Can a field be considered as a set of numbers in which one can add, subtract and multiple but not divide?
In the field of cybersecurity, particularly in classical cryptography, understanding the concept of fields is important for comprehending the inner workings of cryptographic algorithms such as the AES block cipher cryptosystem.
While the assertion that the field be considered as a set of numbers in which one can add, subtract and multiple but not divide can be considered as a simplified definition, it should be noted that it is not entirely accurate when applied to the context of Galois Fields (GF) used in AES.
In the realm of AES, Galois Fields play a fundamental role in the construction and operation of the cipher. Specifically, AES operates over a finite field, denoted as GF(2^8), which consists of 256 elements. Each element in this field can be represented as an 8-bit binary number.
To fully grasp the concept of a Galois Field, it is essential to understand the properties and operations associated with it. In GF(2^8), the addition operation follows the XOR (exclusive OR) operation, which is equivalent to bitwise addition without carrying. For example, if we consider two elements, a = 10101110 and b = 11001010, their sum in GF(2^8) would be a XOR b = 01100100.
Similarly, the multiplication operation in GF(2^8) is performed using a polynomial representation known as the irreducible polynomial. In AES, the irreducible polynomial used is x^8 + x^4 + x^3 + x + 1, which can be represented as 0x1B in hexadecimal notation. Multiplication in GF(2^8) is carried out using this polynomial and the modulo 2 reduction. For instance, if we multiply a = 10101110 and b = 11001010 in GF(2^8), the result would be obtained by multiplying the polynomials and reducing modulo 2 with the irreducible polynomial:
a * b = (x^7 + x^5 + x^4 + x^1) * (x^7 + x^6 + x^3 + x^1)
= x^14 + x^13 + x^10 + x^8 + x^12 + x^11 + x^8 + x^6 + x^11 + x^10 + x^7 + x^5 + x^9 + x^8 + x^5 + x^3
= x^14 + x^13 + x^12 + x^11 + x^9 + x^7 + x^6 + x^5 + x^3 + x^1
= 11110010
The division operation is not applicable in GF(2^8). Instead, the concept of division is replaced by the multiplication with the multiplicative inverse. The multiplicative inverse of an element a is the element b such that a * b = 1. In GF(2^8), the multiplicative inverse of an element can be calculated using the Extended Euclidean Algorithm.
While the simplified definition that the field be considered as a set of numbers in which one can add, subtract and multiple but not divide captures the basic idea of a field, it does not accurately represent all the operations and properties of Galois Fields used in AES. Understanding Galois Fields and their operations, including addition and multiplication, is essential for comprehending the inner workings of AES and other cryptographic algorithms.
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