mirror of
https://github.com/python/cpython.git
synced 2025-11-04 03:44:55 +00:00
94 lines
2.3 KiB
Python
94 lines
2.3 KiB
Python
# 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)
|