Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 16–22.

CRYPTOGRAPHIC ALGORITHM BASED ON

GENERALIZED PELL EQUATION INVOLVING SPHENIC

NUMBERS

J. Kannan^1∗, K. Kaleeswari², and A. Deepshika³

Abstract. One of the key components for the secure transmission of messages is cryp-

tography. Strong algorithms are now required in order to convey a message to another

person. For encryption and decryption, modern academics use several mathematical ideas,

most notably number theory. In this study, we developed a method that ﬁxes d as a prime

integer and n as the square of the sphenic numbers, utilizing the well-known generalized

Pell equation as x²− dy²= n.

Keywords: encryption, decryption, the matrix Q^p∗, generalized Pell equation, Sphenic

numbers.

2010 AMS Subject Classiﬁcation: 11A05, 11A07, 11B37, 11D09, 11D99.

1. Introduction

Number theory - a basket of properties of numbers, especially integers, is one of the

most widely used and most attractive theories in Mathematics. It is more advantageous

subject to master because it encompasses a variety of elements. The creation of Diophan-

tine equations is one example. These equations are polynomial equations with integer

coeﬃcients and two or more unknowns such that the only solutions of interest are the

integer ones.

Diophantine analysis, the mathematical study of Diophantine problems, comprises of

a special type of equation, commonly known as Pell equation [1]. It is one among the

various Diophantine equations and it takes the form

x²− dy²= 1

(1)

where d is a ﬁxed positive integer, not a perfect square. It also has been proved that

equation (1) has inﬁnitely many solutions in positive integers. This Pell equation is

generalized as

x²− dy²= n

(2)

for some integer n.

Let p be a prime and (x_k, y_k) be positive integer solutions of the equation x²− py²= 1.

ꢀ

ꢁ

x₁py₁

Using these solutions we can deﬁne the matrix Q^p∗as Q^p∗=

[3]. The speciality

y₁x₁

of this matrix is its k^thpower can be obtained directly from the k^thsolution (x_k, y_k) of

ꢀ

ꢁ

x_kpy_k

equation x²− py²= 1. That is, (Q^p∗)^k=

.

y_k

x_k

16

CRYPTOGRAPHIC ALGORITHM

17

ꢀ

ꢁ

x₁py₁

y₁x₁

In this paper, we construct a similar matrix Q^p_a^∗

=

for the equation x²₁−

ꢀ

ꢁ

x_rpy_r

y_rx_r

py₁²= a², where a ∈ Z. The r^thpower of this matrix is (Q^p_a^∗)^r= a^r−1

where

(x_r, y_r) is a positive integer solution of x²− py²= a².

In [3], an encryption and decryption algorithms are developed using the Pell equation

x²− py²= 1 and its corresponding matrix Q^p∗. Likewise, in this work, we deal with

the same algorithm with little modiﬁcations but we employ the generalized Pell equation

x²− py²= a²and its corresponding matrix Q^p_a^∗. In particular, here we restrict a as a

Sphenic number, a number which is a product of three distinct primes. The algorithm is

made more secure by the sphenic number that is employed here. In this work, we made

several modiﬁcations to the algorithm. The table below shows the diﬀerences between

the suggested method and the current approach.

Proposed method

Existing method

Cryptographic Algorithm Based on

Generalized Pell Equation involving

Sphenic numbers

Cryptographic algorithm involv-

ing the matrix Q^p∗[3]

In this algorithm, we uses the general- Here the authors use the Pell

ized Pell equation x²− dy²= a²where equation x²− dy²= 1 where x, y

x, y are integers, d as non square posi- are integers , d as a non square

tive integer and a is the sphenic number positive integer

ꢀ

ꢁ

ꢀ

ꢁ

x_rp_r

y_rx_r

x_rp_r

y_rx_r

We use the (Q^p_q^∗)^r= a^r−1

We use the (Q^p_q^∗)^r=

for decryption

2. Notations

In this section, we display the notations and position of characters needed for the

encryption and decryption.

• B - 2n × 2n matrix which is constructed using the given message.

• B_k- k^thblock of B with the size 2 (ie, 2 × 2 matrix).

• b - number of blocks of the matrix B.

ꢂ

2

if b is 1 or 2n (n ∈ N)

• p =

smallest prime divides b if b is 2n + 1(n ∈ N)

ꢂ

b

if b ≤ p

• r =

p if b > p

• d_k- determinant of the matrix B_k.

ꢀ

ꢁ

b_k1b_k2

b_k3b_k4

•

- elements of B_k

• E=[d_k, b_ki]_i=1,2,4- encrypted matrix.

18

J. Kannan, K. Kaleeswari, and A. Deepshika

ꢀ

ꢁ

q₁q₂

q₃q₄

∗

•

- elements of (Q^p_a)^r

• δ -notation for space.

• (x, y) - Positive integer solutions of x²− py²= a²

• a - Sphenic number

2.1. Character’s Position.

A

B

C

D

E

F

G

apr²

apr²+ 1 apr²+ 2 apr²+ 3 apr²+ 4 apr²+ 5 apr²+ 6

H

I

J

K

L

M

N

apr²+ 7 apr²+ 8 apr²+ 9 apr²+10 apr²+11 apr²+12 apr²+13

O

P

Q

R

S

T

U

apr²+14 apr²+15 apr²+16 apr²+17 apr²+18 apr²+19 apr²+20

V

W

X

Y

Z

δ

apr²+21 apr²+22 apr²+23 apr²+24 apr²+25 apr²− 1

3. Algorithm

In this section, we present the encryption and decryption algorithms.

3.1. Encryption.

(1) Using the given text we have to construct the matrix B of order 2n × 2n.

(2) Convert the matrix B into the block matrix B_kof the order 2 × 2.

(3) Finding the number of blocks b and select r using p and b.

(4) p can be choosen in two ways as given above.

(5) Choose any Sphenic number a.

(6) Instead of using the characters in B_kwe apply their positions using the above table

to ﬁnd the elements of B_k.

(7) Find the determinant d_kof the matrix B_k.

(8) Using the elements of B_kand their determinant we construct E.

3.2. Decryption. Using the encrypted matrix E we have to decrypt.

∗

(1) Construct the (Q^p_a)^r.

(2) We have the elements of (Q^p_a)^ras q^′s.

∗

k

(3) Find ω_k1and ω_k2, where ω_k1= q₁b_k1+ q₃b_k2,ω_k2= q₂b_k2+ q₄b_k2.

(4) Solve for t_kusing a^2rd_k= ω_k1(q₂t_k+ q₄b_k4) − ω_k2(q₁t_k+ q₃b_k4).

(5) Now substitute b_k3instead of t_k.

(6) At last we construct B_kand B.

4. Examples

Let us see some examples for two diﬀerent values of a.

CRYPTOGRAPHIC ALGORITHM

19

4.1. Encryption and Decryption of BETA. . Encryption:

ꢀ

ꢁ

B E

T A

(1) B =

(2) There is only one block and so b = 1.

(3) By deﬁnition of p and r, we have p = 2 and r = 1. Choose a = 105.

ꢀ

ꢁ

211 214

229 210

(4) Thus B₁=

and so b^′s are given by b₁₁= 211, b₁₂= 214, b₁₃

=

1k

229, b₁₄= 210.

ꢃ

211 214

229 210

(5) d =

= -4696

1

ꢄ

ꢅ

(6) E =

−4696 211 214 210

Decryption

ꢀ

ꢁ

315 420

210 315

2^∗

105

(1) Q

=

.

(2) Here q₁= 315, q₂= 420, q₃= 210, q₄= 315

(3) ω₁₁= q₁b₁₁+ q₃b₁₂= 111405 and ω₁₂= q₂b₁₂+ q₄b₁₂= 156030.

(4) a¹d₁= ω₁₁(q₂t₁+ q₄b₁₄) − ω₁₂(q₁t₁+ q₃b₁₄) ⇒t₁= 229

(5) b₁₃= t₁= 229

ꢀ

ꢁ

211 214

229 210

(6) Hence we get B₁=

from E.

ꢀ

ꢁ

B E

T A

(7) B =

4.2. Encryption and Decryption of My favourite subject is mathematics. En-

cryption:

ꢆ

ꢉ

M Y

δ

F

A V

ꢇ

ꢊ

O U R I

T

E

δ

ꢇ

ꢊ

δ

S U B J

(1) Now B =

ꢇ

ꢊ

C

T

δ

I

S

ꢇ

ꢈ

ꢊ

ꢋ

M A T H E M

A T C S

I

δ

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

M Y

O U

δ F

A V

T E

(2) Here b = 9. We get B₁=

, B₂=

, B₃=

, B₄=

R I

ꢀ

ꢁ

δ

S

,

C T

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

U B

J E

M A

A T

T H

E M

B₅=

, B₆=

, B₇=

, B₈=

, B₉=

δ

I

S

δ

I

C

S

δ

20

J. Kannan, K. Kaleeswari, and A. Deepshika

(3) We have p = 3 < b and r = 3. Choose a = 30.

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

822 834

824 830

809 815

827 818

810 831

829 814

809 828

812 829

(4) Hence B₁=

, B₂=

, B₃=

, B₄=

,

ꢆ

ꢉ

ꢀ

ꢁ

ꢀ

ꢁ

822 810

810 829

830 811

809 818

819 814

828 809

ꢈ

ꢋ

B₅=

, B₆=

, B₇=

,

ꢀ

ꢁ

ꢀ

ꢁ

829 817

818 812

814 822

828 809

B₈=

, B₉=

(5) The determinants d_kare given by

k

1

2

3

4

5

6

7

8

9

d_k

-4956 -12243 -29559 -1675 22841 -11421 25338 4842 -22090

ꢆ

ꢉ

−4956

822 834 830

ꢇ

ꢊ

−12243 809 815 818

ꢇ

ꢊ

ꢇ

ꢊ

ꢇ−29559 810 831 814ꢊ

ꢇ

ꢊ

−1675

809 828 829

830 811 818

ꢇ

ꢊ

(6) The Encrypted matrix, E =

22841

ꢇ

ꢊ

−11421 819 814 809

ꢇ

ꢊ

25338

4842

822 810 829

829 817 812

ꢇ

ꢊ

ꢇ

ꢈ

ꢊ

ꢋ

−22090 814 822 809

Decryption

ꢀ

ꢁ

702000 1215000

3^∗

30

(1) (Q )³=

405000

702000

k

1

2

3

4

(2) The q_k^′s are

q_k702(10)³1215(10)³405(10)³702(10)³

ω₁₁

ω₁₂

ω₂₁

ω₂₂

ω₃₁

10³(914814) 10³(1584198) 10³(897993) 10³(1555065) 10³(905175)

ω₃₂

ω₄₁

ω₄₂

ω₅₁

ω₅₂

10³(1567512) 10³(903258) 10³(1564191) 10³(911115) 10³(1577772)

(3)

ω₆₁

ω₆₂

ω₇₁

ω₇₂

ω₈₁

10³(904608) 10³(1566513) 10³(905094) 10³(1567350) 10³(912843)

ω₈₂

ω₉₁

ω₉₂

10³(914814) 10³(1584198) 10³(897993)

CRYPTOGRAPHIC ALGORITHM

21

(4) Solving this equation a^2rd_k= ω_k1(q₂t_k+ q₄b_k4) − ω_k2(q₁t_k+ q₃b_k4), we get

k

1

2

3

4

5

6

7

8

9

t_k824 827 829 812 809 828 810 818 828

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

822 834

824 830

809 815

827 818

810 831

829 814

(5) Thus we get B₁=

, B₂=

, B₃=

, B₄=

ꢀ

ꢁ

809 828

812 829

,

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

830 811

809 818

819 814

828 809

822 810

810 829

829 817

818 812

B₅=

, B₆=

, B₇=

, B₈=

,

ꢀ

814 822

828 809

B₉=

from the encrypted matrix E.

ꢆ

ꢉ

M Y

δ

F

A V

ꢇ

ꢊ

O U R I

T

E

δ

ꢇ

ꢊ

δ

S U B J

(6) Hence B =

ꢇ

ꢊ

C

T

δ

I

S

ꢇ

ꢊ

ꢈ

ꢋ

M A T H E M

A T C S

I

δ

Note 1. The R program was also used to validate these examples.

5. Conclusion

∗

In this paper, (Q^p_a)^rconstructed using the solutions of x²− py²= a²using the choices

of a and p. By using very high prime numbers can further secure the process. We encrypt

and decrypt the message using the generalized Pell Equation of the form x²− dy²= n

where d and n are prime (p) and square of a sphenic number (a²) respectively. This

content is the generalization of [3]. We may also extend this with any other Diophantine

Equation or we may use any other choices of a.

REFERENCES

[1] Dickson, L. E.,History of the Theory of Numbers, Volume- II Diophantine Analysis, Dover Publications, New

York (2015).

[2] Gould, H. M., A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonnaci Quart,

19(3), 250-257, (1981).

[3] Kannan, J., Mahalakshmi, M., & Deepshika, A., Cryptographic algorithm involving the matrix Q^p^∗, Korean

J. Math, 30(3), 533-538, (2022).

22

J. Kannan, K. Kaleeswari, and A. Deepshika

[4] Kannan, J., & Manju Somanath, Fundamental Perceptions in Contemporary Number Theory Nova Science

Publisher, New York, (2023).

[5] Kannan, J., Manju Somanath., Mahalakshmi, M., & Raja, K., Encryption decryption algorithm using so-

lutions of pell equation. International Journal for Research in Applied Science and Engineering Technology,

10(1), 1-8, (2022).

[6] Sumeryra, U. C. A. R., Nihal, T.A.S., & Ozgur, N.Y., A new application to coding theory via Fibonacci and

Lucas numbers, Mathematical Sciences and Applications E-Notes, 7(1), 62-70, (2019).

[7] Titu Andreescu., Dorin Andrica., & Ion Cucurezeanu, An introduction to Diophantine equations: a problem

- based approach. Birkhauser Boston, (2010).

(Received, July 21, 2025)

(Revised, August 02, 2025)

¹Department of Mathematics,

Ayya Nadar Janaki Ammal College,

(Autonomous, aﬃliated to Madurai

Kamaraj University), Sivakasi.

Email¹: jayram.kannan@gmail.com

^2,3Ayya Nadar Janaki Ammal College,

(Autonomous, aﬃliated to Madurai

Kamaraj University), Sivakasi.