def verify_1(self,AT, BT, CT, HT, p_n_a, p_n_b, p_n_c, p_n_h):
self.AT = AT
self.BT = BT
self.CT = CT
self.HT = HT
x = symbols('x')
AP = Poly(np.flip(AT, 0), x)
BP = Poly(np.flip(BT, 0), x)
CP = Poly(np.flip(CT, 0), x)
HP = Poly(np.flip(HT, 0), x)
print(HP)
# A(t_0)
A_t_0 = AP.eval(self.t_0)
B_t_0 = BP.eval(self.t_0)
C_t_0 = CP.eval(self.t_0)
H_t_0 = HP.eval(self.t_0)
# verifier
self.v_n_a = cv.mul_point(cv.order + int(A_t_0), G)
self.v_n_b = cv.mul_point(cv.order + int(B_t_0), G)
self.v_n_c = cv.mul_point(cv.order + int(C_t_0), G)
self.v_n_h = cv.mul_point(cv.order + int(H_t_0), G)
print(p_n_a == self.v_n_a, p_n_b == self.v_n_b , p_n_c == self.v_n_c , p_n_h == self.v_n_h )
return (p_n_a == self.v_n_a and p_n_b == self.v_n_b and p_n_c == self.v_n_c and p_n_h == self.v_n_h)
def first_alpha_power_root(poly, irr_poly, p, elements_to_check=None):
poly = Poly([(Poly(coeff, alpha) % irr_poly).trunc(p).as_expr()
for coeff in poly.all_coeffs()], x)
test_poly = Poly(1, alpha)
log.debug(f"testing f:{poly}")
for i in range(1, p**irr_poly.degree()):
test_poly = (Poly(Poly(alpha, alpha) * test_poly, alpha) %
irr_poly).set_domain(GF(p))
if elements_to_check is not None and i not in elements_to_check:
continue
value = Poly((Poly(poly.eval(test_poly.as_expr()), alpha) % irr_poly),
alpha).trunc(p)
log.debug(f"testing alpha^{i} f({test_poly})={value}")
if value.is_zero:
return i
return -1
0
>>> residue(2/sin(x), x, 0)
2
This function is essential for the Residue Theorem [1].
'''
f = x * (x * x + 1)
p = Poly(f, x)
A = -2 + 2 * I
B = 2 + 2 * I
C = 2 - 2 * I
D = -2 - 2 * I
ABCD = [A, B, C, D]
path = list(zip([D] + ABCD, ABCD))
print([p.eval(x, z) for z in ABCD])
logf = log(f)
print([logf.subs(x, z).evalf() for z in ABCD])
ip = [((b - a) / f.subs(x, b)).evalf() for a, b in path]
print(ip, sum(ip))
ress = [residue(1 / f, x, z0) for z0 in [0, I, -I]]
print(sum(ress), ress)
'''A->B : sum(f*dz) = sum(f*dx) = F|(-2->2)
'''
def residue(f, circle, N):
'''residue(f, A->B->.., N) == integrate(f, A->..->A, N)/2pi/1j
'''
def piecewise_symbolic_integral(self, integrand, x, y=None):
"""
Computes the symbolic integral of 'x' of a piecewise polynomial 'integrand'.
The result might be a sympy expression or a numerical value.
Parameters
----------
integrand : list
A list of (lower bound, upper bound, polynomial)
x : object
A string/sympy expression representing the integration variable
"""
res = 0
#logger.debug(f"\t\t\t\tpiecewise_symbolic_integral")
#logger.debug(f"\t\t\t\tlen(integrand): {len(integrand)} --- y: {y}")
for l, u, p in integrand:
symx = symvar(x)
symy = symvar(y) if y else symvar("aux_y")
syml = Poly(to_sympy(l), symy, domain="QQ")
symu = Poly(to_sympy(u), symy, domain="QQ")
#logger.debug(f"\t\t\t\t\tl: {l} --- u: {u} --- p: {p}")
if type(p) != Poly:
symp = Poly(to_sympy(p), symx, domain="QQ")
else:
symp = Poly(p.as_expr(), symx, domain=f"QQ[{symy}]") if y else p
if self.cache is not None: # for cache = True
""" hierarchical cache, where we cache:
- the anti-derivatives for integrands, retrieved by the same
integrand key
- the partial integration term, retrieved by the same
(integrand key, lower / upper bound key) pair
- the whole integration, retrieved by the same
(integrand key, lower bound key, upper bound key) pair
"""
bds_ks = [MPWMI.cache_key(syml)[0],
MPWMI.cache_key(symu)[0]] # cache keys for bounds
bds = [syml.as_expr(),
symu.as_expr()]
p_ks = MPWMI.cache_key(symp) # cache key for integrand polynomial
trm_ks = [(bds_ks[0], p_ks[0]),
(bds_ks[1], p_ks[0])]
if (bds_ks[0], bds_ks[1], p_ks[0]) in self.cache:
# retrieve the whole integration
self.cache_hit[True] += 1
symintegral = self.cache[(bds_ks[0], bds_ks[1], p_ks[0])]
symintegral = symintegral.subs(symintegral.gens[0], symy)
else:
terms = []
for tk in trm_ks: # retrieve partial integration terms
if tk in self.cache:
trm = self.cache[tk]
trm = trm.subs(trm.gens[0], symy)
terms.append(trm)
else:
terms.append(None)
if None not in terms:
self.cache_hit[True] += 1
else:
if p_ks[0] in self.cache: # retrieve anti-derivative
self.cache_hit[True] += 1
antidrv = self.cache[p_ks[0]]
antidrv_expr = antidrv.as_expr().subs(antidrv.gens[0], symx)
antidrv = Poly(antidrv_expr, symx,
domain=f"QQ[{symy}]") if y \
else Poly(antidrv_expr, symx, domain="QQ")
else:
self.cache_hit[False] += 1
antidrv = symp.integrate(symx)
for k in p_ks: # cache anti-derivative
self.cache[k] = antidrv
for i in range(len(terms)):
if terms[i] is not None:
continue
terms[i] = antidrv.eval({symx: bds[i]})
terms[i] = Poly(terms[i].as_expr(), symy, domain="QQ")
for k in p_ks: # cache partial integration terms
self.cache[(bds_ks[i], k)] = terms[i]
symintegral = terms[1] - terms[0]
for k in p_ks: # cache the whole integration
self.cache[(bds_ks[0], bds_ks[1], k)] = symintegral
else: # for cache = False
antidrv = symp.integrate(symx)
symintegral = antidrv.eval({symx: symu.as_expr()}) - \
antidrv.eval({symx: syml.as_expr()})
res += symintegral
#logger.debug(f"\t\t\t\t\tsymintegral: {symintegral}")
return res
Gs = G.subs(x, -x)
Gs = Poly(Gs + pp, x, modulus=p)
A, R = reduction(F, Gs)
res = res * pow(R.coeffs()[0], G.degree(), p) % p
res = -res if R.degree() % 2 else res
R = tru_mod(invs(R.coeffs()[0]) * R)
R = -R if R.degree() % 2 else R
return resultant(G, R, res)
anss = 1
for gs in g_zeros:
anss *= f_pol.eval(x, -gs)
anss = 1
for fs in f_zeros:
anss *= g_pol.eval(x, -fs)
# 나눠:
_, r_pol = reduction(f_pol, Poly(g_pol.subs(x, -x), x))
anss = 1
print(r_pol)
for gs in g_zeros:
anss *= r_pol.eval(-gs)
print(anss)