Actually AES, unlike more ad-hoc block ciphers, is based on the theory of finite fields, including GF(8) that is used for its non-linear component.
The theory of finite fields is based on the theory of prime numbers, because the finite fields are sets of residues modulo a prime number or modulo a power of a prime number.
The theory of finite fields is involved in the design of many other block cipher functions or secure hash functions and also in the design of the most important message-authentication methods, like GCM, which is used to authenticate this HTML page on the HN site.
So prime numbers are important in most cryptographic applications, not only in asymmetric cryptography, like Diffie-Hellman or RSA. Prime numbers are used in one way or another for the transmission of any HTTPS data packet, not only in the key establishment phase of a TLS connection.
> The theory of finite fields is based on the theory of prime numbers, because the finite fields are sets of residues modulo a prime number or modulo a power of a prime number.
It is note quite correct that the finite field of order p^k is the set of residues modulo p^k when k > 1. Instead this field is obtained as a splitting field of the field of order p (which is the set of residues mod p).
The theory of finite fields is based on the theory of prime numbers, because the finite fields are sets of residues modulo a prime number or modulo a power of a prime number.
The theory of finite fields is involved in the design of many other block cipher functions or secure hash functions and also in the design of the most important message-authentication methods, like GCM, which is used to authenticate this HTML page on the HN site.
So prime numbers are important in most cryptographic applications, not only in asymmetric cryptography, like Diffie-Hellman or RSA. Prime numbers are used in one way or another for the transmission of any HTTPS data packet, not only in the key establishment phase of a TLS connection.