Cryptography: Fields

•A field F, sometimes denoted by {F, +, x }, is a set of elements with two binary operations, called addition and multiplication, such that for all a, b, c in F the following axioms are obeyed.

•(A1–M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and M1 through M6.

•(M7) Multiplicative inverse: For each a in F, except 0, there is an element a-1 in F such that aa-1 = (a-1)a = 1.

•In essence, a field is a set in which we can do addition, subtraction, multiplication, and division without leaving the set. Division is defined with the following rule: a/b = a(b-1).

•Familiar examples of fields are the rational numbers, the real numbers, and the complex numbers.

–Set of all integers is not a field, because not every element of the set has a multiplicative inverse; in fact, only the elements 1 and –1 have multiplicative inverses in the integers.

•For every element a in F , except the element designated 0 (which is the identity element for the ’+’ operator), there must also exist in F its multiplicative inverse.

•Note again that a field has a multiplicative inverse for every element except the element that serves as the identity element for the group operator.

Fields – Positive and Negative Examples 

•The set of all real numbers under the operations of arithmetic addition and multiplication is a field.

•The set of all rational numbers under the operations of arithmetic addition and multiplication is a field.

•The set of all complex numbers under the operations of complex arithmetic addition and multiplication is a field.

•The set of all even integers, positive, negative, and zero, under the operations arithmetic addition and multiplication is NOT a field.

•The set of all integers under the operations of arithmetic addition and multiplication is NOT a field.