def get_primitive_polys_gf2(deg, max_num=None, random=True):
# From http://www.seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/theory.html
# Step 1: Generate a polynomial
r = (2 ** deg) - 1
ret = []
for poly in xrange(1 << deg, 1 << (deg + 1)):
# Step 2 (not needed for GF2)
# Step 3: check f(a)!=0 for 1 <= a <= p-1
if (utils.count_bits(poly) % 2) == 0:
continue
# Step 4:
if not berlekamp_factorisation_check(poly):
continue
# Step 5: x**r == a(mod f(x), p) for some integer a (gf(2) means a == 1)
if utils.gf2_mod(1 << r, poly) != 1:
continue
# Step 6: skip
# Step 7:
p_k = list(set(utils.find_prime_factors(r)))
passed = True
for p in p_k:
if utils.gf2_mod(1 << (r/p), poly) == 1:
passed = False
if not passed:
continue
#print poly
ret.append(poly)
print "Found poly = " + bin(poly)
if max_num and len(ret) >= max_num:
return ret
return ret
def berlekamp_factorisation_check(a):
# Check for two or more irreducable factors
# from http://www.diva-portal.org/smash/get/diva2:414578/FULLTEXT01.pdf
# gcd(f(x), f'(x))
# (a >> 1) & 0x77777777 is a quick way of differentiating a gf2 poly < 32 bits
if utils.gf2_gcd(a, (a >> 1) & 0x77777777) != 1:
return False
degree = utils.get_highest_set_bit(a)
polys = [utils.gf2_mod(1 << (2*i), a) ^ (1 << i) for i in xrange(degree)]
for i, div in enumerate(polys[:i]):
for j in range(len(polys[i+1:])):
polys[j+i+1] ^= div
if polys[j+i+1] == 0:
# Matrix of polys is not full rank
return False
return True