Cryptography: Why is Zn not a Field?

•The main reason for why, in general, Zn is only a commutative ring and not a finite field is because not every element in Zn is guaranteed to have a multiplicative inverse.

•In particular, as shown before, an element a of Zn does not have a multiplicative inverse if a is not relatively prime to the modulus n.

•What if we choose the modulus n to be a prime number?

–A prime number has only two divisors, one and itself.

Cryptography: Finite Field

•The stepping stones to understanding the concept of a finite field are:

1.Set

2.Group

3.abelian group

4.Ring

5.commutative ring

6.integral domain

7.field 

 

Groups, Rings and Field

Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra.

•In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set.

•These operations are subject to specific rules, which define the nature of the set.

•By convention, the notation for the two principal classes of operations on set elements is usually the same as the notation for addition and multiplication on ordinary numbers.

•However, it is important to note that, in abstract algebra, we are not limited to ordinary arithmetical operations.