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Calculate Private Key Rsa Given N and E

Reviewed by Calculator Editorial Team

In RSA cryptography, the private key is essential for decryption and digital signatures. This guide explains how to calculate the private key when you know the modulus (n) and public exponent (e).

Introduction

RSA (Rivest-Shamir-Adleman) is a widely used public-key cryptosystem. The private key in RSA is a crucial component that must be kept secret. When you have the modulus (n) and public exponent (e), you can calculate the private key (d) using the Extended Euclidean Algorithm.

The private key is calculated to satisfy the equation:

d ≡ e-1 mod φ(n)

Where φ(n) is Euler's totient function, which counts the integers up to n that are coprime with n.

Formula

The private key (d) is calculated using the following steps:

  1. Compute φ(n) = (p-1)(q-1) where p and q are the prime factors of n.
  2. Find the modular multiplicative inverse of e modulo φ(n). This is the private key d.

Note: For the calculation to work, e and φ(n) must be coprime (gcd(e, φ(n)) = 1).

Calculation Process

To calculate the private key (d) given n and e:

  1. Factorize n to find p and q (the prime factors of n).
  2. Calculate φ(n) = (p-1)(q-1).
  3. Use the Extended Euclidean Algorithm to find d such that e*d ≡ 1 mod φ(n).

The Extended Euclidean Algorithm finds integers x and y such that:

a*x + b*y = gcd(a,b)

In our case, we solve for x where a = e and b = φ(n).

Worked Example

Let's calculate the private key for n = 33 and e = 7.

  1. Factorize n = 33 to find p and q. The prime factors are 3 and 11.
  2. Calculate φ(n) = (3-1)(11-1) = 2*10 = 20.
  3. Find d such that 7*d ≡ 1 mod 20.
  4. Using the Extended Euclidean Algorithm:
    • 20 = 2*7 + 6
    • 7 = 1*6 + 1
    • 6 = 6*1 + 0

    Working backwards: 1 = 7 - 1*6 = 7 - 1*(20 - 2*7) = 3*7 - 1*20

    Thus, d = 3 is the modular inverse of 7 modulo 20.

The private key is d = 3.

FAQ

What is the difference between the public and private key in RSA?

The public key (n, e) is used for encryption and verification, while the private key (d) is used for decryption and signing. The private key must be kept secret.

Can I calculate the private key without knowing the prime factors of n?

No, you need to know the prime factors of n to calculate φ(n), which is required to find the private key.

What happens if e and φ(n) are not coprime?

The private key cannot be calculated if e and φ(n) are not coprime. You would need to choose a different public exponent.

Is RSA secure if the private key is known?

No, if the private key is compromised, the security of the RSA system is broken. The private key must be kept secret at all times.