Find Inverse of e mod n Calculator

Introduction

In modular arithmetic, the inverse of e mod n is the integer d such that:

e * d ≡ 1 (mod n)

This means that e and d are multiplicative inverses of each other modulo n.

How to Find the Inverse of e mod n

There are a few different algorithms that can be used to find the inverse of e mod n. One of the most common is the extended Euclidean algorithm.

The extended Euclidean algorithm works by repeatedly dividing e by n and using the remainders to calculate the coefficients of a linear combination of e and n that equals 1.

For example, to find the inverse of 3 mod 11, we would first divide 3 by 11 and get a remainder of 3. We would then divide 11 by 3 and get a remainder of 2. We would then divide 3 by 2 and get a remainder of 1.

The coefficients of the linear combination of 3 and 11 that equals 1 are -2 and 3, respectively. This means that:

-2 * 3 + 3 * 11 = 1

Therefore, the inverse of 3 mod 11 is -2.

Applications of the Inverse of e mod n

The inverse of e mod n has a number of applications in cryptography, including:

• Encryption: The RSA cryptosystem uses the inverse of e mod n to decrypt messages.
• Digital signatures: The DSA digital signature algorithm uses the inverse of e mod n to verify signatures.
• Key exchange: The Diffie-Hellman key exchange protocol uses the inverse of e mod n to generate shared secrets.

Usability concerns

Most of the time e will be a prime number and therefore we only need to care about finding an inverse for prime numbers. If n is a prime number, there exists a multiplicative inverse d such that e*d ≡ 1 (mod n). This is guaranteed by Fermat's little theorem that states a^n ≡ a (mod p) where p is prime and gcd(a, p) = 1.

Customers' point of view

• How can I use this calculator to find the inverse of e mod n?
• What are the applications of the inverse of e mod n?
• Are there any limitations to this calculator?

FAQs

1. What is the inverse of e mod n?

The inverse of e mod n is the integer d such that:

e * d ≡ 1 (mod n)

2. How do I find the inverse of e mod n?

There are a few different algorithms that can be used to find the inverse of e mod n. One of the most common is the extended Euclidean algorithm.

3. What are the applications of the inverse of e mod n?

The inverse of e mod n has a number of applications in cryptography, including:

• Encryption
• Digital signatures
• Key exchange

4. Are there any limitations to this calculator?

This calculator can only find the inverse of e mod n if n is a prime number.

5. What is the time complexity of this calculator?

The time complexity of this calculator is O(log n), where n is the modulus.

6. What is the space complexity of this calculator?

The space complexity of this calculator is O(1).

7. Can this calculator be used to find the inverse of any integer?

No, this calculator can only find the inverse of integers that are relatively prime to the modulus.

8. What is a relatively prime integer?

Two integers are relatively prime if they have no common factors other than 1.