What is a public key?

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, Introduction to public-key cryptography, Number theory for PKC – Euclidean Algorithm, Euler’s Phi Function and Euler’s Theorem

In the realm of cybersecurity, particularly within the scope of classical cryptography fundamentals, the concept of a public key is central to the understanding and implementation of public-key cryptography (PKC). Public-key cryptography, also known as asymmetric cryptography, is a cryptographic system that employs pairs of keys: public keys, which may be disseminated widely, and private keys, which are known only to the owner. This system underpins many of the secure communications protocols used today.

A public key, as the term suggests, is a cryptographic key that can be shared openly without compromising security. It is used in conjunction with a private key within an asymmetric encryption framework. The public key enables anyone to encrypt a message intended for a specific recipient, who can then decrypt the message using their corresponding private key. This separation of keys into public and private components is what differentiates public-key cryptography from symmetric cryptography, where the same key is used for both encryption and decryption.

The mathematical foundation of public-key cryptography relies heavily on number theory, particularly on the computational difficulty of certain mathematical problems. One of the most well-known public-key cryptographic systems is the RSA algorithm, named after its inventors Rivest, Shamir, and Adleman. The security of RSA is based on the difficulty of factoring large composite numbers, a problem for which no efficient solution is currently known.

Mathematical Foundations

Understanding the role of a public key requires delving into the mathematical principles that underpin public-key cryptography. Three key concepts are particularly relevant: the Euclidean Algorithm, Euler’s Phi Function, and Euler’s Theorem.

Euclidean Algorithm

The Euclidean Algorithm is a method for finding the greatest common divisor (GCD) of two integers. It is based on the principle that the GCD of two numbers also divides their difference. This algorithm is important in public-key cryptography for ensuring that certain numbers used in key generation are coprime, meaning their GCD is 1.

For example, to find the GCD of two integers, say 56 and 15, the Euclidean Algorithm proceeds as follows:
1. Divide 56 by 15, yielding a quotient of 3 and a remainder of 11 (56 = 15 * 3 + 11).
2. Divide 15 by 11, yielding a quotient of 1 and a remainder of 4 (15 = 11 * 1 + 4).
3. Divide 11 by 4, yielding a quotient of 2 and a remainder of 3 (11 = 4 * 2 + 3).
4. Divide 4 by 3, yielding a quotient of 1 and a remainder of 1 (4 = 3 * 1 + 1).
5. Divide 3 by 1, yielding a quotient of 3 and a remainder of 0 (3 = 1 * 3 + 0).

When the remainder reaches 0, the divisor at that step is the GCD. In this case, the GCD of 56 and 15 is 1, indicating that they are coprime.

Euler’s Phi Function

Euler’s Phi Function (φ), also known as Euler’s Totient Function, is a function that counts the number of integers up to a given integer n that are coprime with n. This function is vital in public-key cryptography, particularly in the RSA algorithm, for determining the totient of a product of two prime numbers.

For a prime number p, φ(p) is simply p-1, since all numbers less than p are coprime with p. For two distinct prime numbers p and q, the totient function of their product n = pq is given by:

    \[ φ(n) = φ(pq) = (p-1)(q-1) \]

This property enables the calculation of the totient of large composite numbers, which is a important step in generating the keys for RSA encryption.

Euler’s Theorem

Euler’s Theorem states that for any integer a and n such that a and n are coprime:

    \[ a^{φ(n)} \equiv 1 \pmod{n} \]

This theorem is fundamental in the RSA algorithm, as it provides the mathematical basis for the encryption and decryption processes. Specifically, it ensures that raising a number to the power of the totient function modulo n results in 1, which is a property exploited in the RSA key generation and encryption/decryption algorithms.

RSA Algorithm and Public Key Generation

The RSA algorithm exemplifies the practical application of these mathematical principles. The process of generating a public key in RSA involves several steps:

1. Key Generation:
– Select two large prime numbers, p and q.
– Compute n = pq. The modulus n is used as part of both the public and private keys.
– Calculate the totient φ(n) = (p-1)(q-1).
– Choose an integer e such that 1 < e < φ(n) and e is coprime with φ(n). The integer e is the public exponent.
– Determine d as the modular multiplicative inverse of e modulo φ(n), i.e., d satisfies ed \equiv 1 \pmod{φ(n)}. The integer d is the private exponent.

2. Public Key:
– The public key is composed of the pair (n, e). This key can be distributed openly and used by anyone to encrypt messages intended for the key owner.

3. Private Key:
– The private key is composed of the pair (n, d). This key is kept secret by the owner and used to decrypt messages encrypted with the corresponding public key.

Example of RSA Encryption and Decryption

To illustrate the use of a public key in RSA encryption and decryption, consider a simple example:

1. Key Generation:
– Choose primes p = 61 and q = 53.
– Compute n = pq = 61 \times 53 = 3233.
– Calculate φ(n) = (61-1)(53-1) = 60 \times 52 = 3120.
– Choose e = 17, which is coprime with 3120.
– Compute d, the modular inverse of 17 modulo 3120. Using the Extended Euclidean Algorithm, we find d = 2753.

2. Public Key:
– The public key is (n, e) = (3233, 17).

3. Private Key:
– The private key is (n, d) = (3233, 2753).

4. Encryption:
– To encrypt a message m, convert m to an integer (e.g., m = 65).
– Compute the ciphertext c using the public key: c = m^e \mod n = 65^{17} \mod 3233 = 2790.

5. Decryption:
– To decrypt the ciphertext c, compute the original message m using the private key: m = c^d \mod n = 2790^{2753} \mod 3233 = 65.

The example demonstrates how the public key (n, e) can be used to encrypt a message, while the private key (n, d) is required to decrypt it. The security of the RSA algorithm relies on the computational difficulty of factoring the large number n into its prime factors p and q.

Applications and Security

Public keys are integral to various cryptographic protocols and applications. They are used in secure communications, digital signatures, and key exchange mechanisms. For instance, in the Secure Sockets Layer (SSL) and its successor, Transport Layer Security (TLS), public keys are used to establish secure connections over the internet. Digital signatures use public keys to verify the authenticity and integrity of messages and documents.

The security of public-key cryptography depends on the difficulty of solving specific mathematical problems. In the case of RSA, the security is based on the difficulty of factoring large composite numbers. Other public-key cryptographic systems, such as Elliptic Curve Cryptography (ECC), rely on the hardness of the elliptic curve discrete logarithm problem.

The concept of a public key is foundational to public-key cryptography. It enables secure communication by allowing anyone to encrypt messages that only the intended recipient can decrypt using their private key. The mathematical principles underlying public-key cryptography, including the Euclidean Algorithm, Euler’s Phi Function, and Euler’s Theorem, provide the necessary framework for key generation and encryption/decryption processes. Public-key cryptography is widely used in various applications to ensure the security and integrity of data in the digital world.

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