def square_free_decomposition(poly):
assert type(poly) == ZPoly
print("!")
if len(poly) == 1:
print(poly)
return poly
d = poly.derivative()
if d != ZPoly([0],poly.M):
g = zpoly_gcd(poly,d)
print(g)
if g == ZPoly([1],poly.M):
print("!!!")
print(poly)
return poly
else:
print(poly//g)
return square_free_decomposition(g)
else:
pwr = 1
while True:
for g in all_monic_zpolys(poly.M):
p = g**(poly.M**pwr)
if p == poly and g.derivative() != ZPoly([0],poly.M):
return square_free_decomposition(g)
if len(p) > len(poly):
pwr += 1
break
def all_irreducible_mod2():
"""Sieve of Eratosthenes for polynomials in Z/2Z"""
out = [ZPoly([1,1,1],2)]
yield ZPoly([0,1],2)
yield ZPoly([1,1],2)
all_p = all_zpolys(2)
for i in range(7):
next(all_p)
while True:
poly = next(all_p)
if poly(0) == poly(1) == 1:
for i in out:
if len(i) > len(poly) // 2:
out.append(poly)
yield poly
break
r = poly%i
if r == ZPoly([0],2):
out.append(poly)
yield poly
break
def zpoly_egcd(a, b):
"""Extended Euclidean Algorithm for ZPolys"""
M = a.M
if a == ZPoly([0],M):
return (b, ZPoly([0],M), ZPoly([1],M))
else:
g, y, x = zpoly_egcd(b % a, a)
return (g, x - (b // a) * y, y)
def zpoly_lagrange_interpolation(X,Y,M):
"""Use lagrange polynomial to interpolate points of a zpoly"""
final = ZPoly([0],M)
for x,y in zip(X,Y):
out = ZPoly([y],M)
for m in X:
if m != x:
d = mod_inv(x-m,M)
P = ZPoly([-m,1],M)
out *= P*d
final += out
return final
def all_monic_zpolys(M):
"""Generator for all monic polynomials over M"""
yield ZPoly([1],M)
co = [i for i in range(M)]
l = 1
while True:
P = product(co,repeat=l)
for p in P:
coefs = [i for i in reversed((1,)+p)]
yield ZPoly(coefs,M)
l += 1
def all_zpolys(M):
"""Generator for all polynomials over M"""
co = [i for i in range(M)]
for c in co:
yield ZPoly([c],M)
l = 1
while True:
P = product(co,repeat=l)
for p in P:
for c in co[1:]:
coefs = [i for i in reversed((c,)+p)]
yield ZPoly(coefs,M)
l += 1
def finite_field_example(P):
S = FF2(ZPoly([0, 1], P.M), P)
out = FF2(ZPoly([1], P.M), P)
L = [FF2(P, P)]
elems = (P.M)**(len(P) - 1)
for i in range(1, elems):
if out in L:
raise Exception("Not a finite field")
L.append(out)
out = (out * S)
# for i in L:
# print(i)
return L
def ff_inv(a, R):
"""Modular Multiplicative Inverse"""
a = a % R
g, x, _ = zpoly_egcd(a, R)
if g != ZPoly([1], a.M):
raise ValueError(
f"Multiplicative inverse of {a.full_name} mod {R.full_name} does not exist"
)
else:
return x % R
def __pow__(self, other):
if type(other) != int:
raise TypeError("Power of ZPolyProd must be an integer")
if other < 0:
raise TypeError("Power of ZPolyProd must be non-negative")
if other == 0:
return ZPolyProd(ZPoly([1], self.M), self.M)
if other == 1:
return self
else:
out = self
for i in range(other - 1):
out *= self
return out
def make_shamir_secret(secret,k,n,M):
"""Take a number and produce k ordered pairs such that n can be used to reconstruct the number"""
if secret > M:
raise ValueError("secret cannot be less than M or information will be lost")
if k > n:
raise ValueError("parts needed to reconstruct cannot be greater than total points created")
if len(prime_factorization(M)) != 1:
raise ValueError("Order for finite field must be a prime")
co = [secret] + [randint(0,M-1) for i in range(k-1)]
P = ZPoly( co, M )
pts = []
for i in range(n):
r = randint(0,M-1)
pts.append( (r,P(r)) )
return pts
def __add__(self, other):
A = self.terms.copy()
if type(other) == int:
return self + ZPoly([other], self.M)
if type(other) == ZPoly:
if other.M == self.M:
A.update([other])
else:
raise ValueError("Values of M do not match")
elif type(other) == ZPolySum:
A.update(other.terms)
else:
return NotImplemented
return ZPolySum(A, self.M)
U = RFunc(["1/8"], ["1/3", "5/4"])
rf_minitests = [
(R, "(x + 1) / (3x + 2)"),
(S, "(12x + 28) / (x^2 - 3x + 2)"), #simplify
(T, "x^2 + 5x + 3"), # denominator one
(U, "1/8 / (5/4x + 1/3)"),
(U.primitive_part, "12 / (120x + 32)")
]
###########
## ZPoly ##
###########
M = 29
P = ZPoly([50, 0, -8, 121, 9], M)
Q = ZPoly([25, 29, 8], M)
PP = ZPoly([50, 0, -8, 121, 9])
QQ = ZPoly([25, 29, 1])
z_minitests = [
(P, "9x^4 + 5x^3 + 21x^2 + 21"),
(Q, "8x^2 + 25"),
(P + Q, "9x^4 + 5x^3 + 17"),
(P * Q, "14x^6 + 11x^5 + 16x^4 + 9x^3 + 26x^2 + 3"),
(Q**0, "1"),
(Q**1, "8x^2 + 25"),
(Q**2, "6x^4 + 23x^2 + 16"),
(P // Q, "12x^2 + 26x + 5"),
(P % Q, "17x + 12"),
(P // Q * Q + P % Q, str(P)),
def cast_to_poly(self):
"""Add everything together as a ZPoly"""
out = ZPoly([0], self.M)
for val, mul in self.terms.items():
out += mul * val
return out
pts = make_shamir_secret(secret,min_parts,total_parts,F)
print(f"We will use Shamir's method to break up the secret number {secret} into {total_parts} pieces such any {min_parts} pieces can be used to get the secret.")
print(f"\nTo do this we create a random polynomial of degree {min_parts} that has constant term {secret} over a finite field and choose {total_parts} points on it.")
print(f"\nUsing a finite field of order {F} we get the points:")
for i in pts:
print(i)
print(f"\nPicking three of those we can reconstruct the answer:")
rpts = sample(pts,min_parts)
print(get_shamir_secret(rpts,F))
print("\n\nGCD of ZPolys")
P = ZPoly( [1,2,3], 5) * ZPoly( [2,4], 5)
Q = ZPoly( [1,2,3], 5)
print(P)
print(Q)
print(zpoly_gcd(P,Q))
print("\n\nSquare-Free Decomposition")
P = ZPoly([1,0,2,2,0,1,1,0,2,2,0,1],3)
print(P)
print(square_free_decomposition(P))
print("\nAll monic polynomials over Z/3Z with degree less than 3")
for poly in all_monic_zpolys(3):
if len(poly) > 3:
def cast_to_poly(self):
"""Multiply everything together as a ZPoly"""
out = ZPoly([1], self.M)
for val, pwr in self.terms.items():
out *= val**pwr
return out
# Remove the modulo terms
out = re.sub(" \(mod \d*\)", "", out)
# Remove $ signs
out = out.replace("$", "")
if self.M:
return f"${out}$ [mod {self.M}]"
else:
return f"${out}$"
# Things that are like attributes can be access as properties
pretty_name = property(_pretty_name)
full_name = property(_full_name)
if __name__ == '__main__':
P = ZPoly([1, 2, 3], 19)
Q = ZPoly([2, 4], 19)
C = ZPolyProd([P, Q], 19)
print(P)
print(Q)
print(C)
print(C * Q)
print(C * C)
print(C**3)
print(C * 2)
D = 2 * C * Q
print(D)
print(D**0)
print(D.pretty_name)
print(D.full_name)
f"Multiplicative inverse of {a.full_name} mod {R.full_name} does not exist"
)
else:
return x % R
def ff_div(a, b, R):
if b == R or str(b) == "0":
raise ZeroDivisionError
return (a * ff_inv(b, R)) % R
if __name__ == '__main__':
import random
P = ZPoly([1, 1, 1, 0, 0, 0, 0, 1, 1], 2)
L = finite_field_example(P)
print(
f"\n\nFinite Field of Characteristic 2 with {len(L)} elements\nReducing Polynomial: {P}"
)
print("\n\nAddition and Subtraction and identical")
for i in range(3):
a, b = random.sample(L, 2)
print(f"{a.bitstring()} + {b.bitstring()} = {(a+b).bitstring()}")
print(f"{a.bitstring()} - {b.bitstring()} = {(a-b).bitstring()}")
print()
print("\n\nExamples of Multiplication over the Finite Field")
for i in range(3):