Breaking ECDSA from nonce bits

如果对HNP不太了解，可以先看一下我的另一篇文章HNP

Preview

先简单回顾一下HNP和ECDSA。

Hidden Number problem（HNP） ：有一个对外保密的数\(\alpha\)和对外公开的模数\(n\)。随机的选择\(t_i\)计算\(s_i=\alpha t_i\ mod\ n\)，并且泄漏出\(s_i\)的最高有效位，就可以利用\(CVP\)的方法去恢复\(\alpha\)。

ECDSA ：在有限域\(\mathbb{F}p\)上选择椭圆曲线\(E\left(\mathbb{F}_{p}\right)\)，以及阶为\(n\)的子群的基点\(G\)。私钥为\(0\leq d < n\)，公钥为\(dG=Q\)。
签名：

• 选择nonce为随机数\(k\)
• 计算明文的\(hash\)，\(h=Hash(m)\)
• 计算\(r=(kG)_x\)，x表示\(kG\)的x坐标
• 计算\(s = k^{-1} \cdot (h+d\cdot r)\ mod\ n\)
• 签名为\((r, s)\)

ECDSA as a HNP

由ECDSA的签名过程，重写一下公式。

\[-s^{-1}\cdot h + k\equiv s^{-1}\cdot r \cdot d \pmod{n} \]

与HNP问题对应一下，令

\[\begin{aligned} \alpha=d \\ t_i=s^{-1}\cdot r \\ a_i = s^{-1}\cdot h \end{aligned} \]

由于nonce相较于\(a_i\)来说较小，所以

\[\begin{aligned} MSB_{\alpha,k}(t_i) &= MSB_k(\alpha \cdot t_i\ mod\ n) \\ &= MSB_k(d\cdot s^{-1}\cdot r) \\ &= MSB_k(-a_i + k) \\ &= n-a_i \end{aligned} \]

Solving the HNP with lattices

在之前讲的HNP中，Boneh and Venkatesan给出了这样的格

\[\left[\begin{array}{cccccc} n & 0 & 0 & \cdots & 0 & 0 \\ 0 & n & 0 & \cdots & 0 & 0 \\ & \vdots & & & \cdots & \\ 0 & 0 & 0 & \cdots & n & 0 \\ t_{0} & t_{1} & t_{2} & \cdots & t_{m-1} & 1 / n \end{array}\right] \]

但是在后来的研究中Kannan给出了改进后的格

\[\left[\begin{array}{ccccccc} n & 0 & 0 & \cdots & 0 & 0 & 0 \\ 0 & n & 0 & \cdots & 0 & 0 & 0 \\ & \vdots & & & \vdots & & \\ 0 & 0 & 0 & \cdots & n & 0 & 0 \\ t_{0} & t_{1} & t_{2} & \cdots & t_{m-1} & 2^{\ell} / n & 0 \\ a_{0} & a_{1} & a_{2} & \cdots & a_{m-1} & 0 & 2^{\ell} \end{array}\right] \]

\(\ell\)是nonce的位长度，\(2^{\ell}\)就是nonce的上界。

在这个格中存在一个向量

\[v_k=\left(k_{0}, k_{1}, \ldots, k_{m-1}, 2^{\ell} \cdot \alpha / n, 2^{\ell}\right) \]

用到数第二行向量乘\(\alpha\)加上最后一行，\(t_i\alpha+a_i=-a_i+k_i+a_i=k_i\)。

这个向量在格中是很短的。它的模\(||vk|| \leq \sqrt{m+2}\cdot 2^{\ell}\)。但是格中还有一个更短的向量

\[(0, 0, \cdots,0, 2^{\ell}, 0) \]

所以在利用LLL算法之后，要找的向量并不再第一行。

Example & Demo

from sage.all import *
from hashlib import sha1
import time


# use secp128r1 curve to test
# The parameters:
_p = 0xfffffffdffffffffffffffffffffffff
_b = 0xe87579c11079f43dd824993c2cee5ed3
_a = -0x3
_Gx = 0x161ff7528b899b2d0c28607ca52c5b86
_Gy = 0xcf5ac8395bafeb13c02da292dded7a83
order = 340282366762482138443322565580356624661
E = EllipticCurve(GF(_p), [_a, _b])
G = E(_Gx, _Gy)

# The Lattice attack paramters
# n = ceil(log(order, 2))
# k = ceil(sqrt(n)) + ceil(log(n, 2))
# print(f"The number of significant bits = {k}")
# m = 2*ceil(sqrt(n))
m = 24
print(f"about {m} signatures")


# Ecdsa sign Algorithm
def EcdsaSign(number, k):
    r = ZZ((k*G)[0])
    s = (inverse_mod(k, order) * (number + d*r)) % order
    return (ZZ(r), ZZ(s))

# This function is not used in this test
def string_to_number(str_msg:str, hash:callable):
    """parameters:
    str_msg: the massage be converted to number
    hash: callable hash function
    """
    return int(hash(str_msg.encode()).hexdigest(), 16)

def generator():
    """Return a_i , t_i , msb, h
    a_i = s^(-1) * h
    t_i = s^(-1) * r
    msb = -a_i % order
    h: the message to be signed
    """
    h = randint(0, 2**128) % order
    k = randint(1, 2**120) % order

    r, s = EcdsaSign(h, k)

    a_i = (inverse_mod(s, order) * h) % order
    t_i = (inverse_mod(s, order) * r) % order

    return t_i, a_i, -a_i % order, h


## Solving the HNP with Boneh and Venkatesan's lattice
def build_basis(oracle_inputs):
    """Returns a basis using the HNP game parameters and inputs to our oracle
    """
    basis_vectors = []
    for i in range(m):
        p_vector = [0] * (m+1)
        p_vector[i] = order
        basis_vectors.append(p_vector)
    basis_vectors.append(list(oracle_inputs) + [QQ(1)/QQ(order)])
    return Matrix(QQ, basis_vectors)

def approximate_closest_vector(basis, v):
    """Returns an approximate CVP solution using Babai's nearest plane algorithm.
    """
    BL = basis.LLL()
    G, _ = BL.gram_schmidt()
    _, n = BL.dimensions()
    small = vector(ZZ, v)
    for i in reversed(range(n)):
        c = QQ(small * G[i]) / QQ(G[i] * G[i])
        c = c.round()
        small -= BL[i] * c
    return (v - small).coefficients()

def Boneh_Venkatesan_method():
    # print("[+] using Boneh and Venkatesan's method to recover the private key")
    inputs = []
    answers = []
    for _ in range(m):
        t_i, a_i, msb, h = generator()
        inputs.append(t_i)
        answers.append(msb)

    time0 = time.time()
    lattice = build_basis(inputs)
    u = vector(ZZ, list(answers) + [0])
    v = approximate_closest_vector(lattice, u)

    recovered_alpha = (v[-1] * order) % order

    spend = time.time() - time0
    # print("spend %.3fs" % spend)

    # assert recovered_alpha == d
    if recovered_alpha == d:
        # print("Recovered alpha! Alpha is %d" % recovered_alpha)
        return 1, spend
    else:
        # print("fault to recover!")
        return 0, spend

    


def build_Kannan_basis(t_i, a_i):
    """Returns a basis using the HNP game parameters and inputs to our oracle
    """
    basis_vectors = []
    for i in range(m):
        p_vector = [0] * (m+2)
        p_vector[i] = order
        basis_vectors.append(p_vector)
    basis_vectors.append(list(t_i) + [QQ(2**120)/QQ(order)] + [0])
    basis_vectors.append(list(a_i) + [0] + [QQ(2**120)])
    return Matrix(QQ, basis_vectors)

def Kannan_method():
    # print("\n[+] using Kannan's method to recover the private key")
    inputs = []
    answers = []
    hs = []
    for _ in range(m):
        t_i, a_i, msb, h = generator()
        inputs.append(t_i)
        answers.append(a_i)
        hs.append(h)
    
    time0 = time.time()
    lattice = build_Kannan_basis(inputs, answers)
    # print("Solving SVP using lattice with basis:\n%s\n" % str(lattice))
    L = lattice.LLL()
    # print(L)
    ks = L[1]

    spend = time.time() - time0
    # print("spend %.3fs" % spend)

    if (-answers[0] + ZZ(ks[0])) % order == (inputs[0] * d) % order:
        # print(f"Recovered nonce! {ks}")
        recovered_d = ( (-answers[0] + ZZ(ks[0])) * inverse_mod(inputs[0], order) ) % order
        assert recovered_d == d
        # print("Recovered private key! private key is %d" % recovered_d)
        return 1, spend
    else:
        # print("fault to recover!")
        return 0, spend
    

if __name__ == "__main__":
    method1_spend = []
    method1_success = 0
    method2_spend = []
    method2_success = 0

    for _ in range(20):
        d = randint(1, order-1)
        # print(f"the privet key is {d}")
        pubkey = d*G

        flag0, spend0 = Boneh_Venkatesan_method()
        flag1, spend1 = Kannan_method()

        method1_success += flag0
        method2_success += flag1
        method1_spend.append(spend0)
        method2_spend.append(spend1)
    
    print("[+] Boneh_Venkatesan_method:")
    print(f"successed: {method1_success}/20\ntime spend: {sum(method1_spend)/20}")
    print("[+] Kannan_method:")
    print(f"successed: {method2_success}/20\ntime spend: {sum(method2_spend)/20}")

test result

• 当k的位数很小，比如只有几十位长甚至几位长，然后调小m，会发现Boneh_Venkatesan_method和Kannan_method的时间相差不大，但是前者的正确率更高。

• 当k的位数比较大，比如有一百多甚至一百二十多，为了保证正确率需要调大m，这个时候Kannan_method的正确率以及速度都比Boneh_Venkatesan_method要表现的好。

当然我这里只是20次为一组，测试结果肯定是不太准确的。

Reference

• On Bounded Distance Decoding with Predicate: Breaking the “Lattice Barrier” for the Hidden Number Problem by Martin R. Albrecht and Nadia Heninger
• Intended Solution to NHP in GxzyCTF 2020 by Soreat_u