def main1():
S = PolynomialRing(GF(Integer(13)), names=('x',))
(x,) = S.gens()
R = S.quotient(x**Integer(2) - Integer(3), names=('alpha',))
(alpha,) = R.gens()
print((Integer(2) + Integer(3) * alpha)
* (Integer(1) + Integer(2) * alpha))
def dict_to_pol(dct, bd=global_prec, base_ring=QQ):
R = PolynomialRing(base_ring, "u1, u2, q1, q2")
(u1, u2, q1, q2) = R.gens()
S = R.quotient(u1 * u2 - 1)
(uu1, uu2, qq1, qq2) = S.gens()
l = PrecisionDeg2(bd)
if not hasattr(dct, "__getitem__"):
return dct
return sum([dct[(n, r, m)] * uu1 ** r * qq1 ** n * qq2 ** m
if r > 0 else dct[(n, r, m)]
* uu2 ** (-r) * qq1 ** n * qq2 ** m for n, r, m in l])
def pol_to_dict(pl, bd=global_prec, base_ring=QQ):
R = PolynomialRing(base_ring, "u1,u2,q1,q2")
(u1, u2, q1, q2) = R.gens()
S = R.quotient(u1 * u2 - 1)
(uu1, uu2, qq1, qq2) = S.gens()
l = PrecisionDeg2(bd)
pl_lft = pl.lift()
dct = dict()
for n, r, m in l:
if r >= 0:
cfs = pl_lft.coefficient({u1: r, u2: 0, q1: n, q2: m})
else:
cfs = pl_lft.coefficient({u1: 0, u2: -r, q1: n, q2: m})
dct[(n, r, m)] = cfs
for t in l:
if not t in dct.keys():
dct[t] = 0
return dct
def find_sqrt(a, p):
assert (p - Integer(1)) % Integer(4) == Integer(0)
assert legendre_symbol(a, p) == Integer(1)
S = PolynomialRing(GF(p), names=('x',))
(x,) = S.gens()
R = S.quotient(x**Integer(2) - a, names=('alpha',))
(alpha,) = R.gens()
while True:
z = GF(p).random_element()
w = (Integer(1) + z * alpha)**((p - Integer(1)) // Integer(2))
(u, v) = (w[Integer(0)], w[Integer(1)])
if v != Integer(0):
break
if (-u / v)**Integer(2) == a:
return -u / v
if ((Integer(1) - u) / v)**Integer(2) == a:
return (Integer(1) - u) / v
if ((-Integer(1) - u) / v)**Integer(2) == a:
return (-Integer(1) - u) / v
