What is the difference between a ring-based cryptosystem and a group-based cryptosystem?
A ring is a mathematical object which has two algebraic operations, addition and multiplication. Examples of rings are the integers, the rational numbers, finite fields, and rings of polynomials. Groups, on the other hand, have only one operation. An important example of a group is the collection of non-zero elements in a finite field, where the one operation is multiplication. Another example is the set of points on a mathematical object called an elliptic curve. Aside from NTRUEncrypt, the three most popular pubic key cryptosystems in use today are RSA, El Gamal, and ECC (elliptic curve cryptosystem). All of these are group-based cryptosystems in the sense that the encryption/decryption processes use only one algebraic operation. Thus RSA uses multiplication in a finite ring (the integers modulo m), El Gamal uses multiplication in a finite field, and ECC uses elliptic curve addition. The NTRUEncrypt public key cryptosystem, by way of contrast, makes use of both the addition and the m
