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| import time time.clock = time.time debug = True strict = False helpful_only = True dimension_min = 7 # 如果晶格达到该尺寸,则停止移除 # 显示有用矢量的统计数据 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")
# 显示带有 0 和 X 的矩阵 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 += '~' #print (a)
# 尝试删除无用的向量 # 从当前 = n-1(最后一个向量)开始 def remove_unhelpful(BB, monomials, bound, current): # 我们从当前 = n-1(最后一个向量)开始 if current == -1 or BB.dimensions()[0] <= dimension_min: return BB # 开始从后面检查 for ii in range(current, -1, -1): # 如果它没有用 if BB[ii, ii] >= bound: affected_vectors = 0 affected_vector_index = 0 # 让我们检查它是否影响其他向量 for jj in range(ii + 1, BB.dimensions()[0]): # 如果另一个向量受到影响: # 我们增加计数 if BB[jj, ii] != 0: affected_vectors += 1 affected_vector_index = jj # 等级:0 # 如果没有其他载体最终受到影响 # 我们删除它 if affected_vectors == 0: #print ("* removing unhelpful vector", ii) BB = BB.delete_columns([ii]) BB = BB.delete_rows([ii]) monomials.pop(ii) BB = remove_unhelpful(BB, monomials, bound, ii-1) return BB # 等级:1 #如果只有一个受到影响,我们会检查 # 如果它正在影响别的向量 elif affected_vectors == 1: affected_deeper = True for kk in range(affected_vector_index + 1, BB.dimensions()[0]): # 如果它影响哪怕一个向量 # 我们放弃这个 if BB[kk, affected_vector_index] != 0: affected_deeper = False # 如果没有其他向量受到影响,则将其删除,并且 # 这个有用的向量不够有用 #与我们无用的相比 if affected_deeper and abs(bound - BB[affected_vector_index, affected_vector_index]) < abs(bound - BB[ii, ii]): #print ("* removing unhelpful vectors", ii, "and", affected_vector_index) 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 # nothing happened 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 """ # substitution (Herrman and May) PR.<u, x, y> = 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) * polZ(u, x, y)^kk gg.append(xshift) gg.sort() # 单项式 x 移位列表 monomials = [] for polynomial in gg: for monomial in polynomial.monomials(): #对于多项式中的单项式。单项式(): if monomial not in monomials: # 如果单项不在单项中 monomials.append(monomial) monomials.sort() # y-移位 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) # substitution # 单项式 y 移位列表 for jj in range(1, tt + 1): for kk in range(floor(mm/tt) * jj, mm + 1): monomials.append(u^kk * y^jj) # 构造格 B 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) #约化格的原型 if helpful_only: # #自动删除 BB = remove_unhelpful(BB, monomials, modulus^mm, nn-1) # 重置维度 nn = BB.dimensions()[0] if nn == 0: print ("failure") return 0,0 # 检查向量是否有帮助 if debug: helpful_vectors(BB, modulus^mm) # 检查行列式是否正确界定 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)") # display the lattice basis if debug: matrix_overview(BB, modulus^mm) # LLL if debug: print ("optimizing basis of the lattice via LLL, this can take a long time") #BB = BB.BKZ(block_size=25) BB = BB.LLL() if debug: print ("LLL is done!") # 替换向量 i 和 j ->多项式 1 和 2 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): # 对于i and j, 构造两个多项式 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) # 结果 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) # solutions 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(): ############################################ # 随机生成数据 ########################################## #start_time =time.perf_counter start =time.clock() size=512 length_N = 2*size; ss=0 s=70; M=1 # the number of experiments delta = 299/1024 # p = random_prime(2^512,2^511) for i in range(M): # p = random_prime(2^size,None,2^(size-1)) # q = random_prime(2^size,None,2^(size-1)) # if(p<q): # temp=p # p=q # q=temp N = e = c = hint1 = # p高位 hint2 = # q高位 # print ("p真实高",s,"比特:", int(p/2^(512-s))) # print ("q真实高",s,"比特:", int(q/2^(512-s))) # N = p*q; # 解密指数d的指数( 最大0.292) m = 7 # 格大小(越大越好/越慢) t = round(((1-2*delta) * m)) # 来自 Herrmann 和 May 的优化 X = floor(N^delta) # Y = floor(N^(1/2)/2^s) # 如果 p、 q 大小相同,则正确 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); #A = N+1 P.<x,y> = PolynomialRing(ZZ) pol = 1 + x * (A + y) #构建的方程 # Checking bounds #if debug: #print ("=== 核对数据 ===") #print ("* delta:", delta) #print ("* delta < 0.292", delta < 0.292) #print ("* size of e:", ceil(log(e)/log(2))) # e的bit数 # print ("* size of N:", len(bin(N))) # N的bit数 #print ("* size of N:", ceil(log(N)/log(2))) # N的bit数 #print ("* m:", m, ", t:", t) # boneh_durfee if debug: ##print ("=== running algorithm ===") start_time = time.time() solx, soly = boneh_durfee(pol, e, m, t, X, Y) if solx > 0: #print ("=== solution found ===") 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)) #break print("ss=",ss) #end=time.process_time end=time.clock() print('Running time: %s Seconds'%(end-start)) if __name__ == "__main__": example()
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