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| 1 | +# module 'zmod' |
| 2 | + |
| 3 | +# Compute properties of mathematical "fields" formed by taking |
| 4 | +# Z/n (the whole numbers modulo some whole number n) and an |
| 5 | +# irreducible polynomial (i.e., a polynomial with only complex zeros), |
| 6 | +# e.g., Z/5 and X**2 + 2. |
| 7 | +# |
| 8 | +# The field is formed by taking all possible linear combinations of |
| 9 | +# a set of d base vectors (where d is the degree of the polynomial). |
| 10 | +# |
| 11 | +# Note that this procedure doesn't yield a field for all combinations |
| 12 | +# of n and p: it may well be that some numbers have more than one |
| 13 | +# inverse and others have none. This is what we check. |
| 14 | +# |
| 15 | +# Remember that a field is a ring where each element has an inverse. |
| 16 | +# A ring has commutative addition and multiplication, a zero and a one: |
| 17 | +# 0*x = x*0 = 0, 0+x = x+0 = x, 1*x = x*1 = x. Also, the distributive |
| 18 | +# property holds: a*(b+c) = a*b + b*c. |
| 19 | +# (XXX I forget if this is an axiom or follows from the rules.) |
| 20 | + |
| 21 | +import poly |
| 22 | + |
| 23 | + |
| 24 | +# Example N and polynomial |
| 25 | + |
| 26 | +N = 5 |
| 27 | +P = poly.plus(poly.one(0, 2), poly.one(2, 1)) # 2 + x**2 |
| 28 | + |
| 29 | + |
| 30 | +# Return x modulo y. Returns >= 0 even if x < 0. |
| 31 | + |
| 32 | +def mod(x, y): |
| 33 | + return divmod(x, y)[1] |
| 34 | + |
| 35 | + |
| 36 | +# Normalize a polynomial modulo n and modulo p. |
| 37 | + |
| 38 | +def norm(a, n, p): |
| 39 | + a = poly.modulo(a, p) |
| 40 | + a = a[:] |
| 41 | + for i in range(len(a)): a[i] = mod(a[i], n) |
| 42 | + a = poly.normalize(a) |
| 43 | + return a |
| 44 | + |
| 45 | + |
| 46 | +# Make a list of all n^d elements of the proposed field. |
| 47 | + |
| 48 | +def make_all(mat): |
| 49 | + all = [] |
| 50 | + for row in mat: |
| 51 | + for a in row: |
| 52 | + all.append(a) |
| 53 | + return all |
| 54 | + |
| 55 | +def make_elements(n, d): |
| 56 | + if d = 0: return [poly.one(0, 0)] |
| 57 | + sub = make_elements(n, d-1) |
| 58 | + all = [] |
| 59 | + for a in sub: |
| 60 | + for i in range(n): |
| 61 | + all.append(poly.plus(a, poly.one(d-1, i))) |
| 62 | + return all |
| 63 | + |
| 64 | +def make_inv(all, n, p): |
| 65 | + x = poly.one(1, 1) |
| 66 | + inv = [] |
| 67 | + for a in all: |
| 68 | + inv.append(norm(poly.times(a, x), n, p)) |
| 69 | + return inv |
| 70 | + |
| 71 | +def checkfield(n, p): |
| 72 | + all = make_elements(n, len(p)-1) |
| 73 | + inv = make_inv(all, n, p) |
| 74 | + all1 = all[:] |
| 75 | + inv1 = inv[:] |
| 76 | + all1.sort() |
| 77 | + inv1.sort() |
| 78 | + if all1 = inv1: print 'BINGO!' |
| 79 | + else: |
| 80 | + print 'Sorry:', n, p |
| 81 | + print all |
| 82 | + print inv |
| 83 | + |
| 84 | +def rj(s, width): |
| 85 | + if type(s) <> type(''): s = `s` |
| 86 | + n = len(s) |
| 87 | + if n >= width: return s |
| 88 | + return ' '*(width - n) + s |
| 89 | + |
| 90 | +def lj(s, width): |
| 91 | + if type(s) <> type(''): s = `s` |
| 92 | + n = len(s) |
| 93 | + if n >= width: return s |
| 94 | + return s + ' '*(width - n) |
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