jarvisb1 / BinaryGoppaCode

BinaryGoppaCode

GF2 operation

• Addition
• Multiplication
• Square
• Division
• GCD
• Square root

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다항식 곱셈 검증을 위한 방법

SageMath 프로그램을 이용하여 검증한다.

설치하지 않아도 온라인으로 이용할 수 있다.

https://sagecell.sagemath.org/

bin.<z> = GF(2)[]
poly = z^13+z^4+z^3+z+1
S = bin.quotient(poly, 'X')

#example
A = z^5+z^4+z^3+z^2+z^0
B = z^9+z^8+z^7+z^6+z^3+z^0
C = z^12+z^10+z^9+z^8+z^5+z^1+z^0
S(A*B) == S(C)

irreducible test over GF(2)

bin.<z> = GF(2)[]
A = z^13+z^11+z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z^0
A.is_irreducible()

multiplication test over extension binary field

bin.<z> = GF(2)[]
mod = z^3+z^1+z^0
Fqm.<z> = GF(2^13, modulus = mod)
PR_Fqm.<X> = PolynomialRing(Fqm)
A = (z^0)*X^5+(z^2+z^1+z^0)*X^4+(z^1)*X^3+(z^1)*X^2+(z^3+z^2+z^0)*X^1+(z^3+z^2+z^0)*X^0
B = (z^0)*X^5+(z^3+z^0)*X^4+(z^3+z^1)*X^3+(z^3+z^2+z^1+z^0)*X^2+(z^3+z^2+z^1+z^0)*X^1+(z^3)*X^0
C = (z^0)*X^10+(z^2+z^0)*X^9+(z^2+z^1)*X^8+(z^2+z^0)*X^7+(z^2+z^0)*X^6+(z^1)*X^5+(z^1+z^0)*X^4+(z^2+z^1)*X^3+(z^2+z^1)*X^2+(z^2)*X^1+(z^0)*X^0
A*B == C

monic polynomial over extension binary field

bin.<z> = GF(2)[]
mod = z^13+z^12+z^10+z^9+z^7+z^4+z^0
Fqm.<z> = GF(2^13, modulus = mod)
PR_Fqm.<X> = PolynomialRing(Fqm)
A = (z^12+z^6+z^5+z^4+z^3+z^1+z^0)*X^5+(z^11+z^10+z^9+z^8+z^7+z^6+z^4+z^2)*X^4+(z^12+z^11+z^10+z^9+z^8+z^5+z^3+z^2+z^1+z^0)*X^3+(z^12+z^11+z^9+z^6+z^5+z^4+z^0)*X^2+(z^12+z^11+z^10+z^8+z^7+z^2+z^1+z^0)*X^1+(z^12+z^11+z^9+z^8+z^1)*X^0
B = (z^0)*X^5+(z^12+z^10+z^9+z^6+z^5+z^4+z^1+z^0)*X^4+(z^12+z^11+z^10+z^9+z^8+z^7+z^6+z^4+z^2+z^0)*X^3+(z^11+z^10+z^7+z^3+z^1+z^0)*X^2+(z^12+z^11+z^9+z^6+z^5+z^4+z^3+z^2+z^1+z^0)*X^1+(z^12+z^10+z^9+z^7+z^6+z^4+z^3+z^2+z^1)*X^0
A.monic() == B

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