Number Theory Part 4: Fields

In continuation with my previous blog, this blog will discuss the Feilds

Fields

A field, denoted {F, +, ×}, is an integral domain whose elements satisfy the following additional property:

M7: Multiplicative inverse: For every element a in F, except the element designated 0, there must also exist in F its multiplicative inverse. That is, if a ∈ F and a =! 0, then there must exist an element b ∈ F such that a×b = b×a = 1

Examples:

1. The set of real numbers under addition and multiplication
2. The set of complex numbers under addition and multiplication
3. The set of polynomials with real coefficients under addition and multiplication
4. The set of all integers under the arithmetic addition and multiplication operations is NOT a field.

Finite fields

Finite fields, also known as Galois fields or binary fields, are algebraic structures with a finite number of elements.

Basis of RSA cryptosystem

The set of integers modulo 2, denoted as ℤ₂, is a finite field consisting of elements 0 and 1. In this field, the addition and multiplication operations are modulo 2, meaning the result should be either 0 or 1.

The addition operation in ℤ₂ is defined as follows:

0 + 0 = 0 
0 + 1 = 1 
1 + 0 = 1 
1 + 1 = 0

The addition operation in ℤ₂ forms a group under addition, following the closure, commutativity, associativity, and the existence of an identity element (0).

The multiplication operation in ℤ₂ is defined as follows:

0 × 0 = 0
0 × 1 = 0
1 × 0 = 0 
1× 1 = 1

Like addition, multiplication in ℤ₂ follows closure, commutativity, and associativity properties and has an identity element (1). However, ℤ₂ lacks multiplicative inverses for both 0 and 1, except for 1 itself.

The importance of the finite field ℤ₂, lies in its application to cryptographic algorithms such as the Rivest–Shamir–Adleman (RSA) cryptosystem. In RSA, modular arithmetic operations are extensively used, and ℤ₂ provides a simple and efficient way to perform arithmetic calculations using only two elements. This makes ℤ₂ particularly suitable for binary digits operations, the foundation of many cryptographic systems.

Prime Fields

Prime fields are important in cryptography because they can be used to construct secure cryptographic algorithms. For example, the Diffie-Hellman key exchange protocol uses a prime field to generate a shared secret key between two parties. The ElGamal encryption algorithm also uses a prime field to encrypt and decrypt messages.

Let me give some reasons why prime fields are essential while building cryptographic algorithms:

• The characteristic of a prime field is always 0 or a prime number.
• The multiplicative group of a prime field is cyclic.
• The only subfields of a prime field are the field itself and the field of integers.
• The field of integers modulo p is a prime field if and only if p is a prime number.

The above table summarises the correlation between the algebraic structure and different operations.

Number Theory Part 1: Group

Number Theory Part 2: Ring

Number Theory Part 3: Integral Domain