Coding theory and cryptography have attracted a lot of attention during the last few years because of their influential role in communication. Coding theory has recently helped to relieve demand in computational tasks, such as distributed storage, matrix multiplication, and 5G communication. Cryptography has been threatened by the development of quantum computers. This talk presents a wealth of interconnections between pure mathematics, such as algebra, geometry, and combinatorics, and the areas of coding theory and cryptography. We will see how properties of vanishing ideals (together with Grobner bases, Hilbert functions, footprints, degrees, being decreasing, or being Gorenstein); the study of curves, including Rieman-Roch space associated with special divisors; and the counting of lattice points, such as integer points inside of certain polyhedral, play roles in the coding theory and cryptography tasks mentioned above.