def PolyDivModOverZn(a: poly, b: poly, n: int) -> (poly, poly):
if n < 1:
raise ValueError
if n==1:
raise ZeroDivisionError
a = poly(np.mod(a.coef, n))
b = poly(np.mod(b.coef, n))
a = a.trim()
b = b.trim()
deg_a, deg_b = len(a.coef), len(b.coef)
if deg_a < deg_b:
return poly([0]), a
else:
f = poly(b.coef[::-1]).trim()
g = PolyInverseModOverZn(f, deg_a - deg_b + 1, n)
if g is None:
raise ZeroDivisionError
q = (poly(a.coef[::-1]).trim() * g).truncate(deg_a - deg_b + 1) # q:=rev_n(a)g mod x^(n-m+1)
q = poly(np.mod(q.coef, n))
q = poly(q.coef[::-1]).trim() # q:=rev_(n-m)(q)
if len(q.coef) < deg_a - deg_b + 1:
q.coef = np.concatenate([np.zeros(deg_a - deg_b + 1 - len(q)), q.coef])
bq = poly(np.mod((b * q).coef, n))
r = a - bq
r = poly(np.mod(r.coef, n))
q = poly(np.mod(q.coef, n))
r = r.trim()
q = q.trim()
return q, r
def PolyDivModOverQ(a: poly, b: poly) -> (poly, poly):
if not b.coef.any(): #если полином нулевой
raise ZeroDivisionError
a = a.trim()
b = b.trim()
n, m = len(a.coef), len(b.coef) # степени полиномов
if n < m:
return poly([0]), a
else:
f = poly(b.coef[::-1]).trim() #f:=rev_m(b)
g = PolyInverseModOverQ(f, n - m + 1) #q inverse of f modulo x^(n-m+1)
q = (poly(a.coef[::-1]).trim() * g).truncate(n - m + 1)#q:=rev_n(a)g mod x^(n-m+1)
q = poly(q.coef[::-1]).trim() #q:=rev_(n-m)(q)
if len(q.coef) < n - m + 1:
q.coef = np.concatenate([np.zeros(n - len(q)), q.coef])
r = a - b * q #r:=a-bq
r = r.trim()
q = q.trim()
return q, r
def PolyDivModOverQ(a: Poly, b: Poly) -> (Poly, Poly):
if not b.coef.any():
raise ZeroDivisionError
a = a.trim()
b = b.trim()
n, m = len(a.coef), len(b.coef)
if n < m:
return Poly([0]), a
else:
f = rev(b, m)
g = PolyInverseModOverQ(f, n - m + 1)
q = (rev(a, n) * g).truncate(n - m + 1)
q = rev(q, n - m + 1)
if len(q.coef) < n - m + 1:
q.coef = np.concatenate([np.zeros(n - len(q)), q.coef])
r = a - b * q
r = r.trim()
q = r.trim()
return q, r
def PolyDivModOverZn(a: Poly, b: Poly, mod_r: int) -> (Poly, Poly):
if mod_r < 1:
raise ValueError
if mod_r == 1:
raise ZeroDivisionError
a = Poly(np.mod(a.coef, mod_r))
b = Poly(np.mod(b.coef, mod_r))
a = a.trim()
b = b.trim()
n, m = len(a.coef), len(b.coef)
if n < m:
return Poly([0]), a
else:
f = rev(b, m)
g = PolyInverseModOverZn(f, n - m + 1, mod_r)
if g is None:
raise ZeroDivisionError
q = (rev(a, n) * g).truncate(n - m + 1)
q = Poly(np.mod(q.coef, mod_r))
q = rev(q, n - m)
if len(q.coef) < n - m + 1:
q.coef = np.concatenate([np.zeros(n - m + 1 - len(q)), q.coef])
bq = Poly(np.mod((b * q).coef, mod_r))
r = a - bq
r = Poly(np.mod(r.coef, mod_r))
q = Poly(np.mod(q.coef, mod_r))
r = r.trim()
q = q.trim()
return q, r
def PolyDivModOverZn(a, b: Poly, n: int): #−> (Poly , Poly ):
a = a.trim()
b = b.trim()
if (n < 1):
raise ValueError
if (a.degree() < b.degree()):
return Poly([0]), a
m = a.degree() - b.degree()
revb = Poly(b.coef[::-1])
rrevb = PolyInverseModOverZn(revb, m + 1, n)
if (rrevb is None):
raise ZeroDivisionError("")
reva = Poly(a.coef[::-1])
q1 = reva * rrevb
q1 = q1.truncate(m + 1)
q = Poly(q1.coef[::-1])
q = modPoly(q, n)
bq = b * q
bq = modPoly(bq, n)
r = modPoly(a - b * q, n)
r = Poly(r.coef[::-1])
r.trim()
r = Poly(r.coef[::-1])
return q, r
def PolyInverseModOverQ(f: Poly, r: int): #r=2, предполагаем f0[0] = 1
f = f.trim()
if (r < 1):
raise ValueError
#f = modPoly(f, zn)
f0 = f.coef[0]
if (f0 != Fraction(1, 1)):
f0 = 1 / f0
if (f0 == Fraction(0, 1)):
return None
g = Poly(f0)
l = int(math.ceil(math.log2(r)))
for k in range(1, l + 1):
g = (2 * g - f * (g**2))
g = g.truncate(2**k)
return g
def rev(p: Poly, n: int) -> Poly:
p = p.trim()
result = p.coef[::-1]
if len(p.coef) < n:
result = np.concatenate([np.zeros(n - len(p.coef)), result])
return Poly(result)
