Public Key Cryptography

Understanding Asymmetric Public Key Cryptography

Asymmetric Public Key Cryptography relies on the mathematical relationship between two keys, which is designed in such a way that, knowing the public key, it becomes computationally impractical to deduce the private key. This principle ensures that messages encrypted with the public key can only be decrypted by the corresponding private key, providing a secure means of communication.

How Does Public Key Cryptography Work?

Public Key Cryptography operates under a simple yet profound principle. When a message is sent, it is encrypted with the recipient's public key. This encrypted message can only be decrypted by the matching private key, ensuring that only the intended recipient, who possesses the private key, can access the message. This dual-key system underpins numerous security protocols across the internet, including secure email communication, digital signatures, and the Secure Sockets Layer (SSL) protocol for secure web browsing.

Public Key Cryptography Example

Imagine Alice wants to send a confidential message to Bob. Alice encrypts the message using Bob's public key, which she can obtain from a public directory or directly from Bob. Once encrypted, the message is sent over the internet. Upon receiving it, Bob uses his private key, which only he possesses, to decrypt the message and read its contents. This process ensures that even if the message were intercepted, it would remain secure and unreadable to anyone except Bob.

 
Message before encryption: "Hello, Bob!" 
Encryption with Bob's public key: "89%gHG#...Encrypted Message...KLJ^&5" 
Decryption with Bob's private key: "Hello, Bob!"

Difference Between Public Key and Private Key Cryptography

In the realm of digital security, understanding the difference between public key and private key cryptography is crucial. Both play pivotal roles in securing communications, but they operate on distinct principles. Let's delve into their unique features and how they contribute differently to encryption and decryption processes.

Exploring the Distinct Features

Public Key and Private Key Cryptography are two sides of the encryption coin, each with its own set of characteristics. Public Key Cryptography, also known as asymmetric encryption, utilises two keys: a public key for encrypting messages and a private key for decrypting them. Conversely, Private Key Cryptography, also referred to as symmetric encryption, employs a single key for both encryption and decryption.The table below outlines some of the primary differences between the two methods:

The Roles in Encryption and Decryption

The roles of public and private keys in encryption and decryption highlight the fundamental differences between these two cryptographic systems. In Public Key Cryptography, the public key performs the encryption of data or messages destined for a specific recipient. The corresponding private key, securely held by the recipient, is then used to decrypt the information. This separation of keys enables secure, confidential communication between parties.In contrast, Private Key Cryptography relies on a single shared secret key for both encryption and decryption. This key must be securely exchanged between the communicating parties before it can be used, which presents a challenge in terms of securely distributing the key.The list below summarises the roles played by keys in each type of cryptography:

• In Public Key Cryptography, the public key encrypts data, and the private key decrypts it.
• In Private Key Cryptography, the same shared secret key is used for both encryption and decryption.

The Mathematical Foundation of Public Key Cryptography

At the heart of public key cryptography lies a robust mathematical foundation that enables secure communication over insecure channels. This cryptography form relies on mathematical principles and theorems, such as prime number factorisation and modular arithmetic, to create a secure method of exchanging keys. Understanding these underlying principles is essential for grasping how public key cryptography works.

Underlying Principles and Theorems

Public key cryptography is underpinned by two fundamental mathematical concepts: the computational difficulty of certain mathematical problems and modular arithmetic. These concepts ensure that, even though the public key is openly shared, deciphering the private key from it remains computationally impractical.One such problem is the factorisation of large prime numbers, a task that is easy to perform in one direction and exceedingly difficult to reverse without the private key. Modular arithmetic, on the other hand, facilitates operations within a finite set of numbers and plays a critical role in the encryption and decryption processes.

Let p = 61 and q = 53,
then n = p * q = 3233,
where the factorisation of 3233 back into 61 and 53 is non-trivial without knowing p or q ahead of time.

Public Key Cryptography RSA: A Closer Look

The RSA algorithm, named after Ron Rivest, Adi Shamir, and Leonard Adleman, is one of the first and most widely used public key cryptosystems. RSA operates on the principle that while it is easy to multiply two large prime numbers, factorising their product is difficult.The security of the RSA algorithm is based on the practical difficulty of factorising the product of two large prime numbers, known as the modulus. Let’s detail its working process.

Key Generation:
Choose two prime numbers p and q.
Compute n = p * q.
Choose an encryption key e such that 1 < e < \(\phi(n)\) and e is coprime to \(\phi(n)\), where \(\phi\) is Euler's totient function.
Compute the decryption key d, the modular multiplicative inverse of e modulo \(\phi(n)\).
(public key, private key) = ((e, n), (d, n)).

Implementing Public Key Cryptography in Real Life

In today's digital age, Public Key Cryptography stands as a cornerstone of internet security, facilitating safe and secure communication across an array of platforms. Its implementation in various real-life applications underscores its significance in protecting data and ensuring privacy. Let's explore how this cryptographic method is applied across different domains.

Applications of Public Key Cryptography

Public Key Cryptography serves a variety of functions beyond encrypting messages. Its applications range from digital signatures to secure key exchange mechanisms over the internet. These applications bolster the security of online transactions, email communications, and even the establishment of secure connections via virtual private networks (VPNs).Major areas where Public Key Cryptography is implemented include:

• Secure email communication
• Authentication and digital signatures
• Secure file transfer protocols such as SFTP
• HTTPS protocols for secure browsing
• Blockchain technologies and cryptocurrencies

Public Key Cryptography for Secure Communication

One of the primary uses of Public Key Cryptography is to ensure secure communication over potentially insecure networks like the internet. This method enables the exchange of encrypted messages between entities, ensuring that only the intended recipient can decrypt and read them.

 Message: "Confidential data" 
Encrypted with Bob's public key: "74js...Encrypted Data...kjH7" 
Decrypted with Bob's private key: "Confidential data"