INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XII, December 2025  
The Unseen Engine of Trust A Deep Dive into the RSA Algorithm  
Dr Partha Majumdar  
Department of Computer Science, Swiss School of Business and Management (SSBM), Geneva,  
Switzerland  
DOI : https://doi.org/10.51583/IJLTEMAS.2025.1412000084  
Received: 21 December 2025; Accepted: 27 December 2025; Published: 07 January 2026  
ABSTRACT  
This analysis offers a thorough overview of the RSA algorithm, a key element of modern digital security that  
addresses the historic challenge of secure communication over insecure public channels. Its innovation is based  
on number theory, using a "trapdoor" one-way function reliant on the significant computational difference  
between multiplying large prime numbers and factoring their product. Paired with modular arithmetic and Euler's  
Totient Theorem, this enables the practical use of public-key cryptography, where a public key encrypts data  
that only the matching private key can decrypt. The text highlights RSA's vital role in internet security, especially  
in securing web connections via SSL/TLS and ensuring authenticity and integrity with digital signatures. It  
discusses the cybersecurity arms race, stressing the importance of sufficient key lengths and secure practices to  
fend off classical attacks. The analysis also examines RSA's main vulnerability: the threat from quantum  
computing. Shor's algorithm can efficiently solve the integer factorisation problem, making RSA obsolete and  
prompting a global shift to Post-Quantum Cryptography (PQC). Despite its eventual replacement, RSA's legacy  
as the first framework that enabled trust in the digital world remains a monumental achievement in computer  
science.  
Keywords: RSA algorithm, Public-key cryptography, Integer factorisation, Digital signatures, Quantum threat  
INTRODUCTION:  
The Paradox of Public Secrecy  
In the digital realm, we perform acts of profound trust with a simple click. When we see the small padlock icon  
in a web browser's address bar, we instinctively trust that our credit card numbers, private messages, and sensitive  
data are safe from prying eyes. This trust is not based on faith, but on a revolutionary mathematical concept that  
solved a paradox as old as communication itself: how can two parties, who have never met, establish a secret  
conversation over a public channel that is being actively monitored by adversaries? For centuries, the answer  
was that they could not, at least not without first securely exchanging a secret key through a trusted courier or a  
prior clandestine meeting.  
The algorithm that broke this century-old impasse and became the unseen engine of modern digital trust is known  
as RSA, named for its public inventors, Ron Rivest, Adi Shamir, and Leonard Adleman. Introduced in 1977,  
RSA was the first practical and widely adopted system to realise the dream of public-key cryptography. This  
paradigm fundamentally reshaped our ability to create privacy and verify identity in an increasingly connected  
world. Its impact has been monumental, forming the cryptographic bedrock for everything from the rise of e-  
commerce and online banking to the security of national intelligence communications.  
This report embarks on a comprehensive journey into the world of RSA, structured to align with the principles  
of "Data, Dharma, and Decision." We will first delve into the Data—the elegant and profound principles of  
number theory that provide the algorithm's unshakeable mathematical foundation. Next, we will explore the  
Dharma—the core principles of asymmetric cryptography and the fascinating dual history of RSA's invention,  
a story of both secret government research and open academic collaboration. Finally, we will examine the critical  
Decisions that RSA enables and forces upon us, from its practical applications in securing the internet to the  
urgent, ongoing arms race against new threats, most notably the dawn of quantum computing. This exploration  
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will deconstruct the algorithm from its 18th-century mathematical roots to its potential obsolescence on the  
quantum horizon, revealing the intricate mechanics of the system that secures our digital lives.  
The Elegant Mathematics of Asymmetry (The Data)  
The genius of the RSA algorithm lies not in a complex new invention, but in the clever application of  
mathematical principles that have been understood for centuries. It leverages a profound imbalance in the  
computational difficulty of certain number-theoretic problems, creating a system where locking a message is  
easy for anyone, but unlocking it is practically impossible for everyone except the intended recipient. This  
asymmetry is the source of its power.  
The Primacy of Primes and the One-Way Function  
At the heart of RSA is the prime number—an integer greater than 1 that is divisible only by 1 and itself. These  
numbers are the fundamental building blocks of all integers, a concept formalised in the Fundamental Theorem  
of Arithmetic, which states that every integer greater than 1 can be represented as a product of prime numbers  
in exactly one way (ignoring the order of the factors). This unique factorisation property is a cornerstone of  
number theory and, as it turns out, of modern cryptography.  
The security of RSA is built upon a specific computational task related to primes: integer factorisation. [7] The  
core principle is a "one-way function," a mathematical operation that is easy to perform in one direction but  
exceedingly difficult to reverse. In the case of RSA, this function is the multiplication of two very large prime  
numbers. [6]  
The "Easy" Direction: If one chooses two distinct, large prime numbers, let's call them p and q, calculating  
their product, n = p * q, is computationally trivial. Even for primes that are hundreds of digits long, a modern  
computer can perform this multiplication in a fraction of a second. [1]  
The "Hard" Direction: However, if one is given only the product (known as a semiprime), the reverse  
operation—finding the original prime factors p and q—is a monumentally difficult task for classical computers.  
The best-known classical algorithms for factoring large numbers take a prohibitively long time, potentially  
thousands of years, for the key sizes used in modern applications.  
This vast and predictable gap in computational effort is not merely a mathematical curiosity; it is the essential  
security primitive upon which RSA is built. Historically, security was achieved through physical means or shared  
secrets. The breakthrough of asymmetric cryptography was the realisation that a purely mathematical imbalance  
could serve as a lock. Security could be derived not from hiding information, but from the inherent computational  
limits of even the most powerful machines. This function is not perfectly "one-way," because a reverse path does  
exist. Instead, it is a trapdoor function: the reverse path is computationally infeasible unless one possesses a  
special piece of information—the "trapdoor". [1] In RSA, as we will see, that trapdoor is the knowledge of the  
prime factors themselves.  
The World of Modular Arithmetic  
The operational environment where RSA's mathematics unfolds is known as modular arithmetic, a system of  
arithmetic for integers that "wraps around" upon reaching a certain value—the modulus. It is often intuitively  
explained as "clock arithmetic." On a 12-hour clock, 4 hours after 10 o'clock is not 14 o'clock, but 2 o'clock. In  
mathematical terms, this is expressed as 10 + 4 ≡ 2 (mod 12). The symbol ≡ denotes congruence, meaning that  
14 and 2 have the same remainder when divided by the modulus 12.  
All encryption and decryption operations in RSA are modular exponentiations, taking the form of 푚푒 (mod n)  
or 푐푑 (mod n). This mathematical framework is crucial for two reasons. First, it ensures that the results of  
cryptographic operations—the ciphertext and the decrypted plaintext—always remain within a manageable,  
fixed-size range (from 0 to (n – 1)), regardless of how large the intermediate values of the exponentiation become.  
Second, this finite, cyclical universe of numbers possesses special properties, described by group theory, that  
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Euler's Theorem exploits to guarantee that decryption perfectly and reliably reverses encryption.  
Euler's Totient Theorem: The Key to the Trapdoor  
The mathematical linchpin that makes the RSA trapdoor function work is a theorem from 1763 by Leonhard  
Euler. To understand it, one must first be familiar with the totient function, denoted as ϕ(n) (phi of n).  
Euler's totient function counts the number of positive integers less than or equal to n that are relatively prime to  
n. Two numbers are relatively prime (or coprime) if their greatest common divisor (GCD) is 1. [12] For example,  
ϕ(n) = 4 because only four numbers (1, 5, 7, 11) are less than 12 and share no factors with 12 other than 1.  
The totient function has several properties that are critical for RSA:  
If p is a prime number, then all numbers from 1 to (p – 1) are relatively prime to it, so ϕ(p) = (p - 1).  
If m and n are coprime, the function is multiplicative: ϕ(mn) = ϕ(m) ϕ(n).  
Combining these, for the RSA modulus n=p*q, where p and q are distinct primes, the totient is easily calculated  
as:  
ϕ(n) = ϕ(pq) = ϕ(p) ϕ(q) = (p – 1)(q – 1)  
With this function defined, we can now state Euler's Theorem (also known as the Fermat-Euler theorem or  
Euler's totient theorem). [4] If an integer m and the modulus n are relatively prime, then raising m to the power  
of the totient of n is congruent to 1, modulo n. [1]  
푚휙(푛) ≡ 1 (mod n)  
This theorem provides the mathematical proof for why RSA's encryption and decryption are perfect inverses. In  
RSA, the public exponent e and private exponent d are specifically chosen to be modular multiplicative inverses  
with respect to the totient ϕ(n). This means they satisfy the relationship:  
ed ≡ 1 (mod ϕ(n))  
This congruence implies that there exists some integer k such that ed = kϕ(n) + 1. When a ciphertext c ≡ me  
(mod n) is decrypted, the operation is:  
cd ≡ (me)d (mod n) ≡ med (mod n)  
Substituting the relationship for ed, we get:  
med (mod n) ≡ m(kϕ(n) + 1) (mod n) ≡ mkϕ(n) m1 (mod n) ≡ (mϕ(n))k m (mod n)  
Applying Euler's Theorem, mϕ(n) becomes 1:  
(mϕ(n))k m (mod n) ≡ 1k m (mod n) ≡ m (mod n)  
The original message m is recovered. This elegant proof demonstrates that the decryption process will always  
succeed.  
The critical link between all these concepts is that the entire system hinges on the secrecy of the totient, ϕ(n).  
The public key consists of the pair (n, e). To find the corresponding private key d, an attacker would need to  
solve the congruence ed ≡ 1 (mod ϕ(n)). This is impossible without knowing ϕ(n). And to calculate ϕ(n) = (p –  
1)(q – 1), one must know the secret prime factors p and q. Therefore, the "trapdoor" that allows the owner to  
reverse the one-way function easily is precisely the knowledge of the prime factors of n. An attacker who can  
factor n can break the system; an attacker who cannot is left with an intractable problem.  
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A Secret History: The Invention and Re-Invention of Public-Key Cryptography  
The story of RSA is not a single, linear narrative of invention but a remarkable tale of convergent evolution,  
where the same revolutionary idea was born twice, independently, in two vastly different worlds: the clandestine  
corridors of a government intelligence agency and the open, collaborative environment of a university.  
The Precursors: Diffie, Hellman, and the Public-Key Revolution  
For most of history, cryptography operated under a single paradigm: symmetric encryption. This method uses  
the same secret key to both encrypt and decrypt a message. While effective, it suffers from a fundamental  
logistical challenge known as the key distribution problem: for two parties to communicate securely, they must  
first find a secure way to share the secret key. This often required a trusted courier or a face-to-face meeting, a  
process that was slow, expensive, and impractical for the burgeoning digital age. [1]  
In 1976, this ancient paradigm was shattered. In a groundbreaking paper titled "New Directions in  
Cryptography," researchers Whitfield Diffie and Martin Hellman introduced the world to the theoretical concept  
of public-key cryptography. They envisioned a system with two mathematically related keys: a public key,  
which could be shared openly with the world, and a private key, which would be kept secret by its owner. A  
message encrypted with the public key could only be decrypted by the corresponding private key. This  
asymmetric approach elegantly solved the key distribution problem. They also conceptualised digital signatures,  
but a practical implementation of a "trapdoor one-way function" to make it all work remained an open problem.  
Their paper, however, lit a fire in the academic community and directly inspired the team at MIT that would  
ultimately give the world RSA.  
The Secret Cypher: Clifford Cocks and GCHQ's Classified Discovery  
Unbeknownst to the public academic community, the puzzle posed by Diffie and Hellman had already been  
solved inside the secretive world of British intelligence. In 1973, four years before RSA was publicly announced,  
a young mathematician named Clifford Cocks at the UK's Government Communications Headquarters (GCHQ)  
independently invented an identical algorithm.  
Cocks had recently joined GCHQ after studying number theory at Cambridge and Oxford. He was presented  
with a theoretical paper written in 1969 by his colleague James Ellis, which outlined the concept of "non-secret  
encryption"—what the world would later call public-key cryptography. Ellis had conceived of the possibility but  
had been unable to find a mathematical function to implement it. Cocks, with his background in number theory,  
immediately saw a potential solution in prime factorisation and, within hours, developed the algorithm that  
would later be known as RSA. Shortly after, another GCHQ mathematician, Malcolm Williamson,  
independently developed a method for key exchange equivalent to what is now known as the Diffie-Hellman  
protocol.  
However, this monumental British achievement was immediately classified as top-secret. At the time, with the  
high cost of computing power, the invention was seen more as a theoretical curiosity than a practical tool for  
military use and was never deployed. The work remained hidden from the world for 24 years, finally being  
declassified and revealed to the public in 1997.  
The Public Breakthrough: Rivest, Shamir, and Adleman at MIT  
Back in the public sphere, inspired by Diffie and Hellman's 1976 paper, three colleagues at the Massachusetts  
Institute of Technology (MIT)—computer scientists Ron Rivest and Adi Shamir, and mathematician Leonard  
Adleman—embarked on a quest to find the elusive trapdoor one-way function. Their process was a rigorous  
cycle of creation and destruction: Rivest and Shamir would propose candidate functions, and Adleman's role  
was to find their weaknesses and break them.  
For nearly a year, they struggled, trying and discarding 42 different approaches until they began to believe the  
task was impossible. The breakthrough came in April 1977. After a long day of work followed by a Passover  
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Seder, Rivest lay on a couch, unable to sleep, his mind racing through number theory concepts. Suddenly, the  
idea coalesced. He realised that a function based on modular exponentiation and the difficulty of factoring a  
large semiprime number could provide the properties they needed. He spent the rest of the night formalising the  
algorithm, and by morning, the core of the paper was written.  
The algorithm was named RSA, using the initials of their surnames in the order they appeared on the paper. [1]  
They published their work in 1977 and were granted a U.S. patent in 1983. The parallel invention of the same  
algorithm in two isolated environments is a striking example of convergent discovery. It suggests that the idea's  
time had come, as the necessary mathematical tools and the pressing need for secure digital communication had  
finally aligned. The contrasting fates of these two inventions—one locked away by state secrecy, the other shared  
openly to become a global standard—serve as a powerful illustration of how institutional culture can shape the  
destiny of a technology.  
Deconstructing the RSA Algorithm  
At its core, the RSA cryptosystem involves three distinct processes: the initial generation of a public and private  
key pair, the encryption of a message using the public key, and the decryption of the ciphertext using the private  
key. While the underlying mathematics is profound, the procedural steps are remarkably straightforward.  
Step-by-Step: Key Generation, Encryption, and Decryption  
Key Generation  
The foundation of RSA security is the careful and random generation of a unique key pair. This process is  
performed by any individual or system that wishes to receive encrypted messages. The steps are as follows:  
Choose two distinct large prime numbers, p and q. These primes must be chosen at random from a vast space  
of possibilities and must be kept secret. The security of the entire system rests on the infeasibility of an attacker  
guessing or deriving these two numbers. For a modern 2048-bit key, each prime would be roughly 1024 bits  
long.  
Compute the modulus n = p * q. This product, n, serves as the modulus for all cryptographic operations. Its  
length in bits (e.g., 2048) defines the key size. The value of n is made public as part of the public key.  
Compute the totient ϕ(n) = (p – 1)(q – 1). This value, derived from the secret primes, is essential for calculating  
the private key and must also be kept secret. Modern implementations often use a refinement called Carmichael's  
totient function, λ(n) = lcm(p – 1, q – 1), which can sometimes be a smaller value than ϕ(n) and works just as  
well. However, the original paper and the core concept rely on Euler's totient.  
Choose the public exponent e. This integer must satisfy two conditions: it must be greater than 1 and less than  
ϕ(n) (1 < e < ϕ(n)), and it must be coprime to ϕ(n) (i.e., gcd(e, ϕ(n)) = 1). To improve encryption efficiency, e  
is often chosen to be a small number with a low count of '1's in its binary representation. The most common  
choice in modern applications is the prime number 65537, which is 216 + 1. The value of e is released as the  
second part of the public key.  
Compute the private exponent d. This is the crucial final step. The value d is the modular multiplicative inverse  
of e modulo ϕ(n). It is the unique integer that satisfies the congruence ed ≡ 1 mod(ϕ(n)). This value can be  
calculated efficiently using the Extended Euclidean Algorithm. [12] The resulting number, d, is the core  
component of the private key and must be guarded with absolute secrecy.  
At the end of this process, the public key is the pair (n, e), which can be shared with anyone. The private key  
is the pair (n, d), which must never be revealed. The original primes p and q, and the totient ϕ(n), can be discarded  
after d is computed, as they are no longer needed for operations but could be used to compromise the key if  
discovered.  
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Encryption  
To send a secure message to a recipient, the sender performs the following steps:  
1. Obtain the recipient's authentic public key (n, e).  
2. Convert the plaintext message, m, into an integer such that 0 ≤ m < n. If the message is too long, it is  
broken into smaller blocks, each satisfying this condition. [1]  
3. Compute the ciphertext, c, using the recipient's public key with the formula:  
4. c = me (mod n)  
5. Send the resulting ciphertext c to the recipient.  
Decryption  
Upon receiving the ciphertext c, the recipient uses their own private key (n, d) to recover the original message:  
1. Compute the original message integer, m, using the formula:  
2. m = cd (mod n)  
3. Convert the integer m back into its original plaintext format.  
A Practical Walkthrough with Numerical Example  
To make these abstract steps concrete, consider a simplified example using small prime numbers that can be  
calculated by hand. In a real-world scenario, the primes would have hundreds of digits.  
Let's follow the key generation process using the primes p = 61 and q = 53.  
Step  
Action  
Example Calculation (p = 61, q = 53)  
p = 61, q = 53  
Result  
Choose Primes  
p and q are secret.  
n = 3233 (Public)  
1
2
Compute  
Modulus  
n = p * q = 61 * 53  
Compute Totient  
ϕ(n) = (p – 1) (q – 1) = 60 * 52  
ϕ(n) = 3120 (Secret)  
3
4
Choose Public  
Exponent  
Choose e coprime to 3120. A common  
prime choice is suitable.  
e = 17 (Public)  
Compute Private  
Exponent  
Solve for d in 17 * d ≡ 1 mod(3120).  
Using the Extended Euclidean  
Algorithm, we find the inverse.  
d = 2753 (Secret)  
5
Result  
Key Pairs  
Public Key: (3233, 17)  
Private Key: (3233, 2753)  
Now, suppose a sender wants to encrypt the message "A", which has an ASCII value of 65. So, the plaintext  
message is m = 65.  
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Encryption:  
The sender uses the recipient's public key (3233, 17):  
c = me (mod n) = 6517 (mod 3233)  
This calculation yields:  
c = 2790  
The sender transmits the ciphertext 2790.  
Decryption:  
The recipient uses their private key (3233, 2753) to decrypt the ciphertext:  
m = cd (mod n) = 27902753 (mod 3233)  
This calculation, while enormous, correctly yields:  
m = 65  
The recipient converts the number 65 back to its ASCII representation, recovering the original message "A".  
The Digital Architect: RSA's Modern Applications (The Decision)  
The RSA algorithm is not merely a theoretical construct; it is a foundational technology woven into the very  
fabric of the internet and modern digital security. Its ability to solve the key exchange problem and provide a  
basis for digital identity has made it an indispensable tool in a wide array of applications, forcing organisations  
and individuals to make critical decisions about how they implement trust and privacy online.  
Securing the Web: The Role of RSA in SSL/TLS  
Perhaps the most ubiquitous application of RSA is in the Secure Sockets Layer (SSL) and its successor, Transport  
Layer Security (TLS) protocols—the cryptographic standards that provide the "S" in HTTPS. Every time a user  
connects to a secure website, RSA often plays a crucial role in the initial "handshake" process that establishes a  
secure session.  
The process works through a hybrid cryptosystem, a pragmatic engineering decision that leverages the strengths  
of both asymmetric (RSA) and symmetric (e.g., AES) cryptography. Asymmetric algorithms like RSA are  
computationally intensive and thus too slow for encrypting large volumes of data, such as an entire web browsing  
session. [10] Symmetric algorithms are incredibly fast but require a shared secret key. The TLS handshake uses  
RSA to solve this dilemma: [15]  
1. When a client (your browser) connects to a server (e.g., yourbank.com), the server presents its digital  
certificate.  
2. This certificate, issued by a trusted Certificate Authority (CA), contains the server's public key (n, e) and  
verifies its identity.  
3. The client's browser verifies the certificate's authenticity. It then generates a random symmetric key (called  
a "pre-master secret") that will be used for the rest of the session.  
4. The client encrypts this new symmetric key using the server's RSA public key and sends it to the server.  
5. Because only the server possesses the corresponding private key, it is the only entity that can decrypt the  
message and recover the symmetric key.  
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At this point, both the client and server share the same secret symmetric key. They switch from the slow RSA  
algorithm to a fast symmetric algorithm like AES to encrypt all subsequent data exchanged during the session.  
[11] This hybrid approach provides a best-of-both-worlds solution: the security of public-key cryptography for  
key exchange and the speed of symmetric cryptography for bulk data encryption.  
The Unforgeable Signature: Authentication and Integrity  
Beyond confidentiality, RSA provides a powerful mechanism for ensuring authenticity, integrity, and non-  
repudiation through digital signatures. This application cleverly inverts the normal use of the public and private  
keys to prove identity rather than to conceal information.  
The standard encryption model is for confidentiality: anyone can use a public key to lock a message, but only  
the private key holder can unlock it. For a digital signature, the logic is reversed: only the private key holder can  
"lock" (sign) a document, but anyone with the public key can "unlock" (verify) it. This creates mathematical  
proof of origin. The process is as follows: [15]  
1. The sender (signer) takes the message or file to be signed and creates a fixed-size cryptographic hash of it  
(e.g., using SHA-256). A hash is a unique digital fingerprint of the data.  
2. The sender then encrypts this hash value using their own private key. The result is the digital signature,  
which is appended to the original message.  
3. The recipient receives the message and the signature. To verify it, they first compute their own hash of the  
received message.  
4. Then, they use the sender's public key to decrypt the signature, revealing the original hash computed by  
the sender.  
5. If the two hashes match, the signature is valid. This proves two things:  
Authenticity: Since only the sender has the private key, the signature must have originated from them.  
Integrity: Even a one-bit change to the message in transit would result in a completely different hash, causing  
the verification to fail. This proves the message has not been altered.  
This mechanism is fundamental to securing software downloads (code signing), legal documents, and ensuring  
the authenticity of digital certificates themselves.  
Beyond the Browser: VPNs, Secure Email, and More  
The foundational capabilities of RSA extend far beyond web browsing.  
Virtual Private Networks (VPNs): When a user connects to a corporate network via a VPN, RSA is often used  
during the initial authentication and connection setup phase. Similar to the TLS handshake, it helps authenticate  
the client and server to each other and securely negotiates the symmetric keys that will be used to encrypt all the  
data flowing through the VPN tunnel.  
Secure Email (PGP and S/MIME): Protocols like Pretty Good Privacy (PGP) and Secure/Multipurpose  
Internet Mail Extensions (S/MIME) rely heavily on RSA. To send a confidential email, the message is encrypted  
with the recipient's public key, ensuring that only they can decrypt it with their private key. To prove the sender's  
identity, emails can be digitally signed using the sender's private key. [13]  
The Arms Race: RSA Security and the Quantum Threat (The Dharma)  
The invention of RSA initiated an ongoing arms race between cryptographers and cryptanalysts. The security of  
the algorithm is not absolute; it is a function of contemporary technology, implementation choices, and the  
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discovery of new mathematical attacks. This dynamic forces a continuous re-evaluation of security practices,  
embodying the principle that digital security requires perpetual vigilance.  
The Strength in Numbers: Key Length and Vulnerabilities  
The primary defence of the RSA algorithm is the sheer size of its numbers. The security of an RSA key is directly  
proportional to the difficulty of factoring its modulus, n. This difficulty grows exponentially with the number of  
bits in the key.  
In the early days of RSA, key sizes of 512 or 768 bits were considered secure. Today, advances in computing  
power and factorisation algorithms have rendered such keys obsolete. A 768-bit RSA key was publicly factored  
in 2009, and a 829-bit key was broken in 2020. Consequently, 1024-bit keys, once a common standard, are now  
widely deprecated as they are believed to be within reach of well-funded state-level adversaries.  
Current industry and government standards, such as those from the National Institute of Standards and  
Technology (NIST), mandate a minimum key length of 2048 bits for most applications. For data requiring long-  
term protection (into the 2030s and beyond), 3072-bit or 4096-bit keys are recommended.  
It is crucial to understand that the security strength of asymmetric keys does not scale linearly like that of  
symmetric keys. A 2048-bit RSA key is not "twice as strong" as a 1024-bit key; it is many orders of magnitude  
stronger. The following table compares the equivalent strengths of different cryptographic systems, providing a  
more accurate context for making security decisions.  
Symmetric Key Strength  
(bits)  
Equivalent RSA Key Length  
(bits)  
Equivalent ECC Key Length  
(bits)  
80  
1024  
2048  
3072  
7680  
15360  
160  
224  
256  
384  
521  
112  
128  
192  
256  
Beyond brute-force factorisation, RSA implementations can be vulnerable to other, more subtle attacks: [8]  
Low Public Exponent Attacks: If a small public exponent (like e = 3) is used without proper cryptographic  
padding, and the same short message is encrypted and sent to multiple recipients, an attacker can use the Chinese  
Remainder Theorem to recover the original message. This is why modern systems use standardised padding  
schemes like Optimal Asymmetric Encryption Padding (OAEP).  
Timing Attacks: This is a type of side-channel attack where an adversary meticulously measures the time a  
cryptographic device takes to perform a private key operation. Variations in computation time, which can depend  
on the bits of the secret key d, can leak enough information over many measurements to allow the attacker to  
reconstruct the key.  
Poor Random Number Generation: The security of RSA depends on the unpredictability of its prime factors,  
p and q. If the random number generator used to create them is flawed or predictable, an attacker might be able  
to guess the primes. A more catastrophic failure occurs if two different RSA keys are generated using the same  
faulty random number generator and happen to share one of their prime factors. An attacker can then find this  
common factor simply by computing the greatest common divisor (GCD) of the two public moduli, instantly  
breaking both keys. [6]  
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A final important nuance lies in computational complexity theory. While the integer factorisation problem is  
known to be in the class NP (its solutions can be verified quickly), it is not known to be NP-complete (the  
hardest problems in NP). It is widely suspected to belong to a rare intermediate class known as NP-intermediate.  
[7] This means that even a definitive proof that P≠NP would not guarantee that factorisation remains a  
fundamentally hard problem for classical computers.  
The Quantum Menace: Shor's Algorithm  
The most profound threat to RSA does not come from classical computers but from an entirely new paradigm of  
computation: quantum computing. In 1994, mathematician Peter Shor developed a quantum algorithm that  
fundamentally changed the nature of the integer factorisation problem. [16]  
For a classical computer, the difficulty of factoring grows exponentially with the size of the number. For a  
quantum computer running Shor's algorithm, the problem can be solved in polynomial time. [14] This represents  
a true paradigm shift. While a classical supercomputer might take thousands of years to factor a 2048-bit RSA  
key, a sufficiently large and stable quantum computer could theoretically do so in a matter of hours or days. [3]  
Shor's algorithm does not simply speed up classical factorisation methods. It attacks the problem in a completely  
different way by reframing it. It transforms the task of factoring n into the task of finding the period of a specific  
modular function. While period-finding is also difficult for classical computers, it is a problem that is perfectly  
suited to the unique capabilities of quantum machines, which can use principles like superposition and the  
Quantum Fourier Transform to find the period efficiently. [14]  
This algorithm is not just an incremental improvement; it is a paradigm-killing discovery. It invalidates the core  
assumption of computational difficulty upon which RSA and other public-key systems like Elliptic Curve  
Cryptography (ECC) are built. The existence of Shor's algorithm has forced the entire cryptographic community  
to confront the mortality of its most trusted tools and to begin the urgent search for a new generation of  
cryptography.  
The Next Generation: An Introduction to Post-Quantum Cryptography  
In response to the quantum threat, the field of Post-Quantum Cryptography (PQC) has emerged. Its goal is to  
develop new public-key algorithms that are secure against attacks from both classical and quantum computers.  
[2] These algorithms are not based on integer factorisation or the discrete logarithm problem (which also falls to  
Shor's algorithm), but on different mathematical problems believed to be hard even for quantum computers. [9]  
NIST has been leading a multi-year process to standardise PQC algorithms. The main categories of candidates  
include [3]:  
Lattice-based Cryptography: Based on the difficulty of finding the shortest vector in a high-dimensional  
geometric lattice.  
Code-based Cryptography: Based on the difficulty of decoding a random linear error-correcting code.  
Hash-based Cryptography: Based on the security of cryptographic hash functions.  
Multivariate Cryptography: Based on the difficulty of solving systems of multivariate polynomial  
equations.  
The following table provides a high-level comparison of RSA with its main classical peer, ECC, and the  
symmetric workhorse, AES, to contextualise their roles and vulnerabilities.  
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Feature  
AES (Advanced  
Encryption Standard)  
RSA (Rivest-Shamir-  
Adleman)  
ECC (Elliptic Curve  
Cryptography)  
Symmetric (Single  
Shared Key)  
Asymmetric  
(Public/Private Key  
Pair)  
Asymmetric  
(Public/Private Key  
Pair)  
Type  
Bulk data encryption  
(files, connections)  
Key exchange, Digital  
signatures  
Key exchange, Digital  
signatures  
Primary Use  
128, 192, 256 bits  
Very Fast  
2048, 3072, 4096 bits  
256, 384, 521 bits  
Key Size  
Slow (computationally  
intensive)  
Faster than RSA for  
equivalent security  
Performance  
Grover's Algorithm  
(mitigated by doubling  
key size)  
Shor's Algorithm  
(completely broken)  
Shor's Algorithm  
(completely broken)  
Quantum  
Threat  
The transition to PQC is already underway. A common strategy being deployed is the use of hybrid schemes,  
where a classical algorithm like RSA or ECC is used alongside a new PQC algorithm (like the NIST-selected  
CRYSTALS-Kyber). This ensures that the connection remains secure even if one of the algorithms is broken,  
providing backward compatibility and a bridge to a fully post-quantum future. [5]  
CONCLUSION:  
The Enduring Legacy of an Idea  
The RSA algorithm stands as one of the most significant intellectual achievements of the 20th century. It was  
far more than a clever application of number theory; it was the first practical embodiment of public-key  
cryptography, a concept that fundamentally altered our ability to forge trust and privacy in a world built on public  
networks. By transforming the mathematical asymmetry of integer factorisation into a workable cryptographic  
primitive, RSA provided the essential tools for secure e-commerce, private digital communications, and  
verifiable online identity, thereby enabling the very architecture of the modern internet.  
Its story is a perfect microcosm of the perpetual cycle that defines the field of cybersecurity. A deep mathematical  
principle—the Dharma of factoring's computational difficulty—gave rise to a revolutionary technology. This  
technology, in turn, forced society to make new and critical Decisions about how to build and manage digital  
trust. Now, a new technological horizon—quantum computing—threatens to shatter that original mathematical  
principle, compelling the global community to seek new foundational truths and make the next generation of  
security decisions.  
While the RSA algorithm itself may eventually be relegated to the annals of cryptographic history, succeeded  
by quantum-resistant successors, the paradigm it unleashed is permanent. The revolutionary idea that secrets can  
be established in the open, that identity can be proven with mathematical certainty, and that trust can be  
engineered without a pre-existing relationship is an indispensable and enduring part of our connected civilisation.  
RSA's greatest legacy is not the code or the mathematics, but the demonstration that such a world is possible.  
REFERENCES  
1. Durai Raj Vincent, P. M. (2016, June 1). RSA encryption algorithm - A survey on its various forms and its  
security level. ResearchGate. Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/306215546_RSA_encryption_algorithm_A_survey_on_its_vari  
ous_forms_and_its_security_level  
2. Galbraith, S. (2025, December 1). Post-Quantum Cryptography. ResearchGate. Retrieved December 21,  
www.ijltemas.in  
Page 952  
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
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2025, from https://www.researchgate.net/publication/398244591_Post-Quantum_Cryptography  
3. Grover, L. K. (1996, June 1). Fast quantum mechanical algorithm for database search. ResearchGate.  
Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/2201607_Fast_quantum_mechanical_algorithm_for_database_s  
earch  
4. Ireland, K., & Rosen, M. (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer-  
Verlag New York Inc.  
5. Jain, B., & Mittal, H. K. (2025, May 1). Post-Quantum Cryptography: A Comprehensive Review of Past  
Technologies and Current Advances. ResearchGate. Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/392329453_Post-  
Quantum_Cryptography_A_Comprehensive_Review_of_Past_Technologies_and_Current_Advances  
6. Kulandei, B., & Dhenakaran, S. S. (2017, October 1). An Overview of Cryptanalysis of RSA Public key  
System. ResearchGate. Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/321031748_An_Overview_of_Cryptanalysis_of_RSA_Public_k  
ey_System  
7. Lenstra, A. K., & Lenstra, H. W. (1993). The Development of the Number Field Sieve. Springer. Retrieved  
December 21, 2025, from https://link.springer.com/book/10.1007/BFb0091534  
8. Luo, Z. J., Liu, R., Mehta, A., & Ali, M. L. (n.d.). Demystifying the RSA Algorithm: An Intuitive  
Introduction  
for Novices in  
Cybersecurity.  
Arxiv.  
Retrieved  
December  
21,  
2025, from  
https://arxiv.org/html/2308.02785v2  
9. Mahto, D., & Yadav, D. K. (2017, January 1). RSA and ECC: A comparative analysis. ResearchGate.  
Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/322558426_RSA_and_ECC_A_comparative_analysis  
10. Olutola, A., & Matthew, O. (2023, January 1). Comparative Analysis of Encryption Algorithms.  
ResearchGate. Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/366889851_Comparative_Analysis_of_Encryption_Algorithms  
11. Paul, J. (2025, July 1). USE OF DES, AES, AND RSA IN SECURE COMMUNICATION PROTOCOLS  
(SSL/TLS, VPN, ETC.). ResearchGate. Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/393976056_USE_OF_DES_AES_AND_RSA_IN_SECURE_C  
OMMUNICATION_PROTOCOLS_SSLTLS_VPN_ETC  
12. PITE, J., ZHONG, Y., & ZHU, H. (1977). THE RSA CRYPTOSYSTEM. Massachusetts Institute of  
Technology. Retrieved December 21, 2025, from  
https://math.mit.edu/research/highschool/primes/circle/documents/2024/Honglin.pdf  
13. Rivest, R., Shamir, A., & Adelman, L. (2021, February 1). A Method for Obtaining Digital Signatures and  
Public-Key  
Cryptosystems  
(1978).  
ResearchGate.  
Retrieved  
December  
21,  
2025,  
from  
https://www.researchgate.net/publication/349073609_A_Method_for_Obtaining_Digital_Signatures_and  
_Public-Key_Cryptosystems_1978  
14. Shor, P. W. (2006, June 1). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms  
on  
https://www.researchgate.net/publication/280821144_Polynomial-  
Time_Algorithms_for_Prime_Factorization_and_Discrete_Logarithms_on_a_Quantum_Computer  
a
Quantum  
Computer.  
ResearchGate.  
Retrieved  
December  
21,  
2025,  
from  
15. Trappe, W., & Washington, L. C. (2006). Introduction to Cryptography with Coding Theory (2nd ed.).  
Prentice Hall.  
16. Van Meter, R., & Itoh, K. M. (2004, September 1). Fast Quantum Modular Exponentiation. ResearchGate.  
Retrieved December 21, 2025, from  
https://www.researchgate.net/publication/2193429_Fast_Quantum_Modular_Exponentiation  
www.ijltemas.in  
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