This is not a full deep-dive, but a highly condensed summary to provide a short ant quick grasp of why discrete finite fields and modular arithmetic with primes are essential in modern cryptography.
What is a finite field?
A finite field (also known as a Galois Field, GF) is a set with a fixed number of elements where we can:
- Add
- Subtract
- Multiply
- Divide (except by zero)
And the result always stays within the field.
Why discrete and finite?
Cryptographic systems work on digital data which are discrete bits, not continuous values. A finite number of possible values keeps operations predictable, fast, and secure.
- Computers work with discrete (non-continuous) values like bits and integers, not real numbers.
- Finite means it limits the number of values to ensure predictable and secure computations.
Note: It is essential for algorithms that must behave the same every time, without rounding errors.
The role of modular arithmetic
All operations in finite fields are performed using modulo arithmetic.
For example:
7 + 8 mod 11 = 4
- This “wrap-around” behavior helps keep numbers bounded and secure.
- It also allows encryption algorithms to stay efficient even with very large numbers.
Why use a prime number as the modulus?
When the modulus is a prime number p, the field GF(p) has simple and powerful properties:
- Every non-zero element has a unique inverse (critical for division).
- The math forms a true field, meaning no loopholes or weak points in the system.
- Prime moduli avoid “degenerate” behavior seen with non-prime bases (like GF(15)).
This is why primes are used in RSA, Diffie-Hellman, and Elliptic Curve Cryptography.
Common types
- GF(p): field with a prime number of elements (used in RSA, ECC)
- GF(2^n): field used in AES and binary systems
Where it is used
- Elliptic Curve Cryptography (ECC) – Uses finite fields to define curve equations securely.
- RSA, Diffie-Hellman – Rely on modular exponentiation over prime fields.
- AES – Uses GF(2⁸) for efficient block cipher operations.
- Error correction – In storage and communication (e.g., Reed-Solomon codes).
Benefits in cryptography
- Secure math: Strong foundation based on number theory.
- Hard problems: Discrete logarithms and factorization are tough to reverse in finite fields.
- No rounding: Exact results, critical for reproducible encryption/decryption.
- Performance: Fast, efficient operations on hardware and software.