FACTORIZATION PROBLEM IN ASYMMETRIC ENCRYPTION

SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 97 DOI: https://doi.org/10.5281/zenodo.17773320 FACTORIZATION PROBLEM IN ASYMMETRIC ENCRYPTION Kolkhozbaeva S.B TUIT named after Muhammad al-Khwarizmi, Master‘s student ABSTRACT In the era of rapidly expanding digital networks and cloud computing, securing sensitive information has become a cornerstone of modern computing. Cryptography plays a pivotal role in safeguarding data against unauthorized access, interference, and cyber threats. Among the cryptographic mechanisms, asymmetric encryption algorithms, particularly RSA, rely heavily on the computational difficulty of prime factorization to ensure security. However, traditional factorization methods are often inefficient for large-scale applications, posing potential vulnerabilities. This study explores advanced techniques to enhance data protection by integrating Fermat’s factorization theorem into modern encryption analysis. Keywords: Asymmetric encryption, Symmetric encryption, Cloud computing, Data security, Factorization algorithms, Cryptography ANNOTATSIYA Zamonaviy raqamli tarmoqlar va bulutli hisoblashning tez sur’atlarda rivojlanishi fonida maxfiy ma’lumotlarni himoya qilish zamonaviy hisoblash tizimlarida eng muhim vazifalardan biriga aylandi. Kriptografiya ma’lumotlarni ruxsatsiz kirish, aralashuv va kiberxavflardan himoya qilishda markaziy rol o‘ynaydi. Asimmetrik shifrlash algoritmlari, xususan RSA, xavfsizlikni ta’minlashda katta sonlarni tub sonlarga ajratishning hisoblash jihatdan murakkabligiga tayanadi. Biroq, SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 98 an’anaviy faktorizatsiya usullari katta hajmli ma’lumotlarda samarali ishlamaydi va bu potentsial xavfsizlik zaifliklarini yuzaga keltiradi. Ushbu tadqiqot Fermat faktorizatsiya teoremasini zamonaviy shifrlash tahliliga integratsiya qilish orqali ma’lumotlarni himoya qilishni kuchaytirishni o‘rganadi. Kalit so‘z: Asimmetrik shifrlash, Simmetrik shifrlash, Bulutli hisoblash, Ma’lumot xavfsizligi, Faktorizatsiya algoritmlari, Kriptografiya АННОТАЦИЯ В эпоху стремительного развития цифровых сетей и облачных вычислений защита конфиденциальной информации стала одной из ключевых задач современных вычислительных систем. Криптография играет центральную роль в защите данных от несанкционированного доступа, вмешательства и киберугроз. Среди криптографических механизмов асимметричные алгоритмы шифрования, особенно RSA, основываются на вычислительной сложности разложения больших чисел на простые множители для обеспечения безопасности. Однако традиционные методы факторизации часто неэффективны при работе с большими данными, что создаёт потенциальные уязвимости. В настоящем исследовании рассматриваются передовые методы повышения защиты данных, включая интеграцию теоремы факторизации Ферма в современный анализ шифрования. Ключевые слова:Ассиметричное шифрование, Симметричное шифрование, Облачные вычисления, Безопасность данных, Алгоритмы факторизации, Криптография Introduce Due to the widespread use of the Internet and the sharp increase in the volume of data transmission, the issue of protecting data from hacker attacks, noise, various interferences, and other threats has become highly relevant. Consequently, researchers have devoted considerable attention to the field of cryptography. Cryptography is used SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 99 to protect information, maintain its confidentiality, and ensure authentication during data exchange between users. During transmission, data is converted into a form that cannot be read, and the recipient restores it to its original form. Encrypted data is referred to as cipher-data, and the process of returning this data to its original state is called decryption. Cryptography establishes a set of security objectives to ensure the confidentiality of data: confidentiality, authentication, data integrity, non-repudiation, and access control. Today, cryptography is especially widely applied in cloud computing systems to protect data. Cloud systems provide users with the ability to store, share, and process large volumes of data over the Internet. At the same time, the rapid growth in data volume creates a number of challenges, one of the most important of which is data security. Cryptographic algorithms are divided into two main types: symmetric and asymmetric encryption algorithms. Symmetric algorithms include AES, DES, 3DES, Blowfish, and DSA; they use a single secret key for both encryption and decryption, which ensures high efficiency. Asymmetric algorithms, such as RSA, Diffie-Hellman, DSA, and Elliptic Curve Cryptography, use a pair of public and private keys, providing strong security, though they require more computational resources. In symmetric encryption, the sender and the recipient read data using the same secret key, whereas in asymmetric encryption, the public key is used for encryption and the private key for decryption. Public key-based encryption may be less efficient on small mobile devices because it relies on complex mathematical functions and requires significant computational resources. Therefore, symmetric algorithms operate approximately 1000 times faster than asymmetric algorithms. Asymmetric algorithms utilize a set of standard protocols to ensure security, including SSH, PGP, S/MIME, SSL/TLS, GPG, ZRTP, Internet Key Exchange, and SILC. These protocols ensure a high level of security during data transmission and protect it from various network threats. SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 100 Furthermore, contemporary research investigates methods such as Fermat factorization and prime number identification to enhance the efficiency of asymmetric encryption and, in particular, to improve the security of the RSA algorithm. By combining theoretical knowledge with practical algorithmic improvements, this research contributes to secure, efficient, and scalable data protection in modern cloud systems. Overall, cryptography and encryption algorithms are fundamental tools for preserving data confidentiality, protecting it from various cyberattacks, and enhancing system reliability in cloud computing systems. The combined use of symmetric and asymmetric algorithms increases system efficiency and enables users to securely exchange data. Encryption is a method of protecting sensitive data, allowing users to conceal confidential information and render it readable only with the appropriate key. Symmetric encryption algorithms use a single key to perform both encryption and decryption. The most widely used symmetric algorithms are described below. DES (Data Encryption Standard) — Developed by the National Institute of Standards and Technology (NIST), DES was the first encryption standard. It was created by IBM in the 1970s and adopted as a national standard in 1977. DES provides a standard method for protecting sensitive and unstructured data. Previously, DES was widely used in commercial, military, and other applications. DES initially used a 64bit input block, a 56-bit key, and 8 bits for parity checking. Over time, various attacks revealed vulnerabilities, leading to DES being considered an insecure block cipher. 3DES (Triple Data Encryption Standard) — An enhanced version of DES, where the DES encryption process is applied three times using separate keys at each step, increasing resistance to attacks. 3DES was designed to overcome DES vulnerabilities and provide higher security in computing and network environments. However, 3DES requires more computational resources and is therefore often replaced by AES in modern systems. SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 101 AES (Advanced Encryption Standard) — Developed by NIST to address the weaknesses of DES and 3DES, AES provides high speed, efficiency, and strong security. It supports key lengths of 128, 192, and 256 bits, offering varying levels of protection. Due to its fast performance, efficient algorithmic structure, and wide applicability across platforms, AES is one of the most popular encryption algorithms in global networks and cloud computing systems today. Blowfish — Developed by Bruce Schneier in 1993, Blowfish is a block cipher algorithm designed for speed and high security. It operates with 64-bit blocks and keys ranging from 32 to 448 bits. A key advantage of Blowfish is that it is free and opensource, suitable for diverse software and hardware platforms. However, compared to newer algorithms such as AES, Blowfish has some limitations but remains well-suited for small to medium-scale systems. In summary, symmetric encryption algorithms — DES, 3DES, AES, and Blowfish — enable fast and efficient data protection. They perform encryption and decryption using a single key, making them convenient and rapid solutions for mobile devices and cloud systems. Their main limitation is the requirement for secure key distribution and storage; if the key is compromised, data security is breached. Asymmetric encryption is an advanced method of data protection that uses two types of keys: a public key and a private key. The public key encrypts data, while the private key decrypts it. This approach provides advantages over symmetric encryption in secure data transmission, as it does not require secure key exchange. RSA (Rivest-Shamir-Adleman) — One of the most widely used asymmetric encryption algorithms, developed in 1977 by Rivest, Shamir, and Adleman. It is based on the mathematical problem of factoring large prime numbers. RSA is widely applied for data encryption and digital signature creation. Its security depends on key length: the longer the key, the stronger the encryption. Diffie-Hellman (DH) Key Exchange — A key exchange algorithm allowing two users to establish a shared secret key even without a secure channel. Using modular SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 102 exponentiation, the algorithm minimizes the possibility of key interception during data transmission. Diffie-Hellman is often used alongside other asymmetric algorithms. DSA (Digital Signature Algorithm) — An asymmetric algorithm designed for creating and verifying digital signatures. Adopted as a standard by NIST in 1991, DSA enables users to verify the authenticity and integrity of electronic documents, thereby ensuring non-repudiation. ECC (Elliptic Curve Cryptography) — A relatively new asymmetric algorithm providing high security. ECC operates on elliptic curves and offers the same security as other asymmetric algorithms with shorter key lengths, making it suitable for mobile devices and resource-constrained systems. Popular symmetric encryption algorithms include AES, DES, 3DES, and Blowfish. AES offers high security and speed, DES is an earlier standard now considered weak against modern attacks, 3DES enhances DES security but is slower, and Blowfish is known for its speed and flexibility. Asymmetric encryption algorithms provide high-level security, secure key exchange, and digital signature capabilities. However, they rely on complex mathematical computations, requiring more computational resources than symmetric algorithms. Many security protocols — SSH, PGP, S/MIME, SSL/TLS, GPG, ZRTP, Internet Key Exchange, and SILC — utilize asymmetric encryption to secure networked data. In general, asymmetric encryption algorithms are crucial tools for modern cloud computing and network security. They enable secure data transmission, authentication, and non-repudiation via digital signatures. Symmetric encryption is the most classic and widely used method for protecting data. It uses the same secret key for encryption and decryption, allowing both sender and recipient to read the information using the same key. The security of symmetric encryption depends on keeping the key secret. Its advantages include high speed, efficient processing of large data volumes, low computational resource requirements, and suitability for mobile devices and servers. Algorithms are simple and easy to SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 103 implement. Its disadvantages include the need for secure key storage, potential security breaches if the key is compromised, and complex key exchange over long distances. Symmetric encryption is widely applied in cloud storage systems, databases, and media streaming. Main algorithms include AES, DES, 3DES, and Blowfish. Asymmetric encryption uses two different keys: a public key for encryption (known to everyone) and a private key for decryption (kept secret by the recipient). Advantages include secure key exchange, high-level security, and the ability to support digital signatures and authentication. Disadvantages include high computational requirements, slower encryption of large data volumes, and reduced efficiency on mobile devices. Asymmetric encryption is used for secure Internet communications, email encryption, digital signatures, and key exchange protocols. Main algorithms include RSA, DSA, Diffie-Hellman, and ECC. Comparing symmetric and asymmetric encryption: symmetric encryption uses a single secret key, with encryption and decryption relying on the same key. Its security depends on key secrecy, it is fast, and requires fewer resources. Advantages include speed, efficiency, and ease of implementation; disadvantages include vulnerability if the key is exposed and complex key exchange over long distances. Asymmetric encryption uses two different keys: public key for encryption and private key for decryption. Security relies on mathematical foundations and key pairs; it is slower and resource-intensive. Advantages include high security, safe key exchange, and digital signature support; disadvantages include slow encryption of large data volumes and high computational demand. Symmetric encryption is applied in cloud storage and databases, whereas asymmetric encryption is used in SSL/TLS, email, and digital signature systems. Primary algorithms include AES, DES, 3DES, Blowfish (symmetric) and RSA, DSA, Diffie-Hellman, ECC (asymmetric). SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 104 Aspect Symmetric Encryption Asymmetric Encryption Number of Keys Single secret key Public and private keys Encryption/Decryption Principle Same key is used Encryption with public key, decryption with private key Source of Security Security depends on keeping the key secret Mathematical foundations, key pair Speed Very fast Depends on computational load, slower Resource Requirement Requires fewer resources Requires more resources Advantages Fast, efficient, easy to implement High security, secure key exchange, digital signature capability Disadvantages If the key is compromised, security is broken Encrypting large data is slower, requires more resources Applications Cloud storage, databases, audio-video streams SSL/TLS, email, digital signatures, key exchange Main Algorithms AES, DES, 3DES, Blowfish RSA, DSA, DiffieHellman, ECC In asymmetric encryption, the main algorithms include RSA, DSA, DiffieHellman, and Elliptic Curve Cryptography (ECC). RSA is the most popular, providing high security depending on key length. DSA is optimized for digital signatures, while SCHOLAR ISSN: 2181-4147 VOLUME 3 | ISSUE 13 | 2025 https://t.me/openscholar Multidisciplinary Scientific Journal Noyabr, 2025 105 Diffie-Hellman is used as a secure key exchange protocol. ECC offers high security with relatively shorter key lengths, making it particularly suitable for mobile devices. Symmetric encryption is distinguished by its speed and low resource requirements, but it faces the challenge of secure key exchange. Asymmetric encryption provides higher security and solves the key exchange problem; however, encrypting large volumes of data is slower and demands more resources. Conclusion Ensuring data security in modern cloud computing systems is one of the crucial and urgent challenges. The transmission and storage of large volumes of data over the Internet, along with various hacking attacks, noise, and interference risks, demand the implementation of comprehensive security mechanisms within these systems. From this perspective, cryptography is regarded as a central tool for ensuring information security in cloud environments. Research indicates that symmetric and asymmetric encryption algorithms complement each other with their respective characteristics, advantages, and limitations. Symmetric encryption algorithms—such as AES, DES, 3DES, and Blowfish— allow fast and efficient encryption of large amounts of data because they perform encryption and decryption using a single secret key. Their speed and low computational resource requirements give them a significant advantage in commercial and cloud systems. However, the primary drawback of symmetric encryption is the challenge of securely exchanging the key; if the key is compromised, the security of the entire data set is at risk. Asymmetric encryption algorithms—such as RSA, DSA, Diffie-Hellman, and Elliptic Curve Cryptography—address the key exchange problem and guarantee high security in terms of data confidentiality, authentication, integrity, and non-repudiation. Due to their reliance on complex mathematical functions, these algorithms require substantial computational resources and are relatively slower when encrypting large volumes of data. Therefore, in practice, a hybrid approach is often used: large volumes