ECDSA 安全漏洞剖析：从非 ce 泄露到密钥恢复实战

k = (s₁ - s₂)⁻¹(H(m₁) - H(m₂))x = r⁻¹(ks - H(m))

import ecdsa, olll
# 生成含偏差nonce的签名gen = ecdsa.NIST256p.generatororder = gen.order()priv_key = ecdsa.ecdsa.Private_key(pub_key, secret)nonces = [random.getrandbits(128) + (fixed_bits << 128) for _ in range(6)]sigs = [priv_key.sign(msgs[i], nonces[i]) for i in range(6)]
# 构建格矩阵matrix = [    [order, 0, 0, 0],    [0, order, 0, 0],    [r1*s1_inv, r2*s2_inv, 2^128/order, 0],    [m1*s1_inv, m2*s2_inv, 0, 2^128]]
# LLL格基约简new_matrix = olll.reduction(matrix, 0.75)
# 从格向量恢复私钥for row in new_matrix:    potential_key = (rns1 - r1sn)⁻¹ * (snm1 - s1mn - s1sn*k_diff)    if verify(potential_key):         print("密钥恢复成功!")