This library creates addition and multiplication functions for Galois fields.
The function deffield* returns a list containing the addition and multiplication functions for the given Galois field.
(defun deffield* (&key p (d 1) use-tables)
=> ( #'<+ function> #'<* function>)
The Galois field of order q = (expt p d) is calculated. The value p must be prime.
The value d must be a positive integer.
If use-tables is true (or if use-tables is not specified and q is small), then the addition and multiplication tables are stored in arrays to speed up computations with them.
If use-tables is false (or if use-tables is not specified and q is large), then the addition and multiplication are calculated explicitly on each function invocation.
The Galois field elements are encoded as integers from 0 to q-1. The element a0 + a1*x + a2*x^2 + ... ad*x^d is represented by the integer obtained by evaluating that polynomial for x equal to p.
Here is an example use:
(destructuring-bind (gf+ gf*) (deffield* :p 3 :d 2)
(list (funcall gf+ 7 3)
(funcall gf* 7 3)))
=> (1 4)
Because: 7 represents (1 + 2x) while 3 represent (0 + x). So, 7 + 3 represents (1 + 3x) = (1 + 0x) = 1. Similarly, (1 + 2x) * x = (x + 2x^2) which, modulo (1 + x^2), = (1 + x) which is represnted by 4.
The returned functions behave as expected when given any number of arguments.
(destructuring-bind (gf+ gf*) (deffield* :p 3 :d 2)
(list (funcall gf+)
(funcall gf*)
(funcall gf+ 1 1 3 1)))
=> (0 1 3)
Caveat: Currently, there is no way to specify the irreducible polynomial used in the multiplication.
The library also provides a deffield macro for defining global functions for these operations.
(defmacro deffield (name &key p (d 1) use-tables) ...)
This defines functions based upon the given base-name name.
(deffield gf32 :p 2 :d 5)
(list (gf32+ 12 7)
(gf32* 12 7))
=> (11 1)