def is_prime_power(n):
"""
Prime power is a positive integer power of a single prime number
"""
if len(factor(n)) == 1:
return True
return False
def GF(n, poly=[], var='x', fmtspec="p"):
"""A shorthand for generating finite fields. If poly is not specified then one will be chosen from a list of Conway polynomials."""
if numbthy.is_prime(n):
if len(poly) < 2:
return FiniteField(n, [1], var=var, fmtspec=fmtspec)
else:
return FiniteField(
n, poly, var=var, fmtspec=fmtspec
) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1):
raise ValueError(
'GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]
e = nfactors[0][1] # Prime p, exponent e
if (len(poly) == 0):
try:
poly = readconway('CPimport.txt', p, e)[:-1] # Assume monic
except (IOError, ValueError):
print(" Look for non-Conway primitive polynomial")
poly = findprimpoly(p, e)
else:
if nfactors[0][1] != len(poly):
raise ValueError(
'Polynomial {0} does not have degree {1}'.format(
poly + [1], nfactors[0][1]))
return FiniteField(p, poly, var=var, fmtspec=fmtspec)
def findprimpoly(p, e):
import sys
if sys.version_info[0] > 2:
iterrange = range # Version 3 patch
else:
iterrange = xrange # Version 2 patch
ordfacts = numbthy.factor((p**e) - 1)
print(p**e) - 1, " = ", ordfacts
for thepolynum in iterrange(p + 1, p**e):
# Note: Skip [0,d1,d2,...] as not irred
# Note: Skip [d0,0,0,...,0,1] as not primitive and probably not irred
thepoly = [(thepolynum // (p**i)) % p
for i in range(e)] # length e list of p digits
a = FiniteField(
p, thepoly).gen() # create poly 'x' mod thepoly, check its order
print thepoly, str(a)
if ((a**((p**e) -
1)) == 1): # check for irreducibility, else try next poly
isprim = True
for (thefact, thepow) in ordfacts:
print thefact
if ((a**(((p**e) - 1) / thefact)) == 1): # thepoly not prim
isprim = False
break # Don't need to check any more cofactors
if isprim: return thepoly
print "Oops" # Should throw exception - should never reach this point, all prime/exp have primitive polys
def __init__(self, prime, poly, var='x', orderfacts=None, fmtspec="p"):
"""FiniteField(prime, poly, var='x', orderfacts=None, fmtspec="p")
Create a finite field of order p**d, where d is the degree of the polynomial.
Driving polynomial must be monic and top coeff (i.e. 1) is implicit.
Example:
>>> from finitefield import *
>>> GF9 = FiniteField(3,[2,1]) # Define GF(3^2), polys w/ GF3 coeffs, mod x^2+x+2
>>> a = FiniteFieldElt(GF9,[1,2]) # Define 2x+1 in GF(9)
Define GF(5^8), defined mod z^8+3z^5+z^4+z^2+3z+4
Providing the factored order as (5^8-1) = (2^5)(3)(13)(313)
Default output is coefficients only ('c'), not full polynomials ('z')
>>> GF5e8 = FiniteField(5,[4,3,1,0,1,3,0,0],'z',((2,5),(3,1),(13,1),(313,1)),'c')
>>> '{0}'.format(c**20) # Default format
'20340110'
>>> '{0:p}'.format(c**20) # Force polynomial format
'(2 + 3z^2 + 4z^3 + z^5 + z^6)'"""
self.char = prime
self.degree = len(poly)
self.order = self.char**self.degree
self.modpoly = poly
self.var = var
self.fmtspec = fmtspec # p=polynomial; c=coeffsonly
if(orderfacts == None):
self.facts_order_gpunits = numbthy.factor(self.order - 1)
else:
self.facts_order_gpunits = orderfacts
if((self.char**self.degree-1) != reduce(lambda theprod,primepow:theprod*primepow, [prime**thepow for [prime,thepow] in orderfacts])):
raise ValueError('{0} is not a factorization of ({1}^{2}-1)'.format(orderfacts,self.char,self.degree))
self.reduc_table = [[0 for j in range(self.degree)] for i in range(2*self.degree-1)]
for i in range(self.degree): self.reduc_table[i][i] = 1
self.reduc_table[self.degree] = [(-self.modpoly[j])%self.char for j in range(self.degree)]
for i in range(self.degree+1,2*self.degree-1):
for j in range(self.degree):
self.reduc_table[i][j] = sum(map(lambda k: (-self.modpoly[k]*self.reduc_table[i-self.degree+k][j]), range(self.degree))) % self.char
def isMaximallyIdempotent(n):
factor_list = numbthy.factor(n)
ipList = idempotentPartitions(n, factor_list)
numFactors = len(factor_list)
if len(ipList) == 2**(numFactors - 1) - 1:
return True
return False
def is_square_free(n):
for p in [2, 3, 5, 7, 11, 13]:
if n % p**2 == 0:
return False
factor_list = numbthy.factor(n)
for (p, e) in factor_list:
if (e > 1):
return False
return True
def is_composite_and_square_free(n):
for p in [2, 3, 5, 7, 11, 13]:
if n % p**2 == 0:
return False
factor_list = numbthy.factor(n)
if len(factor_list) == 1:
return False
for (p, e) in factor_list:
if (e > 1):
return False
return True
def __init__(self, prime, poly, var='x', orderfacts=None, fmtspec="p"):
"""FiniteField(prime, poly, var='x', orderfacts=None, fmtspec="p")
Create a finite field of order p**d, where d is the degree of the polynomial.
Driving polynomial must be monic and top coeff (i.e. 1) is implicit.
Example:
>>> from finitefield import *
>>> GF9 = FiniteField(3,[2,1]) # Define GF(3^2), polys w/ GF3 coeffs, mod x^2+x+2
>>> a = FiniteFieldElt(GF9,[1,2]) # Define 2x+1 in GF(9)
Define GF(5^8), defined mod z^8+3z^5+z^4+z^2+3z+4
Providing the factored order as (5^8-1) = (2^5)(3)(13)(313)
Default output is coefficients only ('c'), not full polynomials ('z')
>>> GF5e8 = FiniteField(5,[4,3,1,0,1,3,0,0],'z',((2,5),(3,1),(13,1),(313,1)),'c')
>>> '{0}'.format(c**20) # Default format
'20340110'
>>> '{0:p}'.format(c**20) # Force polynomial format
'(2 + 3*z**2 + 4*z**3 + z**5 + z**6)'"""
self.char = prime
self.degree = len(poly)
self.order = self.char**self.degree
self.modpoly = poly
self.var = var
self.fmtspec = fmtspec
if (orderfacts == None):
self.facts_order_gpunits = numbthy.factor(self.order - 1)
else:
self.facts_order_gpunits = orderfacts
if ((self.char**self.degree - 1) != reduce(
lambda theprod, primepow: theprod * primepow,
[prime**thepow for [prime, thepow] in orderfacts])):
raise ValueError(
'{0} is not a factorization of ({1}^{2}-1)'.format(
orderfacts, self.char, self.degree))
self.reduc_table = [[0 for j in range(self.degree)]
for i in range(2 * self.degree - 1)]
for i in range(self.degree):
self.reduc_table[i][i] = 1
if (self.degree > 1):
self.reduc_table[self.degree] = [(-self.modpoly[j]) % self.char
for j in range(self.degree)]
for i in range(self.degree + 1, 2 * self.degree - 1):
for j in range(self.degree):
self.reduc_table[i][j] = sum(
map(
lambda k: (-self.modpoly[k] * self.
reduc_table[i - self.degree + k][j]),
range(self.degree))) % self.char
def GF(n,name='x',modulus=[]):
if numbthy.is_prime(n):
if len(modulus)<2:
print "Haven't coded prime field GF({0}) yet - sorry".format(n)
raise ValueError("Haven't coded prime field GF({0}) yet - sorry".format(n))
else:
return FiniteField(n,modulus,name) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1): raise ValueError('GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]; e = nfactors[0][1] # Prime p, exponent e
if (len(modulus) == 0):
try:
modulus = readconway('CPimport.txt',p,e)[:-1] # Assume monic
except(IOError,ValueError):
print(" Look for non-Conway primitive polynomial")
modulus = findprimpoly(p,e)
return FiniteField(p,modulus,name)
def GF(n, poly=[], var='x', fmtspec="p"):
"""A shorthand for generating finite fields. If poly is not specified then one will be chosen from a list of Conway polynomials."""
if numbthy.is_prime(n):
if len(poly)<2:
return FiniteField(n,[1],var=var,fmtspec=fmtspec)
else:
return FiniteField(n,poly,var=var,fmtspec=fmtspec) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1): raise ValueError('GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]; e = nfactors[0][1] # Prime p, exponent e
if (len(poly) == 0):
try:
poly = readconway('CPimport.txt',p,e)[:-1] # Assume monic
except(IOError,ValueError):
print(" Look for non-Conway primitive polynomial")
poly = findprimpoly(p,e)
else:
if nfactors[0][1] != len(poly):
raise ValueError('Polynomial {0} does not have degree {1}'.format(poly+[1],nfactors[0][1]))
return FiniteField(p,poly,var=var,fmtspec=fmtspec)
def GF(n, name='x', modulus=[]):
if numbthy.is_prime(n):
if len(modulus) < 2:
return FiniteField(n, [1], name)
else:
return FiniteField(
n, modulus,
name) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1):
raise ValueError(
'GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]
e = nfactors[0][1] # Prime p, exponent e
if (len(modulus) == 0):
try:
modulus = readconway('CPimport.txt', p, e)[:-1] # Assume monic
except (IOError, ValueError):
print(" Look for non-Conway primitive polynomial")
modulus = findprimpoly(p, e)
return FiniteField(p, modulus, name)
def findprimpoly(p,e): import sys if sys.version_info[0] > 2: iterrange = range # Version 3 patch else: iterrange = xrange # Version 2 patch ordfacts = numbthy.factor((p**e)-1) print (p**e)-1, " = ", ordfacts for thepolynum in iterrange(p+1,p**e): # Note: Skip [0,d1,d2,...] as not irred # Note: Skip [d0,0,0,...,0,1] as not primitive and probably not irred thepoly = [(thepolynum//(p**i))%p for i in range(e)] # length e list of p digits a = FiniteField(p,thepoly).gen() # create poly 'x' mod thepoly, check its order print thepoly, str(a) if ((a**((p**e)-1)) == 1): # check for irreducibility, else try next poly isprim = True for (thefact,thepow) in ordfacts: print thefact if ((a**(((p**e)-1)/thefact)) == 1): # thepoly not prim isprim = False break # Don't need to check any more cofactors if isprim: return thepoly print "Oops" # Should throw exception - should never reach this point, all prime/exp have primitive polys
def test_factor(self):
for testcase in ((-15, ((3, 1), (5, 1))),
(1234561000, ((2, 3), (5, 3), (211, 1), (5851, 1)))):
self.assertEqual(numbthy.factor(testcase[0]), testcase[1])
def distinct_factors(n):
factors = numbthy.factor(n)
# print factors
return len(factors)
