Description

The following is a list of primitive irreducible polynomials for generating elements of a binary extension field GF(2m) from a base finite field. The list contains polynomials of degree 2 to 32.

Downloads

primitive_polynomials_GF2.txt

primitive_polynomials_GF2_extended.zip

x^2 + x^1 + 1

x^3 + x^1 + 1

x^4 + x^1 + 1

x^5 + x^2 + 1 x^5 + x^4 + x^2 + x^1 + 1 x^5 + x^4 + x^3 + x^2 + 1

x^6 + x^1 + 1 x^6 + x^5 + x^2 + x^1 + 1 x^6 + x^5 + x^3 + x^2 + 1

x^7 + x^1 + 1 x^7 + x^3 + 1 x^7 + x^3 + x^2 + x^1 + 1 x^7 + x^4 + x^3 + x^2 + 1 x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^7 + x^6 + x^3 + x^1 + 1 x^7 + x^6 + x^4 + x^2 + 1 x^7 + x^6 + x^5 + x^2 + 1 x^7 + x^6 + x^5 + x^4 + x^2 + x^1 + 1

x^8 + x^4 + x^3 + x^2 + 1 x^8 + x^5 + x^3 + x^1 + 1 x^8 + x^6 + x^4 + x^3 + x^2 + x^1 + 1 x^8 + x^6 + x^5 + x^1 + 1 x^8 + x^6 + x^5 + x^2 + 1 x^8 + x^6 + x^5 + x^3 + 1 x^8 + x^7 + x^6 + x^1 + 1 x^8 + x^7 + x^6 + x^5 + x^2 + x^1 + 1

x^9 + x^4 + 1 x^9 + x^5 + x^3 + x^2 + 1 x^9 + x^6 + x^4 + x^3 + 1 x^9 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^9 + x^6 + x^5 + x^4 + x^2 + x^1 + 1 x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1 x^9 + x^8 + x^4 + x^1 + 1 x^9 + x^8 + x^5 + x^4 + 1 x^9 + x^8 + x^6 + x^5 + 1 x^9 + x^8 + x^6 + x^5 + x^3 + x^1 + 1 x^9 + x^8 + x^7 + x^2 + 1 x^9 + x^8 + x^7 + x^3 + x^2 + x^1 + 1 x^9 + x^8 + x^7 + x^6 + x^5 + x^1 + 1 x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + 1

x^10 + x^3 + 1 x^10 + x^4 + x^3 + x^1 + 1 x^10 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^10 + x^8 + x^3 + x^2 + 1 x^10 + x^8 + x^4 + x^3 + 1 x^10 + x^8 + x^5 + x^1 + 1 x^10 + x^8 + x^5 + x^4 + 1 x^10 + x^8 + x^7 + x^6 + x^5 + x^2 + 1 x^10 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^1 + 1 x^10 + x^9 + x^4 + x^1 + 1 x^10 + x^9 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^10 + x^9 + x^8 + x^6 + x^3 + x^2 + 1 x^10 + x^9 + x^8 + x^6 + x^5 + x^1 + 1 x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1

x^11 + x^2 + 1 x^11 + x^5 + x^3 + x^1 + 1 x^11 + x^5 + x^3 + x^2 + 1 x^11 + x^6 + x^5 + x^1 + 1 x^11 + x^7 + x^3 + x^2 + 1 x^11 + x^8 + x^5 + x^2 + 1 x^11 + x^8 + x^6 + x^5 + x^4 + x^1 + 1 x^11 + x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^11 + x^9 + x^4 + x^1 + 1 x^11 + x^9 + x^8 + x^7 + x^4 + x^1 + 1 x^11 + x^10 + x^3 + x^2 + 1 x^11 + x^10 + x^7 + x^4 + x^3 + x^1 + 1 x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^1 + 1 x^11 + x^10 + x^9 + x^8 + x^3 + x^1 + 1

x^12 + x^6 + x^4 + x^1 + 1 x^12 + x^9 + x^3 + x^2 + 1 x^12 + x^9 + x^8 + x^3 + x^2 + x^1 + 1 x^12 + x^10 + x^9 + x^8 + x^6 + x^2 + 1 x^12 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1 x^12 + x^11 + x^6 + x^4 + x^2 + x^1 + 1 x^12 + x^11 + x^9 + x^5 + x^3 + x^1 + 1 x^12 + x^11 + x^9 + x^7 + x^6 + x^4 + 1 x^12 + x^11 + x^9 + x^7 + x^6 + x^5 + 1 x^12 + x^11 + x^9 + x^8 + x^7 + x^4 + 1 x^12 + x^11 + x^9 + x^8 + x^7 + x^5 + x^2 + x^1 + 1 x^12 + x^11 + x^10 + x^5 + x^2 + x^1 + 1 x^12 + x^11 + x^10 + x^8 + x^6 + x^4 + x^3 + x^1 + 1 x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^5 + x^4 + x^3 + x^1 + 1

x^13 + x^4 + x^3 + x^1 + 1 x^13 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^13 + x^9 + x^8 + x^7 + x^5 + x^1 + 1 x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + 1 x^13 + x^10 + x^9 + x^8 + x^6 + x^3 + x^2 + x^1 + 1 x^13 + x^11 + x^8 + x^7 + x^4 + x^1 + 1 x^13 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^13 + x^12 + x^6 + x^5 + x^4 + x^3 + 1 x^13 + x^12 + x^8 + x^7 + x^6 + x^5 + 1 x^13 + x^12 + x^9 + x^8 + x^4 + x^2 + 1 x^13 + x^12 + x^10 + x^8 + x^6 + x^4 + x^3 + x^2 + 1 x^13 + x^12 + x^11 + x^5 + x^2 + x^1 + 1 x^13 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^1 + 1 x^13 + x^12 + x^11 + x^9 + x^5 + x^3 + 1

x^14 + x^8 + x^6 + x^1 + 1 x^14 + x^10 + x^6 + x^1 + 1 x^14 + x^10 + x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1 x^14 + x^11 + x^6 + x^1 + 1 x^14 + x^11 + x^9 + x^6 + x^5 + x^2 + 1 x^14 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + 1 x^14 + x^12 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^1 + 1 x^14 + x^12 + x^11 + x^10 + x^9 + x^7 + x^4 + x^3 + 1 x^14 + x^13 + x^6 + x^5 + x^3 + x^1 + 1 x^14 + x^13 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^14 + x^13 + x^11 + x^6 + x^5 + x^4 + x^2 + x^1 + 1 x^14 + x^13 + x^11 + x^8 + x^5 + x^3 + x^2 + x^1 + 1 x^14 + x^13 + x^12 + x^11 + x^10 + x^7 + x^6 + x^1 + 1 x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^6 + x^5 + 1

x^15 + x^1 + 1 x^15 + x^4 + 1 x^15 + x^7 + 1 x^15 + x^7 + x^6 + x^3 + x^2 + x^1 + 1 x^15 + x^10 + x^5 + x^1 + 1 x^15 + x^10 + x^5 + x^4 + 1 x^15 + x^10 + x^5 + x^4 + x^2 + x^1 + 1 x^15 + x^10 + x^9 + x^7 + x^5 + x^3 + 1 x^15 + x^10 + x^9 + x^8 + x^5 + x^3 + 1 x^15 + x^11 + x^7 + x^6 + x^2 + x^1 + 1 x^15 + x^12 + x^3 + x^1 + 1 x^15 + x^12 + x^5 + x^4 + x^3 + x^2 + 1 x^15 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^2 + 1 x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1

x^16 + x^9 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + 1 x^16 + x^12 + x^3 + x^1 + 1 x^16 + x^12 + x^7 + x^2 + 1 x^16 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^1 + 1 x^16 + x^13 + x^12 + x^11 + x^7 + x^6 + x^3 + x^1 + 1 x^16 + x^13 + x^12 + x^11 + x^10 + x^6 + x^2 + x^1 + 1 x^16 + x^14 + x^10 + x^8 + x^3 + x^1 + 1 x^16 + x^14 + x^13 + x^12 + x^6 + x^5 + x^3 + x^2 + 1 x^16 + x^14 + x^13 + x^12 + x^10 + x^7 + 1 x^16 + x^15 + x^10 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^16 + x^15 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1 x^16 + x^15 + x^11 + x^10 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^16 + x^15 + x^11 + x^10 + x^9 + x^6 + x^2 + x^1 + 1 x^16 + x^15 + x^11 + x^10 + x^9 + x^8 + x^6 + x^4 + x^2 + x^1 + 1

x^17 + x^3 + 1 x^17 + x^3 + x^2 + x^1 + 1 x^17 + x^5 + 1 x^17 + x^6 + 1 x^17 + x^8 + x^4 + x^3 + 1 x^17 + x^8 + x^7 + x^6 + x^4 + x^3 + 1 x^17 + x^10 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + 1 x^17 + x^12 + x^6 + x^3 + x^2 + x^1 + 1 x^17 + x^12 + x^9 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^17 + x^12 + x^9 + x^7 + x^6 + x^4 + x^3 + x^2 + 1 x^17 + x^14 + x^11 + x^7 + x^5 + x^3 + x^2 + x^1 + 1 x^17 + x^15 + x^13 + x^11 + x^9 + x^7 + x^5 + x^3 + 1 x^17 + x^15 + x^13 + x^11 + x^9 + x^7 + x^6 + x^4 + x^2 + x^1 + 1 x^17 + x^16 + x^3 + x^1 + 1

x^18 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^18 + x^7 + 1 x^18 + x^7 + x^5 + x^2 + x^1 + 1 x^18 + x^8 + x^2 + x^1 + 1 x^18 + x^9 + x^7 + x^6 + x^5 + x^4 + 1 x^18 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + x^1 + 1 x^18 + x^9 + x^8 + x^7 + x^6 + x^4 + x^2 + x^1 + 1 x^18 + x^10 + x^7 + x^5 + 1 x^18 + x^10 + x^8 + x^5 + 1 x^18 + x^10 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^18 + x^10 + x^9 + x^3 + 1 x^18 + x^13 + x^6 + x^4 + 1 x^18 + x^15 + x^5 + x^2 + 1 x^18 + x^15 + x^9 + x^2 + 1

x^19 + x^5 + x^2 + x^1 + 1 x^19 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^19 + x^6 + x^2 + x^1 + 1 x^19 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^19 + x^6 + x^5 + x^4 + x^3 + x^2 + 1 x^19 + x^7 + x^5 + x^3 + x^2 + x^1 + 1 x^19 + x^8 + x^7 + x^5 + 1 x^19 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^19 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + x^1 + 1 x^19 + x^9 + x^8 + x^5 + 1 x^19 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^19 + x^9 + x^8 + x^7 + x^4 + x^3 + x^2 + x^1 + 1 x^19 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1 x^19 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1 x^19 + x^16 + x^13 + x^10 + x^7 + x^4 + x^1 + 1

x^20 + x^3 + 1 x^20 + x^9 + x^5 + x^3 + 1 x^20 + x^11 + x^8 + x^6 + x^3 + x^2 + 1 x^20 + x^14 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + 1 x^20 + x^17 + x^14 + x^10 + x^7 + x^4 + x^3 + x^2 + 1 x^20 + x^19 + x^4 + x^3 + 1

x^21 + x^2 + 1 x^21 + x^8 + x^7 + x^4 + x^3 + x^2 + 1 x^21 + x^10 + x^6 + x^4 + x^3 + x^2 + 1 x^21 + x^13 + x^5 + x^2 + 1 x^21 + x^14 + x^7 + x^2 + 1 x^21 + x^14 + x^7 + x^6 + x^3 + x^2 + 1 x^21 + x^14 + x^12 + x^7 + x^6 + x^4 + x^3 + x^2 + 1 x^21 + x^15 + x^10 + x^9 + x^5 + x^4 + x^3 + x^2 + 1 x^21 + x^20 + x^19 + x^18 + x^5 + x^4 + x^3 + x^2 + 1

x^22 + x^1 + 1 x^22 + x^9 + x^5 + x^1 + 1 x^22 + x^14 + x^13 + x^12 + x^7 + x^3 + x^2 + x^1 + 1 x^22 + x^17 + x^9 + x^7 + x^2 + x^1 + 1 x^22 + x^17 + x^13 + x^12 + x^8 + x^7 + x^2 + x^1 + 1 x^22 + x^20 + x^18 + x^16 + x^6 + x^4 + x^2 + x^1 + 1

x^23 + x^5 + 1 x^23 + x^5 + x^4 + x^1 + 1 x^23 + x^11 + x^10 + x^7 + x^6 + x^5 + 1 x^23 + x^12 + x^5 + x^4 + 1 x^23 + x^15 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + 1 x^23 + x^16 + x^13 + x^6 + x^5 + x^3 + 1 x^23 + x^17 + x^11 + x^5 + 1 x^23 + x^17 + x^11 + x^9 + x^8 + x^5 + x^4 + x^1 + 1 x^23 + x^18 + x^16 + x^13 + x^11 + x^8 + x^5 + x^2 + 1 x^23 + x^21 + x^7 + x^5 + 1

x^24 + x^7 + x^2 + x^1 + 1 x^24 + x^21 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^10 + x^9 + x^5 + x^4 + x^1 + 1 x^24 + x^22 + x^20 + x^18 + x^16 + x^14 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + 1

x^25 + x^3 + 1 x^25 + x^3 + x^2 + x^1 + 1 x^25 + x^11 + x^9 + x^8 + x^6 + x^4 + x^3 + x^2 + 1 x^25 + x^12 + x^4 + x^3 + 1 x^25 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^3 + 1 x^25 + x^17 + x^10 + x^3 + x^2 + x^1 + 1 x^25 + x^18 + x^12 + x^11 + x^6 + x^5 + x^4 + x^3 + 1 x^25 + x^20 + x^5 + x^3 + 1 x^25 + x^20 + x^16 + x^11 + x^5 + x^3 + x^2 + x^1 + 1 x^25 + x^23 + x^21 + x^19 + x^9 + x^7 + x^5 + x^3 + 1

x^26 + x^6 + x^2 + x^1 + 1 x^26 + x^19 + x^16 + x^15 + x^14 + x^13 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + 1 x^26 + x^21 + x^18 + x^16 + x^15 + x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^3 + 1 x^26 + x^22 + x^20 + x^19 + x^16 + x^13 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1 x^26 + x^22 + x^21 + x^16 + x^12 + x^11 + x^10 + x^8 + x^5 + x^4 + x^3 + x^1 + 1 x^26 + x^23 + x^22 + x^21 + x^19 + x^18 + x^15 + x^14 + x^13 + x^11 + x^10 +x^9+x^8+x^6+x^5+x^2+ 1 x^26 + x^24 + x^21 + x^17 + x^16 + x^14 + x^13 + x^11 + x^7 + x^6 + x^4 + x^1 + 1

x^27 + x^5 + x^2 + x^1 + 1 x^27 + x^18 + x^11 + x^10 + x^9 + x^5 + x^4 + x^3 + 1 x^27 + x^22 + x^13 + x^11 + x^6 + x^5 + x^4 + x^3 + 1 x^27 + x^22 + x^17 + x^15 + x^14 + x^13 + x^6 + x^1 + 1 x^27 + x^22 + x^21 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 + x^7 + x^5 + 1 x^27 + x^24 + x^19 + x^16 + x^12 + x^8 + x^7 + x^3 + x^2 + x^1 + 1 x^27 + x^24 + x^21 + x^19 + x^16 + x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1 x^27 + x^25 + x^23 + x^21 + x^13 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1 x^27 + x^25 + x^23 + x^21 + x^20 + x^19 + x^18 + x^16 + x^14 + x^10 + x^8 + x^7 + x^4 + x^3 + 1

x^28 + x^3 + 1 x^28 + x^13 + x^11 + x^9 + x^5 + x^3 + 1 x^28 + x^18 + x^17 + x^16 + x^9 + x^5 + x^4 + x^3 + 1 x^28 + x^19 + x^17 + x^15 + x^10 + x^6 + x^3 + x^2 + 1 x^28 + x^22 + x^11 + x^10 + x^4 + x^3 + 1 x^28 + x^24 + x^20 + x^16 + x^12 + x^8 + x^4 + x^3 + 1

x^29 + x^2 + 1 x^29 + x^12 + x^7 + x^2 + 1 x^29 + x^18 + x^14 + x^6 + x^3 + x^2 + 1 x^29 + x^19 + x^16 + x^6 + x^3 + x^2 + 1 x^29 + x^20 + x^11 + x^2 + 1 x^29 + x^20 + x^16 + x^11 + x^8 + x^4 + x^3 + x^2 + 1 x^29 + x^21 + x^5 + x^2 + 1 x^29 + x^23 + x^10 + x^9 + x^5 + x^4 + x^3 + x^2 + 1 x^29 + x^24 + x^14 + x^13 + x^8 + x^4 + x^3 + x^2 + 1 x^29 + x^26 + x^5 + x^2 + 1

x^30 + x^23 + x^2 + x^1 + 1 x^30 + x^24 + x^20 + x^16 + x^14 + x^13 + x^11 + x^7 + x^2 + x^1 + 1 x^30 + x^24 + x^21 + x^20 + x^18 + x^15 + x^13 + x^12 + x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1 x^30 + x^25 + x^24 + x^23 + x^19 + x^18 + x^16 + x^14 + x^11 + x^8 + x^6 + x^4 + x^3 + x^1 + 1 x^30 + x^27 + x^25 + x^24 + x^23 + x^22 + x^19 + x^16 + x^12 + x^10 + x^8 + x^7 + x^6 + x^1 + 1

x^31 + x^3 + 1 x^31 + x^3 + x^2 + x^1 + 1 x^31 + x^13 + x^8 + x^3 + 1 x^31 + x^16 + x^8 + x^4 + x^3 + x^2 + 1 x^31 + x^20 + x^15 + x^5 + x^4 + x^3 + 1 x^31 + x^20 + x^18 + x^7 + x^5 + x^3 + 1 x^31 + x^21 + x^12 + x^3 + x^2 + x^1 + 1 x^31 + x^23 + x^22 + x^15 + x^14 + x^7 + x^4 + x^3 + 1 x^31 + x^25 + x^19 + x^14 + x^7 + x^3 + x^2 + x^1 + 1 x^31 + x^27 + x^23 + x^19 + x^15 + x^11 + x^7 + x^3 + 1 x^31 + x^27 + x^23 + x^19 + x^15 + x^11 + x^10 + x^9 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1

x^32 + x^22 + x^2 + x^1 + 1 x^32 + x^22 + x^21 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 + x^10 + x^8 + x^6 + x^4 + x^1 + 1 x^32 + x^23 + x^17 + x^16 + x^14 + x^10 + x^8 + x^7 + x^6 + x^5 + x^3 + 1 x^32 + x^26 + x^23 + x^22 + x^16 + x^12 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1 x^32 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^17 + x^13 + x^11 + x^10+x^9+x^8+x^7+x^2+x^1 + 1 x^32 + x^28 + x^19 + x^18 + x^16 + x^14 + x^11 + x^10 + x^9 + x^6 + x^5 + x^1 + 1

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