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_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
# 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)
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
# 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)
# module 'zmod'
# module 'zmod'