cryptography deals with discrete structures and algebraic manipulations inside computer hardware and software
As well known, the field of cryptography deals with discrete structures and algebraic manipulations inside computer hardware and software. Finite structures are distinct features that are well studied with a wide range of useful properties which are highly applied in cryptography. Galois field or the limited field is the cornerstone when it comes to the understanding of cryptography. The finite field can be further defined as a set of figures which can be added, subtracted, multiplied, or divided and end up with a figure that exists within our number set. This makes finite field crucial in cryptography as we can be in a position to make use of an extensive collection of numbers.
An efficient computation in finite fields is one of the basic needs for the feasibility of the cryptographic system that is built on them as well as for purposes of cryptanalysis of cryptographic operations. The prime field FP is the most straightforward representation of the finite field, and it behaves as integer modulo P. the arithmetic’s in the prime area forms the base of all other algorithms in other finite fields. An example is arithmetic in the extension field Fpn can be solved through the use modulo built algorithms. In cryptography, the binary prime F2 that works similarly to the Boolean algebra forms the best tool for development and the analysis of symmetric ciphers. This is because most of the symmetric ciphers are easily described using the Boolean functions. The key F2n is used in public cryptography to implement efficient arithmetic, and in symmetric keys cryptography, they are used in designing cipher components.
Finite properties are used in cryptanalysis for application of traditional algebraic analysis to ciphers and to develop an improved means in generating and solving system equations that representing the actions of ciphers. In order to a finite field, the following properties are required. The dot symbol is used to describe the reminder after addition or multiplication of two elements. Another feature is closed, which states that any operation which is performed with features in a set should return an element that was in the original set.
The second property is associate, which states that if an algorithm has (a.b) .c, it can also be indicated as a. (b.c). Identity property says that there is always a neutral element in the algorithm such that a. 1=a. The inverse property states that within a set, there is an additional element: a.(a)^-1=1. Finally, the cumulative property says that the order in which the operation is arranged doesn’t matters such that a.b = b.a. The most crucial feature is the p^m, where p is a prime number, and m represents the chosen problem. For example, a finite field with eleven elements can be easily written as GF (11^1). On the other hand, a finite that has 256 elements, mostly the cipher elements, can be written as GF (2^8)..