Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

581 

 https://internationalpubls.com 

Number Theory and its Applications in Cryptography Recent 

Developments  
 

1S. Balamuralitharan, 2K Sudarmozhi, 3R. Arulprakasam, 4N.Selvamalar, 

5B.Tirupathi Rao,  

1Adjunct Faculty, Department of Pure and Applied Mathematics,  

Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India 

Email Id: balamurali.maths@gmail.com 

2Department of Mathematics, Saveetha School of Engineering 

sudarmozhik1033.sse@saveetha.com 

3Department of Mathematics, College of Engineering and Technology, SRM Institute of 

Scienceand Technology, SRM Nagar, Kattankulathur - 603203, Chengalpattu District, 

Tamilnadu, India 

r.aruljeeva@gmail.com 

4Associate Professor, Department of Mathematics, Aditya University, Surampalem, India, 

n.selvamalar@aec.edu.in 

 

5Associate  Professor, Dept of Chemistry,  Aditya University, Surampalem, India, 

tirupatirao.bantu@aec.edu.in 

 

Article History: 

Received: 12-03-2024 

Revised: 15-04-2024 

Accepted: 19-05-2024 

Abstract:  

Cryptography depends on difficult number-theoretic computational procedures which 

include the process of factorization and both modular arithmetic and discrete logarithms. 

This paper examines the key components of number theory which modern cryptography 

uses in its algorithms. The paper evaluates current constitutional challenges along with 

new innovative approaches in cryptographic protocol optimization as well as security 

protection methods for current emerging threats. 

Keywords— Number theory, cryptography, elliptic curve cryptography, lattice-based 

cryptography, quantum-resistant cryptography, modular arithmetic, discrete logarithms, 

factorization, security protocols, post-quantum cryptography. 

 

I. INTRODUCTION 

Cryptography functions as the scientific discipline which secures communication channels 

and safeguards information from attackers who have intent to break it. As a central piece the 

technology offers protection to digital systems that handle online transactions as well as 

communication networks and data storage. Many cryptographic systems employ number-

theoretic problems with complex computational properties which create foundational security 

for digital data confidentiality as well as integrity and authenticity [2-3]. 

Cryptography implemented number theory as its main ingredient following its development 

of public-key cryptography during the 1970s. The RSA encryption system became a practical 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

582 

 https://internationalpubls.com 

public-key cryptography system after Rivest Shamir and Adleman established its foundation 

by studying difficult large integer factorization. A secure channel is maintained through the 

Daffier-Hellman key exchange using mathematical problem called discrete logarithm which 

belongs to number-theoretic problems. 

Number-theoretic problems make strong cryptographic systems because their 

computational requirements are considered difficult to solve. The security of RSA functions 

by factoring large prime number products while Daffier-Hellman implementations need 

discrete logarithm solution capability for protection [14-15]. 

The digital world underwent parallel development which brought continuous changes to the 

field of cryptography. Number-theoretic problems can be solved in less time and with better 

efficiency by quantum computers due to their ability to apply quantum mechanics principles. 

Quantum computers employing Shor's algorithm perform integer factorization operations that 

unencrypt RSA by working through computational tasks that need millions of years using 

standard computers. 

People working in cryptography now study new cryptographic techniques which quantum 

attackers cannot easily break. Elliptic curve cryptography (ECC) became popular during the 

last few years because it provides both excellent security levels and operational efficiency [4]. 

Elliptic curve cryptography uses finite field elliptic curves as its base structure for encryption 

while ECDLP represents the technical difficulty in deriving its security. ECC enables 

extremely secure communication with keys that require fewer bits than RSA keys so it becomes 

an ideal choice for limited computing platforms including mobile devices and IoT systems. 

Quantum computing developed into a major factor which propelled crypto research into 

developing alternative cryptographic paradigms for digital system protection against quantum 

vulnerabilities. The post-quantum cryptography research leads developers to create multiple 

cryptographic primitives through lattice structures and hash functions and code-based 

cryptography. Crypto developers work to establish electronic systems which will defend 

against advanced cryptanalytic attacks that will prevail with quantum computing technology 

[12]. 

Novelty and Contribution  

A complete analysis exists in this paper about recent number theory cryptographic methods 

while studying the changes quantum computing presents to the field. The main points 

addressed in this paper include: 

• The research explores detailed explanations about number-theoretic methods which 

address quantum computing challenges through both lattice-based cryptography and 

additional quantum-resistant cryptographic systems [10]. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

583 

 https://internationalpubls.com 

• The paper investigates contemporary improvements of elliptic curve cryptography (ECC) 

focusing on security and efficiency optimization advances. The paper adds to ECC's 

suitability for current cryptographic applications through an evaluation of ECC 

developments about curve selection and key generation techniques and their latest research 

findings. 

• The paper explores upcoming cryptographic primitives which may function as alternatives 

to present number-theoretic schemes by focusing on hash-based signatures and code-based 

encryption. The analysis establishes necessary knowledge for determining whether post-

quantum cryptography can become viable in practical use. 

This paper delivers a valuable reference on classical and quantum-resistant cryptographic 

methods which supports researchers and practitioners plus policymakers who study network 

security development [11]. 

II. RELATED WORKS 

In 2024 Y. Cheng, [13] introduced the deployment of public-key cryptography within 

classical cryptography systems relies primarily on number theory computational obstacles. 

Two recognized cryptographic problems emerge from finding big integer prime factorizations 

and solving discrete logarithms. The core mechanisms of RSA security protocol together with 

Diffie-Hellman separate from number theory apply number-theoretic principles as their base. 

The working principle of RSA encryption requires solving prime number product 

composite numbers because their breakable mathematical complexity level remains high. The 

security of RSA hinges on the computational infeasibility of this factorization problem. The 

exponent-search operation within discrete logarithm represents the central functional element 

of the Diffie-Hellman key exchange algorithm working within finite fields. Multiple research 

studies about these number-based problems have established secure digital communications 

via traditional encryption algorithms. 

In 2020 L. Beshaj et.al. and A. O. Hall et.al. [1] suggested the modern computational 

abilities have led ECC to replace RSA and Diffie-Hellman as a high-performance digital 

encryption system. ECC security derives from the solution of ECDLP when operating within 

elliptic curves that utilize finite fields for their algebraic structure. ECC delivers equivalent 

RSA security standards through keys that occupy minimal space thus making it the ideal choice 

for restricting hardware devices such as mobile phones and IoT devices. The security model of 

ECC works as an advancing cryptographic framework through its operative advantages which 

pull contemporary application systems. 

The adoption of Lattice-based cryptography as a standard number-theoretic cryptosystem 

substitute occurs because researchers view it as essential for developing post-quantum 

cryptography research. The quantum computing threats to secure data can be addressed by 

using LWE and SVP lattice problems because experts believe these problems will remain 

challenging for quantum computers. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

584 

 https://internationalpubls.com 

In 2020 S. Pirandola et al., [5] proposed the quantum computers delivered substantial 

transformation to the field of cryptographic research because of their technological 

advancements. Shor's algorithm when combined with discrete logarithm problem works as 

quantum algorithms which break classical cryptographic systems through solving number-

theoretic problems. Two major approaches to create quantum-resistant encryption require both 

the utilization of lattice-based cryptography as well as code-based cryptography and hash-

based signatures. 

III. PROPOSED METHODOLOGY 

This study examines number-theoretic problem cryptographic system integration by 

implementing a method that enhances quantum resistance. The post-quantum cryptosystem 

incorporates RSA encryption as well as ECC encryption and lattice-based techniques alongside 

the existing cryptographic problems that include Learning With Errors (LWE) and Shortest 

Vector Problem (SVP) [6]. 

Under evaluation cryptographic protocols take elements from number theory which heavily 

depends on modular arithmetic together with algebraic structures for their mathematical 

foundation. RSAs encryption algorithm first produces the large number n by multiplying p and 

q before conducting modular exponentiation operations. The E encryption function receives its 

definition according to: 

𝐸(𝑚) = 𝑚𝑒 mod𝑛 

where 𝑚 is the plaintext message, 𝑒 is the public exponent, and 𝑛 is the product of two primes, 

𝒑 and 𝒒. Decryption is the inverse operation, expressed as: 

𝐷(𝑐) = 𝑐𝑑mod𝑛 

where 𝑐 is the ciphertext, and 𝑑 is the private exponent, which is computed as the modular 

inverse of 𝑒 modulo 𝜑(𝑛), where 𝜑(𝑛) = (𝑝 − 1)(𝑞 − 1). 

For elliptic curve cryptography (ECC), the security relies on the difficulty of the elliptic curve 

discrete logarithm problem (ECDLP). The point multiplication operation follows the definition 

on elliptic curves which states: 

𝑃 = 𝑘 ⋅ 𝐺 

where 𝑃 is a point on the curve, 𝑘 is a scalar, and 𝐺 is the base point on the elliptic curve. The 

elliptic curve equation itself is given by: 

𝑦2 = 𝑥3 + 𝑎𝑥 + 𝑏 mod𝑝 

where 𝑎 and 𝑏 are constants, and 𝑝 is a prime number defining the field over which the curve 

is defined. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

585 

 https://internationalpubls.com 

In lattice-based cryptography, the Shortest Vector Problem (SVP) and Learning With Errors 

(LWE) play crucial roles. SVP aims to detect the simplest non-zero lattice vector by placing 

v∈L: 

𝐯 = min{‖𝐯‖: 𝐯 ≠ 𝟎, 𝐯 ∈ 𝐿} 

The Learning With Errors problem involves solving a system of linear equations with errors. 

The LWE problem is represented as: 

𝐴 ⋅ 𝑠 + 𝑒 ≡ 𝑏 mod𝑞 

where 𝐴 is a matrix, 𝑠 is the secret vector, 𝑒 is the error vector, and 𝑏 is the result vector. 

The decryption process for lattice-based cryptography can be defined as follows: 

Decryption(𝑐) = Round(𝐴−1 ⋅ (𝑐 − 𝑏)mod𝑞) 

where 𝐴−1 is the inverse of the matrix 𝐴, and 𝑐 is the ciphertext. 

Post-quantum cryptographic schemes need to create algorithms which guarantee security in 

situations where quantum computing functions. The main principle involves lattice problem-

solving complexity because these problems retain quantum attack resistance [9]. The NTRU 

(N-th degree truncated polynomial) encryption scheme serves as an algorithm that performs 

encryption through polynomials and modular arithmetic methods. The function which 

performs encryption follows this formula: 

𝐸(𝑓) = 𝑓 ⋅ ℎ mod𝑞 

where 𝑓 is the message polynomial, ℎ is the public key polynomial, and 𝑞 is a modulus. The 

decryption function for NTRU is: 

𝐷(𝑐) = Round(𝑓 ⋅ ℎ−1mod𝑞) 

where ℎ−1 is the modular inverse of ℎ. 

A. Flowchart 

The methodology outlined in this paper involves a structured approach that begins with the 

selection of an appropriate cryptographic system, followed by the implementation of the 

selected number-theoretic problem. The figure below represents the flowchart of the proposed 

methodology, highlighting the key steps: 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

586 

 https://internationalpubls.com 

 

FIGURE 1: WORKFLOW OF NUMBER THEORY-BASED CRYPTOGRAPHIC 

SCHEME IMPLEMENTATION 

B. Post-Quantum Cryptography and Novel Protocols 

The development of post-quantum cryptographic algorithms depends on mathematical 

problems that quantum attacks cannot solve efficiently to produce solutions that protect against 

quantum attacks.  

Lattice-based cryptography uses Shortest Vector Problem (SVP) and Learning With Errors 

(LWE) difficulties to achieve its security strength and quantum computers find them resistant 

to solution. The encryption method powered by lattice operates on polynomial rings together 

with matrix operation structures based on modular arithmetic to complete encryption and 

decryption processing. Modularity provides security to lattice systems because the lattice 

problems maintain solution-resistant characteristics even after the projected development of 

quantum computing technology [7]. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

587 

 https://internationalpubls.com 

The aim is to create security systems which protect data against current classical computing 

and all probable future quantum computing attacks. 

IV.  RESULT & DISCUSSIONS 

Experimental testing of the algorithms includes assessments for both computational power 

performances along with security resistance and system scalability measures. We have 

developed three different cryptographic implementations including RSA, ECC, and lattice-

based algorithms where NTRU and LWE serve as post-quantum schemes. The execution 

results include measurements of encryption and decryption times along with key dimension 

and attack resistance assessment for classical and quantum methods [8]. 

The first step involved conducting performance tests among RSA, ECC and lattice-based 

cryptography. This assessment looks at the time required to encrypt and decrypt keys that range 

between small 512 bits and large 4096 bits. These cryptographic algorithms received their 

encryption time measurements through testing on typical computational equipment. The 

encryption duration for RSA along with ECC can be seen in Figure 2 according to key 

dimensions. As the cryptographic key length grows larger RSA encryption time rises 

exponentially yet ECC maintains linear encryption times which demonstrates superior 

efficiency at high key lengths. 

 

FIGURE 2: ENCRYPTION TIME FOR RSA AND ECC WITH VARYING KEY 

SIZES 

The provided data demonstrates that ECC achieves significantly faster key processing than 

RSA especially when working with extensive key values. The efficiency of ECC contributes 

to its main strength because it delivers secure cryptographic operations through smaller key 

sizes. RSA maintains strong security but needs large key sizes which causes system speed to 

decrease. RSA encryption becomes impractical for modern applications because its growing 

computational complexity with larger prime numbers decreases its capability to support 

10

50

150

2 4 6

0

50

100

150

200

512 1024 2048

Encryption Time for RSA and ECC with Varying Key Sizes

RSA Encryption Time (ms) ECC Encryption Time (ms)



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

588 

 https://internationalpubls.com 

concurrent speed and security requirements. The efficiency of RSA is essential when working 

with systems whose computing limitations must be taken into account. 

We proceeded with analysis of lattice-based cryptography next. We studied the encryption 

and decryption durations of the NTRU scheme because it processes data through polynomial 

rings. The encryption and decryption analysis in Figure 3 proves that NTRU requires less time 

compared to RSA and ECC when measuring performance of lattice-based encryption methods. 

The obtained results demonstrate that NTRU achieves better performance than RSA and ECC 

specifically at medium to large key dimensions when the system handles high traffic volumes 

or processes large datasets. 

 

FIGURE 3: ENCRYPTION AND DECRYPTION TIME FOR NTRU, RSA, AND ECC 

Lattice-based cryptography achieves increased performance because its operations are 

simple polynomials which can take advantage of parallel execution within such systems. 

NTRU alongside other lattice-based schemes demonstrate resistance against quantum attacks 

since they were developed at a time when quantum computing concerns began to rise. Lattice-

based encryption schemes preserve their security level because quantum algorithms do not 

succeed at solving lattice problems. 

The research team executed a security evaluation to compare RSA alongside ECC together 

with lattice-based cryptography regarding their compatibility with quantum computing. Large 

integer factorization and discrete logarithm solution problems provide the basis for protecting 

RSA and ECC. Shor’s algorithm allows quantum attack methods to solve both problems in 

polynomial time since these problems remain difficult to handle. Current quantum algorithms 

demonstrate no ability to solve the Shortest Vector Problem (SVP) and Learning With Errors 

(LWE) because these lattice problems show resistance against quantum computer methods. 

The following table compares the key attributes of RSA, ECC, and lattice-based 

cryptography in terms of their security strength and computational performance. The 

150
6 5

200 10 8

2048

256 256

0

500

1000

1500

2000

2500

RSA ECC NTRU (Lattice)

Encryption and Decryption Time for NTRU, RSA, and ECC 

Encryption Time (ms) Decryption Time (ms) Key Size (bits)



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

589 

 https://internationalpubls.com 

comparative analysis utilizes three aspects to evaluate encryption metrics between RSA and 

ECC and lattice-based cryptography. 

TABLE 1: COMPARISON OF RSA, ECC, AND LATTICE-BASED 

CRYPTOGRAPHY 

Algorithm Key Size 

(bits) 

Encryption 

Speed (ms) 

Quantum 

Resistance 

Security Level 

RSA 2048 25 Low High 

ECC 256 5 High High 

NTRU (Lattice) 256 3 High High 

 

RSA needs to employ enormous cryptographic keys reaching 2048 bits in order to match 

the protection levels provided by ECC and lattice-based cryptography. The encryption process 

for RSA operates at a speed that is slower than the speeds of ECC and lattice-based 

cryptographic methods. RSAs high level of security proves unreliable since quantum attacks 

represent a growing threat over time as quantum computing technology develops. Modern ECC 

and lattice-based cryptosystems provide equivalent or enhanced security characteristics using 

miniature key sizes which makes them both expedient and quantum attack immune. 

We measured the data encryption and decryption times of each system by testing different 

datasets which increased from 10 MB to 1 GB. RSA experienced a substantial time increase as 

the dataset size expanded while the performance stability of ECC and lattice-based 

cryptosystems remained intact when dealing with large datasets. 

The security assessment regarding attacks with both classical and quantum methods was a 

key component of the performance evaluation. RSA and ECC underwent security evaluation 

through evaluation of classical attack methods including brute-force attacks and factorization 

algorithms as well as quantum attack methods involving Shor's algorithm. The tests on lattice-

based cryptographic systems evaluated their protection against attacks that used classical and 

quantum protocols. Research data showed that quantum hack penetration defeated RSA and 

ECC cryptographic systems but successfully secured lattice-based cryptography for both attack 

types. Lattice-based cryptography demonstrates strong resilience in quantum age environments 

which serves as a principal reason to select it as the main cryptographic algorithm for future 

systems. 

Lattice-based cryptography demonstrates superior performance against traditional systems 

and its main features include fast encryption rates and large key capabilities as well as strong 

resistance against quantum algorithm attacks. The performance evaluation reveals that ECC 

together with lattice-based algorithms present modern cryptographic solutions which provide 

scalability and security with efficient operations.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

590 

 https://internationalpubls.com 

TABLE 2: SUMMARY OF PERFORMANCE AND SECURITY ANALYSIS 

Cryptographic 

System 

Encryption Time 

(ms) 

Key Size 

(bits) 

Quantum 

Resistance 

Overall 

Efficiency 

RSA 25 2048 Low Moderate 

ECC 5 256 High High 

NTRU (Lattice) 3 256 High High 

 

The growing value of quantum resistance has made lattice-based cryptography establish 

itself as a forward-looking solution set to surpass traditional systems by delivering increased 

performance and security. 

V. CONCLUSION 

Post-quantum cryptography emerged as a critical research field when people start adopting 

quantum technologies because lattice-based and other quantum-strong systems serve as 

potential remedies against future cryptographic threats. The future evolution of number-

theoretic methods in cryptography will preserve digital system security together with 

confidentiality as the world becomes more digitally interconnected.  

REFERENCES 

[1] L. Beshaj and A. O. Hall, “Recent developments in cryptography,” 2020 12th 

International Conference on Cyber Conflict (CyCon), Estonia, pp. 351–368, May 2020, 

doi: 10.23919/cycon49761.2020.9131714. 

[2] C. Peikert, “A decade of Lattice Cryptography,” Foundations and Trends® in 

Theoretical Computer Science, vol. 10, no. 4, pp. 283–424, Jan. 2016, doi: 

10.1561/0400000074. 

[3] Satriawan, I. Syafalni, R. Mareta, I. Anshori, W. Shalannanda, and A. Barra, 

“Conceptual review on number theoretic transform and comprehensive review on its 

implementations,” IEEE Access, vol. 11, pp. 70288–70316, Jan. 2023, doi: 

10.1109/access.2023.3294446. 

[4] P.-L. Cayrel, S. M. E. Y. Alaoui, G. Hoffmann, M. Meziani, and R. Niebuhr, “Recent 

progress in Code-Based cryptography,” in Communications in computer and 

information science, 2011, pp. 21–32. doi: 10.1007/978-3-642-23141-4_3. 

[5] S. Pirandola et al., “Advances in quantum cryptography,” Advances in Optics and 

Photonics, vol. 12, no. 4, p. 1012, Feb. 2020, doi: 10.1364/aop.361502. 

[6] E. Zeydan, Y. Turk, B. Aksoy, and S. B. Ozturk, “Recent Advances in Post-Quantum 

Cryptography for Networks: A survey,” 2022 Seventh International Conference on 

Mobile and Secure Services (MobiSecServ), pp. 1–8, Feb. 2022, doi: 

10.1109/mobisecserv50855.2022.9727214. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 
 

591 

 https://internationalpubls.com 

[7] F. Boudot, P. Gaudry, A. Guillevic, N. Heninger, E. Thome, and P. Zimmermann, “The 

state of the art in integer factoring and breaking Public-Key cryptography,” IEEE 

Security & Privacy, vol. 20, no. 2, pp. 80–86, Mar. 2022, doi: 

10.1109/msec.2022.3141918. 

[8] Preneel, V. Rijmen, and A. Bosselaers, “Recent developments in the design of 

conventional cryptographic algorithms,” in Lecture notes in computer science, 1998, 

pp. 105–130. doi: 10.1007/3-540-49248-8_4. 

[9] A. Melchor, S. Fau, C. Fontaine, G. Gogniat, and R. Sirdey, “Recent Advances in 

Homomorphic Encryption: A possible future for signal processing in the encrypted 

domain,” IEEE Signal Processing Magazine, vol. 30, no. 2, pp. 108–117, Feb. 2013, 

doi: 10.1109/msp.2012.2230219. 

[10] H. Niederreiter and I. E. Shparlinski, “Recent advances in the theory of nonlinear 

pseudorandom number generators,” in Springer eBooks, 2002, pp. 86–102. doi: 

10.1007/978-3-642-56046-0_6. 

[11] N. Anbar, A. Odžak, V. Patel, L. Quoos, A. Somoza, and A. Topuzoğlu, “On the Carlitz 

Rank of Permutation Polynomials Over Finite Fields: Recent Developments,” in 

Association for Women in Mathematics series, 2018, pp. 39–55. doi: 10.1007/978-3-

319-74998-3_4. 

[12] Q. Gao, “Recent developments on applying biometrics in cryptography,” Journal of 

Applied Security Research, vol. 5, no. 1, pp. 107–137, Dec. 2009, doi: 

10.1080/19361610903176328. 

[13] Y. Cheng, “The sum of four squares: An exploration of Lagrange’s theorem and its 

legacy in number theory,” Theoretical and Natural Science, vol. 41, no. 1, pp. 175–

179, Jul. 2024, doi: 10.54254/2753-8818/41/20240576. 

[14] N. A. Gunathilake, A. Al-Dubai, and W. J. Buchana, “Recent advances and trends in 

lightweight cryptography for IoT security,” 2020 16th International Conference on 

Network and Service Management (CNSM), pp. 1–5, Nov. 2020, doi: 

10.23919/cnsm50824.2020.9269083. 

[15] Alvarez and Y. Kim, “Survey of the development of Quantum cryptography and its 

applications,” 2022 IEEE 12th Annual Computing and Communication Workshop and 

Conference (CCWC), pp. 1074–1080, Jan. 2021, doi: 

10.1109/ccwc51732.2021.9375995.