Finite fields are like secret codes for math nerds. They're special number systems with a limited set of elements, always a power of a prime number. Understanding their structure and properties is key to unlocking their power in cryptography and coding theory.
In this section, we'll dive into the nitty-gritty of finite fields. We'll explore how they're built, their unique characteristics, and the cool tricks you can do with their arithmetic. Get ready to level up your finite field game!
Finite Fields and Characteristics
Definition and Properties
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A finite field is a field that contains a finite number of elements, also known as a Galois field
The order (number of elements) of a finite field is always a prime power, denoted as q=pn, where p is a prime number and n is a positive integer
There exists a unique finite field (up to isomorphism) for each prime power order q=pn, denoted as GF(q) or Fq
Characteristic of a Finite Field
The characteristic of a finite field is the smallest positive integer m such that m∗1=0, where 1 is the multiplicative identity of the field
In a finite field of order q=pn, the characteristic is always the prime p
For example, in the finite field GF(23), the characteristic is 2
This means that for any element a in GF(23), 2∗a=0
Constructing Finite Fields
Quotient Rings and Irreducible Polynomials
Finite fields can be constructed as quotient rings of polynomial rings over the integers modulo a prime
An is a polynomial that cannot be factored into the product of two non-constant polynomials over a given field
To construct a finite field of order q=pn, find an irreducible polynomial f(x) of degree n over the field Zp (integers modulo p)
Elements and Arithmetic Operations
The elements of the finite field are the polynomials of degree less than n with coefficients in Zp, with arithmetic operations performed modulo f(x)
For example, in GF(22) constructed using the irreducible polynomial f(x)=x2+x+1, the elements are {0,1,x,x+1}
The multiplicative group of a finite field is cyclic, generated by a
A primitive element is a generator of the multiplicative group, meaning that every non-zero element can be expressed as a power of the primitive element
Arithmetic in Finite Fields
Addition and Subtraction
in a finite field is performed by adding the coefficients of the corresponding terms of the polynomials modulo the characteristic p
For example, in GF(22), (x+1)+(x)=1 because 1+1≡0(mod2)
Subtraction in a finite field is performed by subtracting the coefficients of the corresponding terms of the polynomials modulo the characteristic p
For example, in GF(22), (x+1)−(x)=1 because 1−1≡0(mod2)
Multiplication and Division
in a finite field is performed by multiplying the polynomials and then reducing the result modulo the irreducible polynomial f(x) and the characteristic p
For example, in GF(22) with f(x)=x2+x+1, (x+1)∗(x)=x2+x≡1(modf(x))
Division in a finite field is performed by multiplying the dividend by the inverse of the divisor, where the inverse is computed using the extended Euclidean algorithm
For example, in GF(22) with f(x)=x2+x+1, (x+1)/(x)=(x+1)∗(x+1)≡1(modf(x)) because (x+1) is the inverse of (x)
Properties of Finite Fields
Primitive Elements
Every finite field has a primitive element, which is a generator of the multiplicative group of the field
The number of primitive elements in a finite field of order q is ϕ(q−1), where ϕ is Euler's totient function
For example, in GF(23), there are ϕ(7)=6 primitive elements
Primitive elements are used in various applications, such as constructing pseudo-random number generators and designing error-correcting codes
Trace and Norm Functions
The trace and norm functions are important maps between finite fields and their subfields, which can be used to analyze the structure and properties of the fields
The TrFq/Fp(a) of an element a in Fq is the sum of its conjugates over the Fp
The NFq/Fp(a) of an element a in Fq is the product of its conjugates over the subfield Fp
These functions have various properties, such as linearity and multiplicativity, which can be used to solve equations and study the structure of finite fields