Galois Fields

• galois_fields — Overview of the Galois Fields toolbox: finite fields GF(p) and GF(p^n)
• Fields
  • gf_characteristic — Characteristic of a finite field
  • gf_check_field — Checks that a variable is a field created by gf_create_field
  • gf_create_field — Creates the finite field GF(p^n) from a primitive polynomial
  • gf_degree — Degree of a finite field over its prime subfield
  • gf_elements — Elements of a finite field as Scilab polynomials
  • gf_order — Number of elements of a finite field
  • gf_primitive_elements — Primitive elements (generators of the multiplicative group) of a finite field
• Primitive and minimal polynomials
  • gf_default_primitive_polynomial — Default primitive polynomial of a given degree over GF(p)
  • gf_is_primitive_polynomial — Tests whether a polynomial over GF(p) is primitive
  • gf_minimal_polynomial — Minimal polynomial over GF(p) of a field element
  • gf_primitive_polynomial — Primitive polynomial of a field as a Scilab polynomial
  • gf_primitive_polynomials — All monic primitive polynomials of a given degree over GF(p)
• Element arithmetic
  • gf_add — Adds elements of a finite field
  • gf_are_equal — Tests whether two arrays have the same size and entries
  • gf_divide — Divides elements of a finite field
  • gf_is_one — Tests whether an array represents the unit element
  • gf_is_zero — Tests whether all coefficients of an array are zero
  • gf_multiply — Multiplies elements of a finite field
  • gf_power — Raises elements of a finite field to an integer power
  • gf_subtract — Subtracts elements of a finite field
• Polynomials over GF(p)
  • gf_poly_add — Adds polynomials over GF(p)
  • gf_poly_divide — Division with remainder of polynomials over GF(p)
  • gf_poly_multiply — Multiplies polynomials over GF(p)
  • gf_poly_power — Raises a polynomial over GF(p) to an integer power
  • gf_poly_subtract — Subtracts polynomials over GF(p)
• Operation tables
  • gf_addition_table — Addition table of a prime field
  • gf_division_table — Division table of a prime field
  • gf_multiplication_table — Multiplication table of a prime field
  • gf_subtraction_table — Subtraction table of a prime field
• Conversions
  • gf_to_exponential — Converts elements of GF(p^n) to exponential notation α^k
  • gf_to_polynomial — Converts rows of coefficients to Scilab polynomials
• Integer utilities
  • gf_is_prime — Tests whether integers are prime
  • gf_mod — Non-negative remainder of integers modulo m