Write the simplest polynomial function with the given zeros. a) 1 - i and 2 b) 1+5 and 3 c) 2i, 2, and 2 Complex and Irrational Roots of a Polynomial: The value x=a is a root of a polynomial p(x) if p(a)=0.
root =tuple(R(root[var])forvarinf.variables()) roots.append(root)returnrootsreturn[] a_high =3960604425233637243960750976884707892473356737965752732899783806146911898367312949419828751012380013933993271701949681295313483782313836179989146607655230162315784541236731368582965456428944524621026385297377746108440938677401125816586119588080150103855075450874...
next() #p4 = 0x81a722c9fc2b2ed061fdab737e3893506eae71ca6415fce14c0f9a45f8e2300711119fa0a5135a053e654fead010b96e987841e47db586a55e3d4494613aa0cc4e4ab59fc6a958b5 kbits = pbits - 576 p4 = p4 << kbits PR.<x> = PolynomialRing(Zmod(n)) f = x + p4 x0 = f.small_roots(X=2...
方程分解 $N$. #!/usr/bin/sage -python from sageall import * from Crypto.Util.numberimport long_to_bytes import gmpy2 _p = p0 - (p0&(2**668-2**444)) PR = PolynomialRing(Zmod(N1), 'x') x = PR.gen() f = 2**444 * x + _p f = f.monic() r = f.small_roots(X...
This example shows how to write class-based unit tests to qualify the correctness of a function defined in a file in your current folder. ThequadraticSolverfunction takes as inputs the coefficients of a quadratic polynomial and returns the roots of that polynomial. If the coefficients are specifi...
01522403673111515117219716055561941951891570977025178643791gift=(gift//2024)-2x1=(gift>>86)<<86e=0x10001PR.<x>=PolynomialRing(RealField(1000))f=x*(x1-x)-nphigh=int(f.roots()[0][0])print(phigh,phigh.bit_length())PR.<x>=PolynomialRing(Zmod(n))f=phigh+...
#sage import time from Crypto.Util.number import * """ Setting debug to true will display more informations about the lattice, the bounds, the vectors... """ debug = True """ Setting strict to true will stop the algorithm (and return (-1, -1)) if we don't have a correct upper...
(P,p,k):x,y,z=PPR.<c>=PolynomialRing(Zmod(p))f=x^3+c*y^3+c^2*z^3-3*c*x*y*z-1c_list=[ZZ(x)forx,repinf.roots()]PR.<c>=PolynomialRing(Zmod(p^k))f=x^3+c*y^3+c^2*z^3-3*c*x*y*z-1c_list=hensel_lifting(f,p,k,c_list)c=ZZ(c_list[0])returnc_list...
append(root) return roots return [] def boneh_durfee(): print('Boneh Durfee') beta = 0.397 bounds = (floor(N^beta) // 2, floor(N^0.5)) R = Integers(e) P.<k, s> = PolynomialRing(R) f = k * (N^2 + N*2*s+(2*s)^2-N+2*s+1) + 1 print(small_roots(f, bounds, ...
(Herrman and May) PR. = PolynomialRing(ZZ) #多项式环 Q = PR.quotient(x*y + 1 - u) # u = xy + 1 polZ = Q(pol).lift() UU = XX*YY + 1 # x-移位 gg = [] for kk in range(mm + 1): for ii in range(mm - kk + 1): xshift = x^ii * modulus^(mm - kk) * po...