How is the Extended Euclidean Algorithm utilized in encryption?

The Extended Euclidean Algorithm is a fundamental tool in number theory that plays a significant role in various encryption methods, particularly in public-key cryptography. Its primary application in this context is to find modular inverses, which are essential for algorithms like RSA.

In RSA, two large prime numbers are selected, and their product forms the modulus for both the public and private keys. The encryption and decryption processes involve exponentiation modulo this prime product. Specifically, for decryption to work correctly, it’s necessary to compute the modular inverse of the encryption exponent with respect to the Euler's totient of the modulus. The Extended Euclidean Algorithm efficiently determines this modular inverse, enabling the computation of the private key based on the public key.

The other options do not align with the practical use of the Extended Euclidean Algorithm in encryption. Generating random keys typically involves cryptographic secure pseudorandom number generators rather than number-theoretic algorithms. While encryption and decryption processes might use modular arithmetic, they do not directly rely on the Extended Euclidean Algorithm itself for real-time operations. Additionally, although digital signatures are important in cryptography, their creation and verification processes predominantly use hash functions and asymmetric key techniques rather than directly involving the Extended Euclidean Algorithm. Therefore,