Finite rings and fields are the building blocks of algorithms in cryptography and error-correcting codes. The intractable problems in such structures allow us to build several public-key cryptographic algorithms including RSA, Diffie-Hellman and ElGamal. Also since the invention of full-homomorphic encryption algorithms rings based on polynomial rings have become very important. In this talk, I will review arithmetic algorithms in number rings and polynomial rings used in cryptography.