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| import time time.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 print (nothelpful, "/", BB.dimensions()[0], " vectors are not helpful")
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[ii, jj] = gg[ii].monomial_coefficient(monomials[jj]) * monomials[jj](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() == [1]: 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 = 69207225407236621802315929835231678761546030648552499878532449478584182354765750349071726491300234635799981022731725455349420914234822062855723904939138000102040435210706843712478106458961468791872716857992483073814316706027260218386995042614451566024972455009936823034721213885693157803402838690192435869721 e = 28439197921283357831697812537770489393495780585893113255835906777860388696994349687910509232020125501124985537099309478678733953591875352794038209770419925216539701941346792691704315717440469781000758533118851176304883130375842134875219545766782891367082825940026559693057872966937790726617783138946733512771 c = 22634701644450101524194718626550730546669791908217195025458791096208664618277869132516992188391372685210476489439282043033169958992171845152117468239445520601245104073454741171223045094363461153069787573765111331214431209598625611554915848071794889073522221012875111880946316417640573688399584093700714982302 hint1 = 654543761191063613807 hint2 = 819778612327847774041
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()
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