Why in FF GF(8) irreducible polynomial itself does not belong to the same field?
In the field of classical cryptography, particularly in the context of the AES block cipher cryptosystem, the concept of Galois Fields (GF) plays a important role. Galois Fields are finite fields that are used for various operations in AES, such as multiplication and division. One important aspect of Galois Fields is the existence of irreducible polynomials, which are polynomials that cannot be factored into lower-degree polynomials over the same field.
In the case of GF(8), which is a Galois Field with 8 elements, the irreducible polynomial used is x^3 + x + 1. This polynomial is chosen because it satisfies the necessary properties for constructing the Galois Field. However, it is important to note that this irreducible polynomial itself does not belong to the same field.
To understand why the irreducible polynomial does not belong to GF(8), we need to consider the definition of a field. In mathematics, a field is a set of elements with two binary operations, usually addition and multiplication, that satisfy certain properties. One of these properties is closure, which means that the result of an operation on any two elements in the field is also an element of the field.
In the case of GF(8), the elements of the field are represented by polynomials of degree less than 3 with coefficients in GF(2), which is the binary field. The addition operation in GF(8) is performed modulo 2, which means that the coefficients of the polynomials are added modulo 2. The multiplication operation, on the other hand, is performed modulo the irreducible polynomial x^3 + x + 1.
Now, let's consider the irreducible polynomial x^3 + x + 1. If we try to add or multiply this polynomial with any other polynomial in GF(8), we will not obtain a polynomial that satisfies the closure property. For example, if we add x^3 + x + 1 with x^2, we get x^3 + x^2 + x + 1. This polynomial has a degree greater than 2, so it does not belong to GF(8).
Similarly, if we multiply x^3 + x + 1 with x^2, we get x^5 + x^3 + x^2. To bring this polynomial into GF(8), we need to perform the multiplication modulo x^3 + x + 1. However, since x^5 has a degree greater than 3, we cannot reduce it modulo x^3 + x + 1 to obtain a polynomial in GF(8).
Therefore, the irreducible polynomial x^3 + x + 1 does not belong to GF(8) because it does not satisfy the closure property of the field. It is important to understand this distinction because the irreducible polynomial is used in AES for various operations, but it is not an element of the field itself.
To summarize, in the context of GF(8) and the AES block cipher cryptosystem, the irreducible polynomial x^3 + x + 1 is used for constructing the Galois Field. However, this irreducible polynomial itself does not belong to GF(8) because it does not satisfy the closure property of the field. Understanding this distinction is important for correctly implementing and analyzing the AES algorithm.
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