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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
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()
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