Why do we use Group, Ring and Fields in Information Security?

Groups, rings, and fields are the important elements of a branch of mathematics called as abstract algebra, or modern algebra. In abstract algebra, it is concerned with sets on whose elements and it can operate algebraically; that is, it can combine two elements of the set, perhaps in multiple ways, and it can obtain a third element of the set.

Group

A group (G) is indicated by {G,∙}. It is a group of elements with a binary operation ′ ∙ ′ that satisfies four properties. The properties of Group are as follows −

• Closure − If a and b are elements of G, therefore c = a ∙ b is also an element of set G. This can define that the result of using the operations on any two elements in the set is another element in the set.

• Associativity − If a, b, and c are element of G, therefore (a ∙ b) ∙ c = a ∙ (b ∙ c), means it does not substance in which order it can use the operations on higher than two elements.

• Identity − For all a in G, there occur an element e in G including e ∙ a = a ∙ e = a.

• Inverse − For each a in G, there occur an element a’ known as the inverse of a such that a ∙ a′ = a′ ∙ a = e.

A group is an abelian group if it satisfies the following four properties more one additional property of commutativity.

Commutativity − For all a and b in G, we have a ∙ b = b ∙ a.

Ring − A ring R is indicated by {R, +, x}. It is a set of elements with two binary operations, known as addition and multiplication including for all a, b, c in R the following axioms are kept −

• R is an abelian group regarding addition that is R satisfies properties A1 through A5. In the method of additive group, it indicates the identity element as 0 and the inverse of a as − a.

• (M1): Closure under multiplication − If and b belong to R, then ab is also in R.

• (M2): Associativity of Multiplication − a(bc)=(ab)c for all a, b, c in R.

• (M3): Distributive Laws −

     a(b+c)=ab + ac for all a, b, c in R

     (a+b)c=ac+bc for all a, b, c in R

• (M4): Commutative of Multiplication − ab=ba for all a, b in R.

• (M5): Multiplicative identity − There is an element 1 in R including a1=1a for all a in R.

• (M6): No zero divisors − If a, b in R and ab = 0, therefore a = 0 or b = 0.

Field − A field F is indicated by {F, +, x}. It is a set of elements with two binary operations known as addition and multiplication, including for all a, b, c in F the following axioms are kept −

• F1 is an integer domain that is F satisfies axioms A1 through A5 and M1 through M6.

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