The RSA algorithm is named after its developers—Ronald Rivest, Adi Shamir, and Leonard Adleman—derived from the initials of their surnames. It was developed in 1977 by these three academics. Belonging to the class of public-key (asymmetric) encryption algorithms, RSA has become one of the foundational components of modern digital security architecture. The algorithm was patented in the United States in 1983, with its protection period expiring in 2000. RSA is widely used both for ensuring data confidentiality and for performing critical functions such as authentication and digital signing.

Fundamental Principle and Cryptographic Security Basis

The security foundation of the RSA algorithm lies in the computational difficulty of the integer factorization problem. Specifically, factoring a number (the modulus) that is the product of two large prime numbers is considered infeasible in practice using classical computation methods due to the extensive time required. This problem is regarded as the core mathematical structure that guarantees the security of RSA, owing to its computational hardness. In this context, RSA provides a secure infrastructure for both encryption and decryption processes, as well as for digital signing and verification mechanisms.

^{Visual Representing the RSA Algorithm (Created with Artificial Intelligence)}

Key Generation Steps

The RSA algorithm employs a structure that uses two mathematically related but functionally distinct keys: a public key and a private key. The key generation process is carried out through the following steps:

1. Two large, randomly selected and mutually independent prime numbers,p and q, are chosen.
2. The modulus value is calculated.(N = p x q)
3. With the help of Euler’s totient function φ(N)=(p−1)×(q−1) the value is obtained.
4. An integer e is chosen such that it is coprime with φ(N).This value is a component of the public key.
5. The value d, which is the modular multiplicative inverse of the chosen e modulo φ(N),is computed. his operation is typically performed using the extended Euclidean algorithm.

As a result:

• Public key: (e,N)
• Private key:(d,N)

Encryption and Decryption Process

In the RSA algorithm, encryption and decryption are based on modular exponentiation. The plaintext is first converted into an appropriate numerical form, and then the following operations are performed:

• Encryption:$C = M^e mod N$; Here M represents the plaintext and C represents the ciphertext.
• Decryption: $M = C^d mod N$; As a result of this process, the original message is recovered.

Thanks to this structure, only the party possessing the private key can decrypt the plaintext, thereby ensuring data security.

Performance Evaluation and Implementation Challenges

The primary disadvantage of the RSA algorithm is the high computational cost of encryption and especially decryption processes. This issue becomes more pronounced with the use of large key sizes. As a result, the following approach is generally preferred in practice: large data blocks are encrypted using symmetric encryption algorithms (such as AES), while the keys used by these algorithms are protected using RSA.

To mitigate performance issues, certain optimization techniques are employed:

• Montgomery multiplication: for accelerating modular multiplication operations
• Exponentiation algorithms: such as square-and-multiply or 3-bit exponentiation methods to reduce the number of operations

Application Areas

The RSA algorithm is widely used in various fields for both secure data transmission and digital identity authentication. Its main areas of application include:

• Internet security: SSL/TLS protocols (especially HTTPS)
• Email security: PGP (Pretty Good Privacy), S/MIME (Secure/Multipurpose Internet Mail Extensions)
• Digital signatures: Electronic document verification and integrity checking
• Electronic authentication: E-government systems and secure financial transactions

Future Threats and the Risk of Quantum Computing

Although the RSA algorithm offers strong security within the framework of classical computing, it may become vulnerable with the advancement of quantum computing technologies. In particular, quantum algorithms like Shor’s algorithm, which can perform prime factorization in polynomial time, could weaken RSA’s foundational security. Consequently, research into quantum-resistant (post-quantum) encryption algorithms has accelerated. Lattice-based, code-based, and hash-based cryptographic methods are at the core of these efforts.