Private Key Generation in RSA Encryption
Learn how to calculate the private key (d) using the modular multiplicative inverse and understand the Extended Euclidean Algorithm.
Understanding the Private Key
The private key (d) is a critical component of RSA encryption that allows the decryption of messages. It is mathematically related to the public exponent (e) and must be kept secret.
Definition: The private key (d) is the modular multiplicative inverse of the public exponent (e) with respect to φ(n). This means that (d × e) mod φ(n) = 1.
Mathematical Relationship
The private key d is calculated using the formula: d ≡ e⁻¹ (mod φ(n)), which means d is the modular multiplicative inverse of e modulo φ(n).
Finding d in RSA
To find the private key d, we need to solve the congruence equation: d⋅e ≡ 1 (mod φ(n)). This means finding a value d such that when multiplied by e and taken modulo φ(n), equals 1.
Example:
Given e = 3 and φ(n) = 40, find d where:
3⋅d ≡ 1 (mod 40)
Testing values systematically:
- When d = 1: 3⋅1 ≡ 3 (mod 40) ≠ 1
- When d = 2: 3⋅2 ≡ 6 (mod 40) ≠ 1
- When d = 3: 3⋅3 ≡ 9 (mod 40) ≠ 1
- ...
- When d = 27: 3⋅27 ≡ 1 (mod 40) ✓
Therefore, d = 27 satisfies the congruence equation 3⋅d ≡ 1 (mod 40).
Using the Extended Euclidean Algorithm
Rather than testing values manually, we can efficiently find d using the Extended Euclidean Algorithm. This algorithm helps us solve the congruence equation d⋅e ≡ 1 (mod φ(n)).
Mathematical Foundation:
The congruence equation d⋅e ≡ 1 (mod φ(n)) can be rewritten as:
d⋅e - k⋅φ(n) = 1
where k is some integer. This is exactly what the Extended Euclidean Algorithm solves.
For our example where e = 3 and φ(n) = 40:
1. First, apply the algorithm:
40 = 3 × 13 + 1
2. Working backwards:
1 = 40 - 3 × 13
1 = 40 - 3 × 13
Therefore: d = -13 ≡ 27 (mod 40)
This confirms our earlier result that d = 27 is the solution to 3⋅d ≡ 1 (mod 40).
Extended Euclidean Algorithm Step-by-Step
The Extended Euclidean Algorithm calculates the modular multiplicative inverse through a series of divisions and substitutions.
Algorithm Steps:
- Start with φ(n) and e
- Apply the Euclidean Algorithm to find gcd(φ(n), e)
- Work backwards through the equations to find the coefficients
- Extract the coefficient of e, which is the modular multiplicative inverse
- Adjust the result if necessary to ensure it's positive
Example with e = 7, φ(n) = 40:
Step 1: Apply the Euclidean Algorithm
40 = 7 × 5 + 5
7 = 5 × 1 + 2
5 = 2 × 2 + 1
2 = 1 × 2 + 0
Step 2: Work backwards
1 = 5 - 2 × 2
1 = 5 - 2 × (7 - 5 × 1)
1 = 5 - 2 × 7 + 2 × 5
1 = 3 × 5 - 2 × 7
1 = 3 × (40 - 7 × 5) - 2 × 7
1 = 3 × 40 - 3 × 7 × 5 - 2 × 7
1 = 3 × 40 - 7 × (3 × 5 + 2)
1 = 3 × 40 - 7 × 17
Therefore, d = -17 ≡ 23 (mod 40)
Interactive Visualization
Enter values in the calculator below to see the Extended Euclidean Algorithm in action
Private Key Calculator Tool
Calculate the private key (d) given the public exponent (e) and φ(n).
Enter values to calculate the private key
Private Key Verification Tool
Verify that your calculated private key (d) works correctly with the public exponent (e).
Practice Exercises
Test your understanding of private key generation in RSA encryption.
Knowledge Check
1. What is the mathematical relationship between the private key (d) and the public exponent (e)?
This is not the correct relationship between d and e. Think about how d and e work together in modular arithmetic to produce 1 when multiplied.
2. Which algorithm is commonly used to calculate the modular multiplicative inverse?
3. Why must the private key be kept secret in RSA encryption?