# module 'zmod'
# Compute properties of mathematical "fields" formed by taking
# Z/n (the whole numbers modulo some whole number n) and an
# irreducible polynomial (i.e., a polynomial with only complex zeros),
# e.g., Z/5 and X**2 + 2.
#
# The field is formed by taking all possible linear combinations of
# a set of d base vectors (where d is the degree of the polynomial).
#
# Note that this procedure doesn't yield a field for all combinations
# of n and p: it may well be that some numbers have more than one
# inverse and others have none. This is what we check.
#
# Remember that a field is a ring where each element has an inverse.
# A ring has commutative addition and multiplication, a zero and a one:
# 0*x = x*0 = 0, 0+x = x+0 = x, 1*x = x*1 = x. Also, the distributive
# property holds: a*(b+c) = a*b + b*c.
# (XXX I forget if this is an axiom or follows from the rules.)
import poly
# Example N and polynomial
N = 5
P = poly.plus(poly.one(0, 2), poly.one(2, 1)) # 2 + x**2
# Return x modulo y. Returns >= 0 even if x < 0.
def mod(x, y):
return divmod(x, y)[1]
# Normalize a polynomial modulo n and modulo p.
def norm(a, n, p):
a = poly.modulo(a, p)
a = a[:]
for i in range(len(a)): a[i] = mod(a[i], n)
a = poly.normalize(a)
return a
# Make a list of all n^d elements of the proposed field.
def make_all(mat):
all = []
for row in mat:
for a in row:
all.append(a)
return all
def make_elements(n, d):
if d == 0: return [poly.one(0, 0)]
sub = make_elements(n, d-1)
all = []
for a in sub:
for i in range(n):
all.append(poly.plus(a, poly.one(d-1, i)))
return all
def make_inv(all, n, p):
x = poly.one(1, 1)
inv = []
for a in all:
inv.append(norm(poly.times(a, x), n, p))
return inv
def checkfield(n, p):
all = make_elements(n, len(p)-1)
inv = make_inv(all, n, p)
all1 = all[:]
inv1 = inv[:]
all1.sort()
inv1.sort()
if all1 == inv1: print 'BINGO!'
else:
print 'Sorry:', n, p
print all
print inv
def rj(s, width):
if type(s) is not type(''): s = `s`
n = len(s)
if n >= width: return s
return ' '*(width - n) + s
def lj(s, width):
if type(s) is not type(''): s = `s`
n = len(s)
if n >= width: return s
return s + ' '*(width - n)