Find Inverse of e mod n Calculator

Calculating the inverse of e modulo n is a common task in cryptography and number theory. The inverse of e mod n is a number x such that (e * x) % n = 1.

Applications of finding the inverse of e mod n

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

• Key exchange: The Diffie-Hellman key exchange protocol uses the inverse of e mod n to establish a shared secret key between two parties.
• Digital signatures: The RSA digital signature algorithm uses the inverse of e mod n to verify signatures.
• Encryption: The ElGamal encryption algorithm uses the inverse of e mod n to encrypt messages.

How to find the inverse of e mod n

There are a few different algorithms for finding the inverse of e mod n. One common algorithm is the extended Euclidean algorithm.

The extended Euclidean algorithm works by finding the greatest common divisor (GCD) of e and n. Once the GCD is found, the inverse of e mod n can be calculated using the following formula:

x = (inverse_of_e mod n) = (1 / GCD) * (y mod n)

where y is the remainder of e divided by the GCD.

Common mistakes to avoid when finding the inverse of e mod n

There are a few common mistakes that people make when finding the inverse of e mod n. These mistakes include:

• Not checking if e and n are coprime. The inverse of e mod n only exists if e and n are coprime.
• Using the wrong algorithm. The extended Euclidean algorithm is the most efficient algorithm for finding the inverse of e mod n.
• Making arithmetic errors. It is important to be careful when performing the arithmetic operations in the extended Euclidean algorithm.

Step-by-step approach to finding the inverse of e mod n

1. Check if e and n are coprime. This can be done using the Euclidean algorithm.
2. Find the GCD of e and n using the extended Euclidean algorithm.
3. Calculate the inverse of e mod n using the formula:

x = (inverse_of_e mod n) = (1 / GCD) * (y mod n)

where y is the remainder of e divided by the GCD.

Conclusion

Finding the inverse of e mod n is a common task in cryptography and number theory. There are a few different algorithms for finding the inverse of e mod n, but the extended Euclidean algorithm is the most efficient. It is important to be careful when performing the arithmetic operations in the extended Euclidean algorithm to avoid making mistakes.