Status:

This is C++ library for general Galois fields based on Art Owen's code in Statlib. This library supports arithmetic over a finite field of any prime power, GF(p^n) where p is a prime and n >= 1, and it does so relatively efficiently by relying on addition and multiplication tables.

The library consists primarily of two classes:

• The Galois Field class precomputes various tables for efficient arithmetic on a Galois field with q elements. This is essentially Owen's GF C struct converted to a C++ class with a constructor and a few convient member functions.
• The Galois Element class takes a pointer to a Field and enables arithmetic over a Galois field using operator overload.

For example the following code performs simple arithmetic over GF(q = 2^2 = 4):

Galois::Field gf(4);
std::cout << gf << std::endl;

Galois::Element A(&gf,3);
Galois::Element B(&gf,2);
Galois::Element C = A * B;
Galois::Element D = (B - 1) / C;

std::cout << "A = " << A << std::endl;
std::cout << "B = " << B << std::endl;
std::cout << "C = A * B = " << C << std::endl;
std::cout << "D = (B - 1) / C = " << D << std::endl;

This would output:

For GaloisField(4) p=2 n=2
x**n = (1,1)


GF(4) Polynomial coefficients:
  0  0 0 
  1  1 0 
  2  0 1 
  3  1 1 


GF(4) Addition Table
   0 1 2 3
   1 0 3 2
   2 3 0 1
   3 2 1 0


GF(4) Multiplication table
   0 0 0 0
   0 1 2 3
   0 2 3 1
   0 3 1 2


GF(4) Reciprocals
 0 -1
 1 1
 2 3
 3 2


GF(4) Negatives
 0 0
 1 1
 2 2
 3 3


GF(4) Square roots
 0 0
 1 1
 2 3
 3 2

A = 3
B = 2
C = A * B = 1
D = (B - 1) / C = 3

The test subdirectory contains some unit tests, which can also serve as additional usage examples.

This library is adapted directly from Art Owen's code in Statlib.

Other Galois field libraries served as inspiration, including:

• GaloisCPP, by Saied H. Khayat
• liboa, by Robert Carnell