def fieldAdd(array1,array2):
"""Given two field elements, defines the additive operation between them over GF(2), since we are working over GF(2), this is the same as subtraction
"""
a = truthtable.fromBinary(array1[::-1])
b = truthtable.fromBinary(array2[::-1])
added = a^b
ans = truthtable.binary(added, max(len(array1),len(array2)))[::-1]
return ans[:primpoly.polyDeg(ans)+1]
def divAlg(array1,array2):
""" Given two polynomials, performs the division algorithm to find the quotient and remainder of the polynomial of first the polynomial divided by the second over the field GF(2)
"""
dif= polyDeg(array1) -polyDeg(array2)
if dif < 0:
return [0], array1
a = [0 for i in range(dif)]
q = truthtable.binary(2**(dif),len(array1))[::-1]
m = a + array2
r = truthtable.binary((truthtable.fromBinary(m[::-1]))^(truthtable.fromBinary(array1[::-1])), len(array1))[::-1]
newQ,newR = divAlg(r,array2)
return [truthtable.binary((truthtable.fromBinary(newQ[::-1]))^(truthtable.fromBinary(q[::-1])), len(q))[::-1],newR]
def bigFPolyDeg(array):
"""Given a polynomial in F_{2^{n}}[x], returns its degree
"""
for i in range(len(array)-1,-1,-1):
if truthtable.fromBinary(array[i]) > 0:
return i
return 0
def fieldInv(array, dpoly):
"""Given a field element, finds its inverse in the field defined by dpoly
"""
if truthtable.fromBinary(array)==0:
raise Exception("No inverse for zero element")
field = fieldGen(len(dpoly)-1)
for element in field:
if fieldMult(element, array, dpoly) == [1]:
return element[:primpoly.polyDeg(element)+1]
break
def divChecker(array1, array2):
""" Given two polynomials, determines whether one divides the other
"""
if polyDeg(array1) == polyDeg(array2):
return False
a = divAlg(array1, array2)[1]
j = truthtable.fromBinary(a[::-1])
if j == 0:
return True
else:
return False
def irrChecker(array):
""" Given a particular polynomial, prints true if it is irreducible false if it is not
"""
d = polyDeg(array)
for i in range(2,2**(d)+1):
j = truthtable.binary(i,len(array))[::-1]
a = divAlg(array,j)[1]
b = truthtable.fromBinary(a[::-1])
if b == 0:
return False
return True