题目描述

ECDSA

签名

假设我们的私钥为$d_A$而公钥为$Q_A$，$Q_A=d_A\cdot G$，接下来就是签名的过程，要签名的消息为$m$

1. 取 $e=HASH(m)$
2. 取$e$的左边的$L_n$个bit长度的值为$z$，$L_n$即为前面提到的参数里$n$的比特长度
3. 从 $[1,n-1]$范围内，随机选择一个整数$k$
4. 利用$k$得到椭圆曲线上的一点$(x1，y1)=k\cdot G$
5. 然后计算$r\equiv x_1(modn)$，如果如果$r=0$则返回步骤3重新选择$k$
6. 计算$s=k^{-1}(z+r\cdot d_A)(modn)$，如果$s=0$则返回步骤3重新选择$k$
7. 得到的数字签名即为$(r，s)$

验证

取点$P=(X_p,Y_p)=s^{-1}\cdot z\cdot G+s^{-1}\cdot r\cdot Q_A$
若$X_p=r$，则签名有效，否则无效

Simple Proof
$P=(X_p,Y_p)=s^{-1}\cdot z\cdot G+s^{-1}\cdot r\cdot Q_A$
因为$Q_A=d_A\cdot G$，故有
$$P=s^{-1}\cdot z\cdot G+s^{-1}\cdot r\cdot Q_A$$
$$=s^{-1}\cdot z\cdot G+s^{-1}\cdot r\cdot d_A\cdot G$$
$$=s^{-1}\cdot G\cdot (z+r\cdot d_A)$$
又有$$s=k^{-1}(z+r\cdot d_A)(modn)$$
即$$s^{-1}=k\cdot (z+r\cdot d_A{)}^{-1}(modn)$$
代入得到$$P=k\cdot G$$

利用冲突的随机数恢复私钥

从上面的签名过程我们可以看到最关键的地方就在于随机数k，对于一个固定的椭圆曲线，一个确定的k就意味着一个确定的r，所以如果有两个相同的私钥签署的签名出现了相同的r就代表着在生成随机数时取到了相同的k，看到这里想必你也明白了我们题目的交易签名的问题出在哪了，这两笔交易的r值相同，代表在它们签名时使用的随机数k是相同的，而这就是我们恢复私钥的关键
我们不妨设这两个签名的$z$与$s$分别为$z_1$，$z_2$与$s_1$，$s_2$
则有
$$s_1-s_2=k^{-1}(z_1+d_A\cdot r)-k^{-1}(z_2+d_A\cdot r)$$
$$=k^{-1}(z_1-z_2)$$
那么$$k=\frac{(z_1-z_2)}{(s_1-s_2)}$$
通过$k$，可以计算出$d_A=(s\cdot k-z)/r$

P2PKH

PubkeyScript是一张记录了交易记录的指令列表，它控制了下一名使用者如何解锁已接收的比特币并传送。收款人会制造一个signature script，而该文件必须满足最后一个发送者创建的PubkeyScript的参数。
PubkeyScript的参数：

1. 公钥哈希(Public Key Hash) (比特币地址）
2. 电子签署(ScriptSig: (r,s)+pubkey)

ASM是汇编代码

1. OP_PUSHBYTES_71指压入栈中一个71字节大小的数据
2. nSequence用以记录该笔交易是否可以上链
3. Previous output Script用以验证当前用于支付的比特币的来源
4. ScriptPubKey是一个脚本，用以验证当前用户有能力使用这个UTXO中的比特币来支付，即证明身份
   OP_DUP复制栈顶数据
   OP_HASH160先后进行两种hash操作然后压入栈
   OP_PUSHBYTES_20将签名的hash值压入栈
   OP_EQUALVERIFY比较计算签名hash和刚压入栈的hash来验证有效性，有效返回1否则返回0
   OP_CHECKSIG检测栈顶的2个元素，pub key和signature是否能对应的上。对应的上，说明这个签名的私钥，和收款人的公钥可以对上。有资格花这笔钱
   OP_RETURN 用来当注释，携带一些信息
   Transaction hex则包含所有信息连接在一起

求解过程

提取r,s,z

脚本来源

# -*-coding:utf-8-*-
"""
@author: iceland
"""
import sys
import hashlib
import argparse
from urllib.request import urlopen

# # ==============================================================================
# parser = argparse.ArgumentParser(
#     description='This tool helps to get ECDSA Signature r,s,z values from Bitcoin rawtx or txid',
#     epilog='Enjoy the program! :)    Tips BTC: bc1q39meky2mn5qjq704zz0nnkl0v7kj4uz6r529at')
#
# parser.add_argument("-txid", help="txid of the transaction. Automatically fetch rawtx from given txid", action="store")
# parser.add_argument("-rawtx", help="Raw Transaction on the blockchain.", action="store")
#
# if len(sys.argv) == 1:
#     parser.print_help()
#     sys.exit(1)
# args = parser.parse_args()
# # ==============================================================================
#
# txid = args.txid if args.txid else ''
# rawtx = args.rawtx if args.rawtx else ''
#
# if rawtx == '' and txid == '':
#     print('One of the required option missing -rawtx or -txid');
#     sys.exit(1)


# ==============================================================================

def get_rs(sig):
    rlen = int(sig[2:4], 16)
    r = sig[4:4 + rlen * 2]
    #    slen = int(sig[6+rlen*2:8+rlen*2], 16)
    s = sig[8 + rlen * 2:]
    return r, s


def split_sig_pieces(script):
    sigLen = int(script[2:4], 16)
    sig = script[2 + 2:2 + sigLen * 2]
    r, s = get_rs(sig[4:])
    pubLen = int(script[4 + sigLen * 2:4 + sigLen * 2 + 2], 16)
    pub = script[4 + sigLen * 2 + 2:]
    assert (len(pub) == pubLen * 2)
    return r, s, pub


# Returns list of this list [first, sig, pub, rest] for each input
def parseTx(txn):
    if len(txn) < 130:
        print('[WARNING] rawtx most likely incorrect. Please check..')
        sys.exit(1)
    inp_list = []
    ver = txn[:8]
    if txn[8:12] == '0001':
        print('UnSupported Tx Input. Presence of Witness Data')
        sys.exit(1)
    inp_nu = int(txn[8:10], 16)

    first = txn[0:10]
    cur = 10
    for m in range(inp_nu):
        prv_out = txn[cur:cur + 64]
        var0 = txn[cur + 64:cur + 64 + 8]
        cur = cur + 64 + 8
        scriptLen = int(txn[cur:cur + 2], 16)
        script = txn[cur:2 + cur + 2 * scriptLen]  # 8b included
        r, s, pub = split_sig_pieces(script)
        seq = txn[2 + cur + 2 * scriptLen:10 + cur + 2 * scriptLen]
        inp_list.append([prv_out, var0, r, s, pub, seq])
        cur = 10 + cur + 2 * scriptLen
    rest = txn[cur:]
    return [first, inp_list, rest]


# ==============================================================================
def get_rawtx_from_blockchain(txid):
    try:
        htmlfile = urlopen("https://blockchain.info/rawtx/%s?format=hex" % txid, timeout=20)
    except:
        print('Unable to connect internet to fetch RawTx. Exiting..')
        sys.exit(1)
    else:
        res = htmlfile.read().decode('utf-8')
    return res


# =============================================================================

def getSignableTxn(parsed):
    res = []
    first, inp_list, rest = parsed
    tot = len(inp_list)
    for one in range(tot):
        e = first
        for i in range(tot):
            e += inp_list[i][0]  # prev_txid
            e += inp_list[i][1]  # var0
            if one == i:
                e += '1976a914' + HASH160(inp_list[one][4]) + '88ac'
            else:
                e += '00'
            e += inp_list[i][5]  # seq
        e += rest + "01000000"
        z = hashlib.sha256(hashlib.sha256(bytes.fromhex(e)).digest()).hexdigest()
        res.append([inp_list[one][2], inp_list[one][3], z, inp_list[one][4], e])
    return res


# ==============================================================================
def HASH160(pubk_hex):
    return hashlib.new('ripemd160', hashlib.sha256(bytes.fromhex(pubk_hex)).digest()).hexdigest()


# ==============================================================================

txn = '010000000153db4e56f159c0679818ef8ce814ce8fcaad12b854da7e582fb5f19266945f63000000006a47304402200e1e942f62d61cc25117d71bc2da4b523bd720dc7feec77551a0b152eb042cd7022030d7d78612b765dff96dd14fc5d723e06a8fa61b42a93410236273baf82f7f15012102572263bbac032e37cf96fe7664fb799f56353108c032807cc23ca557fb60b394ffffffff02d0070000000000001976a914b1c75a61c0461cd92c124e00ee275a600aa096b288ac0000000000000000056a0343544600000000'
# if rawtx == '':
#     rawtx = get_rawtx_from_blockchain(txid)

print('\nStarting Program...')

m = parseTx(txn)
e = getSignableTxn(m)

for i in range(len(e)):
    print('=' * 70,
          f'\n[Input Index #: {i}]\n     R: {e[i][0]}\n     S: {e[i][1]}\n     Z: {e[i][2]}\nPubKey: {e[i][3]}')

求dA

#-*-coding:utf-8-*-
r1 = 0x0e1e942f62d61cc25117d71bc2da4b523bd720dc7feec77551a0b152eb042cd7
s1 = 0x5611099541793c7681a9f8b48364d6e7088e16afe7b7b6244c52e94d28252a3b
z1 = 0x5ecd4154a2db20480d7715d6e47a772aaf596e11e6f16a4d58e9f0d260294660

r2 = 0x0e1e942f62d61cc25117d71bc2da4b523bd720dc7feec77551a0b152eb042cd7
s2 = 0x30d7d78612b765dff96dd14fc5d723e06a8fa61b42a93410236273baf82f7f15
z2 = 0x50f1c6205aab5f8dac7b505f91dfc437b1b13cd00f12a492570a040a27c38e25

assert r1 == r2
r = r1
def inverse_mod( a, m ):
    """Inverse of a mod m."""
    if a < 0 or m <= a: a = a % m
    c, d = a, m
    uc, vc, ud, vd = 1, 0, 0, 1
    while c != 0:
        q, c, d = divmod( d, c ) + ( c, )
        uc, vc, ud, vd = ud - q*uc, vd - q*vc, uc, vc

    assert d == 1
    if ud > 0: return ud
    else: return ud + m


def derivate_privkey(p, r, s1, s2, z1, z2):
    z = z1 - z2
    s = s1 - s2
    r_inv = inverse_mod(r, p)
    s_inv = inverse_mod(s, p)
    k = (z * s_inv) % p
    d = (r_inv * (s1 * k - z1)) % p
    return d, k

p  = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141

privatekey,k=derivate_privkey(p,r,s1,s2,z1,z2)

利用私钥解密AES

import base64
cipher = base64.b64decode(b"4w/VLHqPZi/epoOGvjoY9TZWhDtYpL3iLsUTyvzghJM=")
privatekey = 0xF41AA419CB6BD43F322D403F40728CE9784CD0B465F409322A76A3DF0A984A29
from Crypto.Cipher import AES
from Crypto.Util.number import *
key = long_to_bytes(privatekey)[0:16]
iv = long_to_bytes(privatekey)[16:]
aes = AES.new(key=key,iv=iv,mode=AES.MODE_CBC)
flag = aes.decrypt(cipher.encode())
print(flag)

参考链接