Here are some references with applications. The first one is not about algebraic number theory but deserves to be consulted by anyone who wants to find a list of ways that simple concepts in number theory have a quasi-wide range of practical uses. The other second and third references are uses of actual algebraic number theory.
Schroeder's "Number Theory in Science and Communication" has many examples of ways in which elementary number theory can be applied (not just to cryptography).
Mollin's book "Algebraic Number Theory" is a very basic course and each chapter ends with an application of number rings in the direction of primality testing or integer factorization.
Chapter 16 of Washington's book on cyclotomic fields (2nd ed.) starts with a section on the use of Jacobi sums in primality testing.
Suitable references to the role of primality testing and integer factorization in cryptography can convey the practical interest in such methods.
