Galois Field Arithmetic
Finite fields underlie modern cryptography, error-correcting codes, and digital communications. Explore their multiplication tables, generator cycles, and polynomial structure.
What is a Galois Field?
A Galois field GF(q) (also written Fq) is a finite set of elements closed under addition and multiplication, where every nonzero element has a multiplicative inverse.
Fields exist only when q = pn for a prime p. The prime fields GF(p) use ordinary arithmetic modulo p. The binary extensions GF(2n) use polynomials over GF(2) reduced modulo an irreducible polynomial.
Structure & Applications
The nonzero elements of GF(q) form a cyclic multiplicative group of order q−1. A primitive element (generator) cycles through all q−1 nonzero elements by repeated multiplication.
GF(28) is the field used in AES encryption. GF(2n) fields power Reed-Solomon codes in CDs, DVDs, and QR codes. The color in the multiplication table encodes the element value — patterns reveal deep algebraic structure.
Reading the Table
The multiplication table shows a×b for all a,b in the field. Colors cycle through the full spectrum as element values rise from 0 to q−1. Notice the diagonal (a²) and how every row/column is a permutation of nonzero elements.
The addition table for GF(2n) is XOR on the binary representation — a checkerboard of XOR sums.
Generator Cycle
The cycle view traces powers of a primitive element g: g¹, g², g³, … arranged on a circle. Each dot represents gk, colored by its value. The connections show how multiplication by g rotates the cycle.
Every nonzero field element appears exactly once — confirming the cyclic group structure. Changing the primitive polynomial changes which element is g but preserves the overall structure.