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