In addition to encryption, prime numbers are also used for error correction, such as 929 which is used to create a finite field (numbers modulo 929) used for the error correction on PDF417 bar codes. However, most error correction schemes use finite fields based on "prime" polynomials that use 1 bit coefficients (so add and subtract effectively become xor). AES encryption uses Rijndael S-box on 8 bit bytes, finding the multiplicative inverse of that byte modulo x^8 + x^4 + x^3 + x + 1 (hex 11B) (division by a 9 bit polynomial produces an 8 bit remainder). For a software implementation, typically a 256 byte lookup table is used. However in hardware, which may include 10 or more S-box'es in parallel, there's been a lot of effort made to reduce the gate count well below the hardware equivalent of a lookup table, using some interesting properties of fields based on 1 bit coefficients, in this case being able to map an 8 bit field into two 4 bit fields and then into four 2 bit fields. There are a lot (but not anastronomically large number) of possible mappings, and a brute force approach to simply try them all and select the one that needs the fewest number of gates has been done.
The point here is that prime numbers and finite field math at one time were just exercises in higher level mathematics, but once there was a commercial application for this stuff, a lot more people and more effort became involved, and the was significant advancement in the commercial aspect for this branch of mathematics.