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+# 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) <> type(''): s = `s`
+ n = len(s)
+ if n >= width: return s
+ return ' '*(width - n) + s
+
+def lj(s, width):
+ if type(s) <> type(''): s = `s`
+ n = len(s)
+ if n >= width: return s
+ return s + ' '*(width - n)