3 = −86, D4 = −60

P =



−180 4 216 23
306 18 24 21
−86 8 6 8
−60 8 12 3




Decryption: (
Q8

∗)4
=

(
x4 8y4
y4 x4

)
=

(
577 1632
204 577

)
⇒ q11 = 577, q12 = 1632, q21 = 204, q22 = 577.

∴ One can find,

p11 p12 p21 p22 p31 p32 p41 p42
5572 15760 15282 43224 5840 16518 7064 19980

On solving the equation

Dj = pj1(q12sj + q22gj4) − pj2(q11sj + q21gj4),

one can get s1 = 17, s2 = 3, s3 = 25, s4 = 7.

Thus g13 = s1 = 17, g23 = s2 = 3, g33 = s3 = 25, g43 = s4 = 7.

Thus,

G1 =

(
4 16
17 23

)
, G2 =

(
18 24
3 21

)
, G3 =

(
8 6
25 8

)
, G4 =

(
8 12
7 3

)
from P.

Hence

G =



A M O U
N T ∗ R
E C E I
V E D ∗




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Algebraic coding theory using Pell equation x2 − 8y2 = 1

3 Conclusions
The paper identifies a few particular cryptographic applications that can be

utilised to demonstrate and comprehend the fundamental idea of the Fibonacci Q
matrix. Fibonacci, often spelled Bonacci, Leonard of Pisa, or Leonardo Bigollo
Pisano, was an Italian mathematician who was born in the Republic of Pisa. He is
regarded as ”the most talented Western mathematician of the Middle Ages” and
had a significant impact on number theory. Moreover, Pell’s equation refers to
any Diophantine equation of the type x2 − 8y2 = 1 , where n is a given positive
non-square integer and integer solutions are sought for x and y.

In this work, using the solutions of the Pell equationx2 − 8y2 = 1 and QD∗ is
defined. This contributes to the decryption algorithm. In encryption, the message
must first be transformed into an even single matrix, then into slabs of size 2.
The fact that the entries QD

∗
get bigger and bigger depends on the strict secrecy.

One may search the decryption algorithm for the solutions of other well-known
equations with the new matrix QD

∗
. On the basis of this scheme various more

algorithms can be developed.

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