How to Set Up an RSA Key Pair for Public-Key Encryption

Introduction to RSA Key Pair Setup

Setting up an RSA key pair is a foundational skill in cybersecurity, essential for enabling secure, encrypted communication. This guide will walk you through each step of the RSA setup process, drawing from Martin’s Chapter 5: Public-key encryption, Section 5.2.1 to provide a solid foundation. This tutorial also considers various learning approaches to help you understand public-key encryption, from quick overviews to in-depth exploration.

What is an RSA Key Pair?

An RSA key pair includes:

• Public Key: Shared openly for encrypting data.
• Private Key: Kept secret for decrypting data.

Together, these keys enable secure communication by allowing encrypted data to be exchanged safely over public channels.

Key Parts of the RSA Key Pair

• n (product of two prime numbers): Used in both the public and private keys.
• e (public exponent): A value that works with n to form the public key.
• d (private exponent): Used to decrypt data and is kept confidential.

Step-by-Step Guide to Setting Up an RSA Key Pair

Step 1: Generate Two Large Primes (p and q)

Start by selecting two large prime numbers, p and q. In RSA encryption, these primes should ideally be at least 1,024 bits long for adequate security, with larger bit sizes like 2,048 bits often recommended.

1. Generate p and q, ensuring they are large prime numbers.
2. Multiply them to calculate n = p × q. The product n will be part of the public key.

Example: Let p = 47 and q = 59, so n = 47 × 59 = 2,773.

Step 2: Select a Public Exponent (e)

Next, choose e, an integer that meets specific properties for compatibility with RSA:

• e must be co-prime with (p−1)×(q−1)(p – 1) \times (q – 1)(p−1)×(q−1) (no shared factors except 1).
• Common values for e are 3 or 65537 due to efficiency in encryption.

Example: Let’s select e = 17, as it satisfies the required properties.

Step 3: Formulate the Public Key

The public key is composed of n and e, both of which are safe to share publicly. With this key, anyone can encrypt messages meant only for you.

Example: The public key is (2,773,17)(2,773, 17)(2,773,17).

Step 4: Calculate the Private Key (d)

To complete the RSA setup, calculate the private key d, which allows you to decrypt messages encrypted with the public key. Use the Extended Euclidean Algorithm to find d, satisfying the equation:d×e≡1 (mod (p−1)×(q−1))

This means d is the multiplicative inverse of e in modulo arithmetic, and only you should have access to it.

Example: Using p = 47, q = 59, and e = 17, you can calculate d = 157.

Example RSA Key Pair Setup

This example uses small numbers to simplify understanding (not recommended for actual encryption):

1. Generate primes: p = 47, q = 59.
2. Calculate n: n=p×q=2,773.
3. Choose e: e=17e = 17e=17.
4. Public Key: (2,773, 17).
5. Calculate d: d=157d = 157d=157.
6. Private Key: d = 157.

In this setup:

• Public Key: (2,773, 17)
• Private Key: 157 (kept confidential)

Note on Practical Use

In real scenarios, p and q must be much larger (1,024 or 2,048 bits) to secure against factorization attacks, as smaller values are easily broken.

Studying RSA Setup: Options for Different Learning Styles

Based on this week’s material, here are two approaches to studying RSA:

1. Overview Approach: Focus on key steps (finding p, q, n, e, and d) without delving into the math, useful for those who prefer a big-picture view.
2. In-Depth Exploration: For those comfortable with math, explore the Extended Euclidean Algorithm and number theory concepts for deeper insights.

For detailed explanations and additional examples, refer to Martin’s Chapter 5: Public-key encryption, Section 5.2.1.