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An Approach for Cipher Communication Utilizing the Concepts of 

Elliptic Curve Cryptography 

 

S. Sarvalakshmi1*, CH. Suneetha2 
1*. Research Scholar in Mathematics, GITAM (Deemed to be University), Visakhapatnam, India 

2. Associate Professor in Mathematics, GITAM (Deemed to be University), Visakhapatnam, India 

1*Corresponding Author Mail: radhika1483@gmail.com 

 

Article History: 

Received: 16-07-2024 

Revised: 30-08-2024 

Accepted: 11-09-2024 

Abstract:  

Message encryption is a process of protection of the original data in transmission. Several 

cryptosystems were designed in the history for different environments. Public key or 

Asymmetric cryptosystem has become more popular in recent era where the key space 

consists both public and private keys. Elliptic curve cryptography dominates public key 

cryptosystems like Diffie Hellman key exchange and RSA. ECC is more powerful and 

alternative technique of RSA, with a simple replacement of exponent calculations of large 

prime numbers to mathematical representation of Elliptic curve points. Besides the 

computation ECC works with shorter key length than RSA. The present paper describes 

an encryption algorithm using elliptic curves over finite fields. Here a group of authentic 

users transmit. 

Keywords: Elliptic curve cryptography, Elliptic curve points, Encryption Decryption, 

Generator. 

 

1. Introduction: 

Diffie Hellman Key Exchange Algorithm is a compelling protocol which utilizes exponentiation of 

huge groups to start an encrypted communication between two unknown communicators. It has been 

proposed as an internet standard and significantly enhances internet security. However, the algorithm 

affects from a security flaw known as a man-in-the-middle attack.  Later, the Diffie Hellman Key 

Exchange algorithm was modified and published by several authors, on the other hand, RSA is a public 

key cryptosystem comes from the challenge of factorizing big integers into their prime. Even if the 

key size doubles, the RSA algorithm's strength will not increase although the public key operations 

being faster. Elliptic curve cryptography depends of the hardness of Elliptic Curve Discrete Logarithm 

Problem (ECDLP). The concept of elliptic curve Utilization in public key cryptography trailblazers 

were Koblitz and Miller. The main motive for attentive attraction of the researchers towards ECC is 

yields the level of security as RSA. 

2. Literature Survey: 

Numerous researchers employed elliptic curves over finite fields for security applications and 

researched the benefits of ECC over traditional cryptosystems. ECC-based text-based encryption 

techniques were proposed by several authors. When it comes to (Secure Sockets Layer) SSL and TLS 

certificates, Benjamin Clement Sebastian and Ugar Alpay Cenar [1] researched and compared the 

substantial advantages of ECC over RSA.  Elliptic curve cryptography was used to implement text 

encryption by L. Dolendro and K. Manglem [2]. The writers of this chapter provided a conventional 



Communications on Applied Nonlinear Analysis 

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way of mapping the characters to affine points in the curve by matching the ASCII values of the plain 

text characters. A proposed encryption method using elliptic curve cryptography was made by Renee 

Brady et al. [3]. They used Unicode to encrypt the text in that study and explored the drawbacks of 

sending text communications through platforms like Twitter. The text is encoded using the Koblitz 

method as a point on an elliptic curve. For users of online social networks, S. Thiraviya and S. Britto 

[4] suggested better elliptic curve cryptography. They used encryption, decryption time analysis, plain 

text, and cipher text size analysis in that paper. A. Shamir [5] was the first to suggest an Identity-based 

cryptosystem as an alternative to conventional public key infrastructure for resolving authenticity 

problems. A pre-communication requirement is part of the authentication system described by Aziz 

and Diffie [6]. The use of a nonsingular matrix to map the identical characters in the message to 

different positions on the curve was recommended by F. Amounas and E.H. EI Kinani [7]. Elliptic 

curves were suggested for encryption and decryption by Kolhekar M. and Jadhav A. [8], A. Gera and 

K. Agrawal [9]. 

Elliptic Curve Arithmetic 

An irreducible cubic with a flex can be affinely transformed into a Weierstrass equation:  

𝑄2 + 𝑎1𝑃𝑄 + 𝑎3𝑄 =  𝑃3 + 𝑎2𝑃2 + 𝑎4𝑃 + 𝑎6, where 𝑎1𝑎2, 𝑎3,𝑎4, 𝑎6 are real and P∈ 𝑅, Q∈ 𝑅.The 

equationy2 = x3 +ax + b, 4a3+27b2≠ 0 over the finite field Fq defines an elliptic curve for cryptography 

purposes. 

The points on the curve taking ∞, infinity point as identity element an abelian group with respect to 

point addition is formed 

Point addition on the Curve: If Ƿ(a1,b1) and Ɋ(a2,b2) are two points on the curve, Ƿ +Ɋ is obtained 

by taking the image or reflection of the point obtained by intersecting the line through Ƿ, Ɋ with the 

curve. The reflection is considered about X-axis. For opposite points Ƿ and– Ƿ the vertical line is 

intersected through the curve.  

The identity is Ƿ + (-Ƿ) = ∞. For Ƿ(a1,b1),Ɋ(a2,b2) Ƿ +Ɋ is Ɍ(a3,b3) 

𝑎3 = (
𝑏2−𝑏1

𝑎2−𝑎1
)

2

− 𝑎1 − 𝑎1 − 𝑎2

        

  

𝑏3 = (
𝑏2−𝑏1

𝑎2−𝑎1
) (𝑎2 − 𝑎3) − 𝑏1

        

  

 

Point multiplication or point doubling on the curve: - for the point Ƿ (a1, b1), 2 Ƿ is the image of 

the point obtained by intersecting the tangent through Ƿ with the curve. The image is about X- axis 

[1,2]. 

For Ƿ ≠ - Ƿ, 2 Ƿ is given by (a3, b3) 

𝑎3 = (
3𝑎1

2+𝑎

2𝑏1
)

2

− 2𝑎1   

𝑏3 = (
3𝑎1

2 + 𝑎

2𝑏1
) (𝑎1 − 𝑎3) − 𝑏1 

 



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Figure 1: Elliptic curve arithmetic 

Discrete Logarithmic Problem on Elliptic Curve (ECDLP): -  

Above Figure 1, Elliptic Curve Cryptography is safeguarded by a hard problem called Elliptic Curve 

Discrete Logarithmic Problem (ECDLP) [4,5,6]. For two given points P, Q on the curve such that Q = 

xP for ‘x’ a big random number it is difficult to find x = 𝑙𝑜𝑔𝑝Q for given P, Q. 

Methodology: 

An Elliptic curve 𝐸𝑝(𝑎, 𝑏), with prime p having as many points as possible and generators are selected 

by one of the members of asset of communicating team. In curve selection the following main features 

should be carefully observed. 

i.The order of the curve 𝐸𝑝(𝑎, 𝑏) must not contain small prime factors. 

ii.The curve is supposed to be not super singular. 

iii. It is non anomalous; order is not equal to p #𝐸𝑝(𝑎, 𝑏) ≠p 

In 𝐸𝑝(𝑎, 𝑏)  with p as prime, to have the cyclic nature of the group and to treat each point as a generator. 

All the points on the curve are mapped to alphabet, numeral and alpha numeral characters and a 

common look up table available for all the members. This procedure is set up phase for the group set 

up phase trusted external services can be availed. 

Individual set up phase: 

The curve generator and common look up table are available for all the members of the community. 

Every authentic member of the group can share messages with every other member. For that a pair of 

authentic users of the group has to arrange a pre setup. If two users Robert and James of the group 

want to transmit messages before transmission, they set up private and public key system. 

Robert choose two random points R1, R2 on the public curve and two large random prime numbers 

∝1, ∝2 (less than the order). 

He computes      R3 = ∝1𝑅2+∝2R1 

R4 = (∝1∝2) R3 

He communicates R3, R4 to James. 



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Similarly, James selects two random points J1, J2 and large random numbers 𝛽1, 𝛽2(less than the 

order), computes     J3 = 𝛽1𝐽2 + 𝛽2𝐽1 

J4 = (𝛽1𝛽2)J3   and communicates to Robert. 

The secret key set for Robert is (R1, R2, ∝1, ∝2) 

The public key set for Robert to communicate is (R3, R4) 

In the same way key set for James is (J1, J2, 𝛽1, 𝛽2). public key set to communicate with Robert is (J3, 

J4). 

Encryption: 

If James wants send message to Robert, he uses his secret keys and Robert’s public keys for encrypting 

the messages. 

Encryption stages are 

1. Consider message M consisting of characters 𝑚1, 𝑚2, 𝑚3--- 𝑚𝑛. All the characters are mapped 

to elliptic curve points with the help of common look up table P1, P2, P3---𝑃𝑛. 

2. An additional parameter 𝛾 other than secret key set is chosen by James to encrypt the plaintext. 

Here 𝛾 is a big number smaller than the order of the curve different for different plaintext characters. 

3. He encrypts the first point P1 using  𝛾1as E11 = 𝛾1R3 

E12 = P1+ 𝛾1R4 

So, the point P1 of the message is converted to pair of points (E11, E12). 

4. Similar series of stages 2, 3 is applied for all the plaintext elliptic curve points P1, P2, P3---𝑃𝑛. 

(En1, En2) is cipher pair and 𝛾nis large random number for n= 1,2,3--- , where 

En1= 𝛾nR3 

En2= 𝑃𝑛 + 𝛾nR4 

The cipher pair for P1 is (E11, E12), P2 is (E21, E22), P3is (E31, E32), -----, 𝑃𝑛 is (En1, En2). 

5. All the points E11, E12, E21, E22, E31, E32, ----, En1, En2 are reverse mapped to text characters, 

which is the cipher text C communicated to Robert in a public channel. 

Decryption: 

  Robert converts the cipher text characters into points on the curve and  

1. Considers the points pair wise (E11, E12), (E21, E22), (E31, E32) --- (En1, En2). 

2. Decrypts each pair using his own secret keys and James’s public keys to get original message. 

P1= E12-∝1∝2E11 

P2= E22-∝1∝2E21 

-------------------- 

𝑃𝑛= En2-∝1∝2En1  

Proof of the decryption 

                   𝑃𝑛= En2-∝1∝2En1 

                     = 𝑃𝑛 + 𝛾nR4- ∝1∝2𝛾nR3 



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                     = 𝑃𝑛 + 𝛾n(∝1∝2) R3- ∝1∝2𝛾nR3 

                     = 𝑃𝑛 + 𝛾n(∝1∝2) R3- ∝1∝2𝛾nR3 

                     = 𝑃𝑛  

If Robert wants to send massage to James, he uses James’s public keys for encryption 

En1= 𝛾nJ3 

En2= 𝑃𝑛+ 𝛾nJ4 

James decrypts the pair to 𝑃𝑛as 𝑃𝑛= En2- 𝛽1𝛽2 En1 

Example: 

Consider the elliptic curve 𝐸31(5,3) and there are 40 number of points on this curve. Take any point 

on this curve let it be G= (1,3). Now we calculate 2G,3G, ---- 40G. Create a table by using points on 

the curve and assign character values, special symbols and numerals to those points.  

 For given message compare each letter with that table. 

Take the message MRPG COLLEGE VIZIANAGARAM 

By comparing each letter with the table, we get 13G 18G 16G 7G 28G 3G 15G 12G 12G 5G 7G 5G 

28G 22G 9G 26G 9G 1G 7G 1G 18G 1G 13G  

Here R1= (3,13) R2= (4,5) and ∝1= 13 ∝2= 17 

R3 = 13(4,5) + 17(3,13) 

R3 =104G+ 306G 

R3= 410 G (mod 31) = 7G 

R4= (13*17) 7G= 28 G 

J1= (6,1)     J2= (7,3) 𝛽1=19 𝛽1=23 

J3=19(7,3) + 23(6,1) 

J3= 589G+299G 

J3=888G (mod 31) = 20G 

J4= (19*23)20G=29G 

We have En1= 𝛾nJ3 and En2= 𝑃𝑛+ 𝛾nJ4 

Also, we consider different 𝛾nfor different characters in our message 

Here we take 𝛾n=6, 2, 4, 9, 11, 8, 5, 3, 7, 10, 12, 14, 15, 21, 16, 17, 22, 24, 19, 18, 25, 27, 29, 20, 23 

for n= 1, 2, 3, ---25. Then by using above formula we calculate 

(E11, E12) = (27,1) (E21, E22) = (9,14) (E31, E32) = (18,8) (E41, E42) = (25,20) (E51, E52) = (5,12) 

(E61, E62) = (3,12) ---- (En1, En2) = (26,29), for n= 1, 2, 3, ---25. 

Now compare each point with the table to get the cipher text 



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AINRHYTELCLGE#FPWNAHU#QKJHJWF.OOHGS.DSMIVV WZ# 

Example for repeated words 

Take the message GITAMGITAMGITAM 

Here we take 𝛾n= 7, 3, 5, 10, 12, 9, 6, 4, 8, 11, 13, 15, 16, 22, 17, for n= 1, 2, 3, ---15. 

Then by using above formula we calculate 

(E11, E12) = (17,5) (E21, E22) = (19,14) (E31, E32) = (18,1) (E41, E42) = (36,4) (E51, E52) = (35,33)  

(E61, E62) = (16,22) ---- (En1, En2) = (12,14), n= 1, 2, 3, ---15. 

Now compare each point with the table to get the cipher text 

QESNRA5D42PV7S8M6 OENOM322$XLN 

Table 1: Table for time calculation 

Message size in KB Execution Time in Sec. 

0.1 0.03816 

0.2 0.04918 

0.3 0.06110 

0.4 0.07234 

0.5 0.08318 

0.6 0.09812 

0.7 0.1121 

0.8 0.2314 

0.9 0.3145 

1 0.4285 

Execution time graph: 

 

Figure 2: Execution time graph 

 



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Figure 3: conventional ECC for different 

sizes of data (number of characters) 
 

Figure 4: Total time required for conventional 

ECC & Proposed method 

The above table 1 and figures 2,3,4 show encryption and decryption time of the present technique and 

conventional ECC for different sizes of data (number of characters). Present technique shows 

approximately same execution time when compared with conventional elliptic curve cryptography. 

So, the computational cost is not more than conventional ECC and the present algorithm is as fast as 

conventional ECC algorithm. 

Security Analysis: 

Here 𝛾 value is different for encrypting different message characters. Due to these same characters in 

the original message will be mapped to different cipher characters. In the first example given above 

there are repeated characters (M, R, L, A, G) in the original message. First ‘R’ letter is mapped to ‘AI’ 

whereas second ‘R’ letter is mapped to ‘IV’. Similarly, the same character ‘L’ repeated side by side is 

mapped to ‘FP’ and ‘WN’ in the cipher text. Since different random numbers are used for 𝛾(less than 

generator) the present encryption has achieved a good required feature to face several active and 

passive attacks. In the history of cryptography several authors suggested multiple encryptions using 

the same key. The proposed method is as starts as available multiple encryption algorithms with 

comparatively less computational overhead. Fig 6 shows the repetition of the English alphabets in the 

cipher text for 1 KB of data. 

In ECC the cipher text space twice than the message space. So, English alphabets repetition will not 

give any information about the cipher text, because each character maps to two cipher characters. So, 

linear cryptanalysis and differential cryptanalysis are tough to execute for the present cipher. This hard 

problem protects ECC against several active and passive attacks. In Pollard –Rho attack ECDLP is 

cracked by exhaustive search of cipher text points (points on elliptic curve). But in the present 

technique conventional elliptic curve cryptographic algorithm (similar to Elgamal encryption) is 

modified by considering more number of points and big random numbers to construct public and secret 

keys. In addition, each message character is encrypted using a parameter 𝛾 (random number less than 

generator). This 𝛾acts as a safe guard for the cipher text. Pollard –Rho and Pohlig- Hellman attacks 

are quite impossible. 𝛾values are chosen by encryptor for different message characters. These 

𝛾values are inherent in the cipher text. 

 



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Even for the repeated words message (SSSS------ SUSUSU------) different 𝛾 values are used to 

encrypting the characters. This is the basic strength of the present cipher to endure and avoid Pollard 

–Rho attack. Underlying  𝛾 values is helping the present cipher to overcome the weakness of ECC. So, 

the ECC algorithm presented here is as safe as conventional public key cryptographic techniques with 

small key size. 

Conclusion: 

The process of utilizing the concept of Elliptical Curve Cryptography (ECC) delineated to restraint 

intrusions in the communication of sensitive information securely. The encryption technique proposed 

here offers same level of security as the conventional public key cryptographic techniques as it is 

highly difficult to calculate the secret keys of both the legitimate entities. 

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[2] Laiphrakpam Dolendro Singh and Khumanthem Manglem Singh “Implementation of Text Encryption using Elliptic 

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