Significance of Extended Euclidean Algorithm in Cryptography
The significance of the EEA is that it gives an efficient way to compute modular inverses. Since computing modular inverses is an extremely common task in practical cryptography (among many others, you need it to derive the decryption exponent of an RSA keypair from the encryption exponent), the fact that it can be performed efficiently, as demonstrated by the existence of the EEA, is fundamental. If it couldn't, many of the cryptographic schemes in use today just wouldn't work.