The Frey-Rück Attack is a cryptographic attack used to solve the Discrete Logarithm Problem (DLP) in the context of elliptic curve cryptography. It can be used to solve the DLP in the cases where the elliptic curve has a composite order.

Here’s a Python implementation of the Frey-Rück Attack using the `sympy`

and `tinyec`

libraries:

```
import math
from sympy import mod_inverse
from tinyec.ec import SubGroup, Curve
def prime_factors(n):
i = 2
factors = []
while i * i <= n:
if n % i:
i += 1
else:
n //= i
factors.append(i)
if n > 1:
factors.append(n)
return factors
def frey_ruck_attack(E, P, Q, n):
factors = prime_factors(n)
nP, nQ = 1, 1
for f in factors:
P_f = P * (n // f)
Q_f = Q * (n // f)
while (nP * P_f).is_infinite() and (nQ * Q_f).is_infinite():
nP *= f
nQ *= f
d = mod_inverse(nP, nQ)
l = (nP * Q - nQ * P) * d % n
return l
if __name__ == "__main__":
# Example parameters
a = -1
b = 1
p = 751
G = (0, 1)
n = 727
h = 1
E = Curve(a, b, p, SubGroup(p, G, n, h))
P = E(-1, 0)
Q = E(13, 1)
l = frey_ruck_attack(E, P, Q, n)
print("The discrete logarithm of Q with respect to P is:", l)
```

In this example, we’re using a simple elliptic curve with a composite order. You can replace the curve parameters (a, b, p, G, n, h) and points P and Q to test the attack on other curves and points.

Make sure you have the `sympy`

and `tinyec`

libraries installed, which can be done using pip:

`pip install sympy tinyec`

Keep in mind that this attack is not applicable to all elliptic curves, and it is not efficient for curves with a large prime order, which are usually used in elliptic curve cryptography.

**You can get detailed information and examples by studying the materials: https://cryptodeeptech.ru/frey-ruck-attack**

The Frey-Rück attack is a method to create a false signature for a Bitcoin transaction. It’s a form of transaction malleability attack. Here’s a simple Python script that demonstrates this attack:

```
from bitcoin import *
from bitcoin.network import *
from bitcoin.wallet import *
# Funds to spend (replace these with actual transaction details)
tx_hex = "your_transaction_hex_here" # The transaction you want to modify
private_key = PrivateKey.from_wif('your_private_key_here')
scriptPubKey = Script([OP_DUP, OP_HASH160, b'\<your_scriptPubKey_hex_here>', OP_EQUALVERIFY, OP_CHECKSIG])
# Decode the original transaction
tx = Transaction().parse(HexCoder.decode(tx_hex))
# Find the input index associated with your key
input_index = None
for i, txin in enumerate(tx.vin):
if txin.scriptSig == scriptPubKey:
input_index = i
break
if input_index is None:
print("Could not find relevant input index.")
exit(1)
# Modify the transaction
tx.vin[input_index].scriptSig = CScript([OP_DUP, OP_DUP, OP_HASH160, bytes(20), OP_EQUALVERIFY, OP_CHECKSIG])
# Serialize and encode the modified transaction in hex
modified_tx_hex = tx.serialize().hex()
print("Original transaction hash:", tx.get_hash())
print("Modified transaction hash:", tx.serialize().get_hash())
# Sign the modified transaction
tx.sign_input(input_index, private_key, scriptPubKey)
# Broadcast the modified transaction (optional, you can use any method to broadcast)
# For instance, using a bitcoin RPC connection:
# rpc_connection = BitcoinRPCConnection(host='localhost', port=8332, user='your_rpc_user', passwd='your_rpc_password')
# rpc_connection.send_raw_transaction(modified_tx_hex)
print("Modified signed transaction:\n", modified_tx_hex)
```

Replace ‘your_transaction_hex_here’ with the hex-encoded transaction you want to modify, ‘your_private_key_here’ with the private key associated with the transaction input you want to tweak, and ‘your_scriptPubKey_hex_here’ with the hex-encoded scriptPubKey of the key.

This script finds the input in the transaction associated with your private key, changes the scriptSig to create a new signature hash, signs it with your private key, and then prints the modified signed transaction.

Please note that this attack is possible because Bitcoin transaction signatures don’t depend on the entire transaction. However, this kind of attack can be mitigated by using sequence-locked scripts (BIP68) and witness commitment transactions (BIP143), which are now standard in the Bitcoin network. Additionally, this script doesn’t actually broadcast the modified transaction – it only demonstrates how the Frey-Rück attack can be implemented. Broadcasting modified transactions could potentially lead to issues with the Bitcoin network and could be illegal depending on your jurisdiction.

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