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 | import timetime.clock = time.time
 
 debug = True
 
 strict = False
 
 helpful_only = True
 dimension_min = 7
 
 def helpful_vectors(BB, modulus):
 nothelpful = 0
 for ii in range(BB.dimensions()[0]):
 if BB[ii,ii] >= modulus:
 nothelpful += 1
 
 
 
 def matrix_overview(BB, bound):
 for ii in range(BB.dimensions()[0]):
 a = ('%02d ' % ii)
 for jj in range(BB.dimensions()[1]):
 a += '0' if BB[ii,jj] == 0 else 'X'
 if BB.dimensions()[0] < 60:
 a += ' '
 if BB[ii, ii] >= bound:
 a += '~'
 
 
 
 
 def remove_unhelpful(BB, monomials, bound, current):
 
 if current == -1 or BB.dimensions()[0] <= dimension_min:
 return BB
 
 
 for ii in range(current, -1, -1):
 
 if BB[ii, ii] >= bound:
 affected_vectors = 0
 affected_vector_index = 0
 
 for jj in range(ii + 1, BB.dimensions()[0]):
 
 
 if BB[jj, ii] != 0:
 affected_vectors += 1
 affected_vector_index = jj
 
 
 
 
 if affected_vectors == 0:
 
 BB = BB.delete_columns([ii])
 BB = BB.delete_rows([ii])
 monomials.pop(ii)
 BB = remove_unhelpful(BB, monomials, bound, ii-1)
 return BB
 
 
 
 
 elif affected_vectors == 1:
 affected_deeper = True
 for kk in range(affected_vector_index + 1, BB.dimensions()[0]):
 
 
 if BB[kk, affected_vector_index] != 0:
 affected_deeper = False
 
 
 
 if affected_deeper and abs(bound - BB[affected_vector_index, affected_vector_index]) < abs(bound - BB[ii, ii]):
 
 BB = BB.delete_columns([affected_vector_index, ii])
 BB = BB.delete_rows([affected_vector_index, ii])
 monomials.pop(affected_vector_index)
 monomials.pop(ii)
 BB = remove_unhelpful(BB, monomials, bound, ii-1)
 return BB
 
 return BB
 
 """
 Returns:
 * 0,0   if it fails
 * -1,-1 如果 "strict=true",并且行列式不受约束
 * x0,y0 the solutions of `pol`
 """
 def boneh_durfee(pol, modulus, mm, tt, XX, YY):
 """
 Boneh and Durfee revisited by Herrmann and May
 
 在以下情况下找到解决方案:
 * d < N^delta
 * |x|< e^delta
 * |y|< e^0.5
 每当 delta < 1 - sqrt(2)/2 ~ 0.292
 """
 
 
 PR.<u, x, y> = PolynomialRing(ZZ)
 Q = PR.quotient(x*y + 1 - u)
 polZ = Q(pol).lift()
 
 UU = XX*YY + 1
 
 
 gg = []
 for kk in range(mm + 1):
 for ii in range(mm - kk + 1):
 xshift = x^ii * modulus^(mm - kk) * polZ(u, x, y)^kk
 gg.append(xshift)
 gg.sort()
 
 
 monomials = []
 for polynomial in gg:
 for monomial in polynomial.monomials():
 if monomial not in monomials:
 monomials.append(monomial)
 monomials.sort()
 
 
 for jj in range(1, tt + 1):
 for kk in range(floor(mm/tt) * jj, mm + 1):
 yshift = y^jj * polZ(u, x, y)^kk * modulus^(mm - kk)
 yshift = Q(yshift).lift()
 gg.append(yshift)
 
 
 for jj in range(1, tt + 1):
 for kk in range(floor(mm/tt) * jj, mm + 1):
 monomials.append(u^kk * y^jj)
 
 
 nn = len(monomials)
 BB = Matrix(ZZ, nn)
 for ii in range(nn):
 BB[ii, 0] = gg[ii](0, 0, 0)
 for jj in range(1, ii + 1):
 if monomials[jj] in gg[ii].monomials():
 BB(UU,XX,YY)
 
 
 if helpful_only:
 
 BB = remove_unhelpful(BB, monomials, modulus^mm, nn-1)
 
 nn = BB.dimensions()[0]
 if nn == 0:
 print ("failure")
 return 0,0
 
 
 if debug:
 helpful_vectors(BB, modulus^mm)
 
 
 det = BB.det()
 bound = modulus^(mm*nn)
 if det >= bound:
 print ("We do not have det < bound. Solutions might not be found.")
 print ("Try with highers m and t.")
 if debug:
 diff = (log(det) - log(bound)) / log(2)
 print ("size det(L) - size e^(m*n) = ", floor(diff))
 if strict:
 return -1, -1
 else:
 print ("det(L) < e^(m*n) (good! If a solution exists < N^delta, it will be found)")
 
 
 if debug:
 matrix_overview(BB, modulus^mm)
 
 
 if debug:
 print ("optimizing basis of the lattice via LLL, this can take a long time")
 
 
 BB = BB.LLL()
 
 if debug:
 print ("LLL is done!")
 
 
 if debug:
 print ("在格中寻找线性无关向量")
 found_polynomials = False
 
 for pol1_idx in range(nn - 1):
 for pol2_idx in range(pol1_idx + 1, nn):
 
 
 
 PR.<w,z> = PolynomialRing(ZZ)
 pol1 = pol2 = 0
 for jj in range(nn):
 pol1 += monomials[jj](w*z+1,w,z) * BB[pol1_idx, jj] / monomials[jj](UU,XX,YY)
 pol2 += monomials[jj](w*z+1,w,z) * BB[pol2_idx, jj] / monomials[jj](UU,XX,YY)
 
 
 PR.<q> = PolynomialRing(ZZ)
 rr = pol1.resultant(pol2)
 
 
 if rr.is_zero() or rr.monomials() == :
 continue
 else:
 print ("found them, using vectors", pol1_idx, "and", pol2_idx)
 found_polynomials = True
 break
 if found_polynomials:
 break
 
 if not found_polynomials:
 print ("no independant vectors could be found. This should very rarely happen...")
 return 0, 0
 
 rr = rr(q, q)
 
 
 soly = rr.roots()
 
 if len(soly) == 0:
 print ("Your prediction (delta) is too small")
 return 0, 0
 
 soly = soly[0][0]
 ss = pol1(q, soly)
 solx = ss.roots()[0][0]
 return solx, soly
 
 def example():
 
 
 
 
 start =time.clock()
 size=512
 length_N = 2*size;
 ss=0
 s=70;
 M=1
 delta = 299/1024
 
 for i in range(M):
 
 
 
 
 
 
 N = 64421669931279763032337930351371074312763623230334264672272991963587968145469789623226631192910548697947848779021944177739911084383786634595255436873350170397825307930618697902052402139072111389681839113510986333829074390665317265603346122882832568330112945567680656219994967593371244275156478322204419261819
 e = 31010691142874094705557092175690153458422282630834301371323563655985170293061551914791533335820551677589839843761852199680128997193432075880230502409814741638209947794142932041357373199521605405986705517942961028228255066560857365694517578822510622110740926010357756915595155038827023753366174321207668419113
 c = 44228757856694834445202355486356318880471120936109992694489050154150768403667631244534149277159399976972028214341072460353357561691320731420869678207365539080035139895735181119706727564445399237943178679966032596522818504363414526390960660239215638736706163678206959509331192083842100728618312517781111784803
 hint1 =  597218086872775768003
 hint2 =  836339982627318976124
 
 
 
 
 
 
 
 
 
 
 m = 7
 t = round(((1-2*delta) * m))
 X = floor(N^delta)
 Y = floor(N^(1/2)/2^s)
 for l in range(int(hint1),int(hint1)+1):
 print('\n\n\n l=',l)
 pM=l;
 p0=pM*2^(size-s)+2^(size-s)-1;
 q0=N/p0;
 qM=int(q0/2^(size-s))
 A = N + 1-pM*2^(size-s)-qM*2^(size-s);
 
 P.<x,y> = PolynomialRing(ZZ)
 pol = 1 + x * (A + y)
 
 
 
 
 
 
 
 
 
 
 
 
 if debug:
 
 start_time = time.time()
 
 
 solx, soly = boneh_durfee(pol, e, m, t, X, Y)
 
 
 if solx > 0:
 
 if False:
 print ("x:", solx)
 print ("y:", soly)
 
 d_sol = int(pol(solx, soly) / e)
 ss=ss+1
 
 print ("=== solution found ===")
 print ("p的高比特为:",l)
 print ("q的高比特为:",qM)
 print ("d=",d_sol)
 
 if debug:
 print("=== %s seconds ===" % (time.time() - start_time))
 
 print("ss=",ss)
 
 end=time.clock()
 print('Running time: %s Seconds'%(end-start))
 if __name__ == "__main__":
 example()
 
 |