Hello ninjas, in this blog, we will discuss Galois Fields, also known as Finite Fields.

This is a mathematical concept that is essential to understand Cryptography. Fields are a definite set of numbers in a given range that remain in the same group under a given binary operation. For instance, F is a field having elements (a, b, c, d â€¦) for binary operations â€˜+â€™ and â€˜*.â€™ Then, the addition(+) and multiplication(*) of any two numbers from the set (a, b, c, dâ€¦) will always give a result that exists in the set (a, b, c, d).

This concept is essential for Cryptography as we have to deal with limited numbers where each set contains massive numbers.

Fields

A field is a set with two Binary operations, + and *. A field F must have the following attributes:

(F, +) must be an Abelian Group. This means that it should satisfy the following properties:

Closure: a âˆ— b âˆˆ F âˆ€ a,b âˆˆ F

Associative: (aâˆ—b)âˆ—c =aâˆ—(bâˆ—c) âˆ€ a,b,c âˆˆ F

Identity: aâˆ—e=eâˆ—a=a âˆ€ a âˆˆ F (where e=identity)

Inverse: a^{-1}âˆ—a=aâˆ—a^{-1}=e

Commutative: aâˆ—b=bâˆ—a âˆ€ a,b âˆˆ F

(F-{0}, âˆ—) must be an Abelian group

The distributive law should hold: aâˆ—(b+c)= (aâˆ—b) + (aâˆ—c)

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Galois Fields or Finite Fields

Finite fields or Galois fields satisfy all the conditions of a field and are written as GF(p^{m}) where p is a prime number and m is any number that we chose. Hence, a finite field has p*m elements. If m=1, then the field can be classified as Prime Field. If m>1, we obtain Extension fields.

Prime finite fields

Prime finite fields are denoted by GF(p) and have 1^{p} elements.

A prime field is the minor subfield of the field.

It can perform all the arithmetic functions with given binary operators and follow all the rules. The only exception sometimes is with â€˜0â€™.

At the same time, referring to the prime field as F we mean that 0 is not included.

Extension Fields

Extension fields are certainly more complex than prime fields as they have m>1.

This results in polynomials being a part of the subfield.

For m=2, GF=(p^{2})

And for m=n , GF=(p^{n})

Please note that the polynomials encountered in the GF=(p^{n}) have all their coefficients expressed in GF=(p). This means that for reducible polynomials, the modulus of the elements belongs to the original prime field set.

Let's take the case where m=3 and p=2.

This will give in a total of 8 parts.

And the polynomial will take the form:

a_{1}x_{2}+a_{2}x+a_{3}

Here, a1,a2, and a3 can only be â€˜0â€™ or â€˜1â€™

Hence, the GF(2^{3})={0, 1, x, x^{2}, x+1, x^{2}+1, x^{2}+x,x^{2}+x+1}

We can perform arithmetic operations on this set and modulus the result to simplify.

Our original set GF(2)={1,0}

Let's take two elements x^{2} and x^{2}+x+1 and add them

This equals, (1+1)x^{2}+x+1 (1)

Now, since (1+1)%2=0

(1) Becomes: x+1

Now, x can be 0 or 1

If x=1, (1+1)%2=0 which is contained in the original set GF(2)

If x=0, 1+0= 1, which is again contained in the original set GF(2)

Therefore all the reducible polynomial forms are always contained in Prime subfields.

All the irreducible polynomials can be converted into reducible forms.

Binary Fields

Binary fields are binary-number-filled fields. They may refer to a field that may store any type of information, such as data, text, graphics, pictures, audio, video, or binary digits for mathematical purposes.

Frequently asked questions

What is an Abelian group?

An abelian group is a commutative group that satisfies all the conditions of a group.

What is a Ring?

A ring is a set R with two binary operations, + and *. Here, (R,+) is an Abelian group, and (R,*) is a semigroup.

What is the use of Finite fields in cryptography?

In cryptography, we generally deal with a massive set of numbers. Hence, the concept of field is valid as any operation on elements of a field will always result in a component contained in the field.

What is semigroup?

A semigroup follows closure and associativity properties.

Conclusion

We hope this blog successfully helped you understand the concept of Galois fields.