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Technical documentation

brief introduction
Quantum cryptocurrency is an innovative digital currency based on quantum computing and quantum key distribution (QKD) technology, aimed at providing extremely high security and privacy protection. By utilizing the principles of quantum mechanics, our encryption algorithm can resist attacks from traditional and quantum computers, ensuring the immutability of transactions and the security of data.

Technical architecture

1. Distributed ledger
We use blockchain technology to implement distributed ledgers, where each node saves a complete copy of the ledger. The consensus mechanism adopts a hybrid PoS (Proof of Stack) and PoW (Proof of Work) algorithms to ensure the security and efficiency of the network.

PoW (Proof of Work) algorithm

The PoW algorithm requires participants (miners) to solve complex computational problems in order to obtain accounting rights. This challenge is usually finding a hash value that satisfies specific conditions.

Algorithm steps

1.Hash calculation:
Given a block head H to be verified, miners need to find a random number nonce such that the hash value Hash (H+nonce) satisfies specific conditions (such as the first few bits being zero).

2.Formula:
Hash(H+nonce)<target
Among them, target is the target value adjusted based on network difficulty

Among them:
Block_header contains relevant information about the block (such as the hash, timestamp, transaction data, etc. of the previous block)
The target is the difficulty target

1.Initialize nonce=0.
2.Calculate hash value:

3.If the hash satisfies the condition that hash<target, find a valid nonce; otherwise, increase nonce and repeat step 2.

PoS (Proof of Stack) algorithm

The PoS algorithm selects the bookkeeper based on the amount and duration of coins held. The accounting probability of each node is directly proportional to the number of cryptocurrencies it holds.

Algorithm steps

1.Equity selection:
Each node calculates its accounting weight based on the amount of coins it holds, stack, and time period

2.Formula:

Among them, stack, i is the amount of coins held by node i, and time, i is the time for which node i holds coins.

Among them:

Stake (validator) is the equity held by the validator (i.e. the amount of currency held)

Sum (stack (all_validators)) is the total amount of equity held by all validators

Ⅱ.Quantum encryption algorithm

Quantum Key Distribution (QKD)

Quantum key distribution is one of our core technologies. QKD uses the quantum state of photons to distribute keys and achieves secure key exchange through quantum entanglement or quantum teleportation. Common QKD protocols include BB84 and E91 protocols.

BB84 protocol

The BB84 protocol is the earliest quantum key distribution protocol proposed by Charles Bennett and Gilles Brassard in 1984. This protocol uses the polarization state of photons to transmit keys, and the receiver measures the photon state through randomly selected bases to ensure the security of key distribution.

Algorithm steps：

1. Preparation stage:
Alice prepares a random bit sequence and randomly selects one of the four polarization states to send each bit:

2. Measurement and base exchange:

Bob randomly selects two bases (e.g. standard basis And diagonal basis ）One is to measure each received photon and record the measurement results And the basis used.

Alice and Bob exchange base information through classical communication channels, but do not exchange specific bit values.

3. Screening stage:

Alice and Bob discard the measurement results that do not match the base, retain the bits that match the base, and form the initial key and

4. Error correction and privacy amplification:

Alice and Bob use classic algorithms for error correction and privacy amplification, eliminating any potential errors and improving the security of the key.

Ⅲ.Quantum secure public key encryption

We use public key encryption algorithms based on lattice theory, which are considered to be resistant to quantum computing attacks. The following is the specific LWE (Learning With Errors) encryption algorithm.

1.LWE encryption algorithm
Key generation:
Choose a secret vector And an error vector .
Generate public key matrix And public key vector .

2.Encryption:
Clear text Convert to vector
Select a random vector
Calculate ciphertext

3.Decryption:
Use private key S to decrypt ciphertext C and obtain plaintext M.

mathematical formula

Key generation:
Public key: S

Encryption:

Select random vector r
Cryptography:

Decryption:

Decryption operation:

Ⅳ.Detailed algorithm validation

1. LWE algorithm validation
To verify the correctness of the LWE encryption and decryption algorithm, assuming we have:

n =8， m =12，q =256
Select random matrix A , secret vector s , and error vector e
calculate b = A · s +e
Decrypt to obtain the original message .

import numpy as np

# Generate public and private keys
def generate_lwe_keys(n, m, q):
A = np.random.randint(q, size=(m, n))
s = np.random.randint(q, size=n)
e = np.random.randint(q, size=m)
b = (np.dot(A, s) + e) % q
return A, b, s

# Encrypted message
def encrypt_lwe(A, b, message, q):
r = np.random.randint(q, size=A.shape[0])
u = (np.dot(A.T, r)) % q
v = (np.dot(b.T, r) + message) % q
return u, v

# Decrypt message
def decrypt_lwe(u, v, s, q):
return (v - np.dot(u.T, s) % q) % q

# Parameter setting
n, m, q = 8, 12, 256
A, b, s = generate_lwe_keys(n, m, q)
message = 42

# encipher
u, v = encrypt_lwe(A, b, message, q)

# decrypt
decrypted_message = decrypt_lwe(u, v, s, q)

# verify
assert decrypted_message == message, "Decryption failure"
print("Primary message:", message)
print("Decrypted message:", decrypted_message)

The above code implements the public key generation, encryption, and decryption process of the LWE algorithm, and verifies the consistency between the encrypted message and the original message.

2.2 BB84 Protocol Validation
To verify the correctness of the BB84 protocol, assuming that Alice and Bob have the same base, they should be able to generate the same key.

import numpy as np

def bb84_key_exchange(alice_bits, alice_bases, bob_bases):
key = []
for ab, bb, bit in zip(alice_bases, bob_bases, alice_bits):
if ab == bb:
key.append(bit)
return key

# hypothesis Alice 和 Bob Use the same base
alice_bits = np.random.randint(2, size=10)
alice_bases = np.random.randint(2, size=10)
bob_bases = alice_bases

# Generate a shared key
shared_key = bb84_key_exchange(alice_bits, alice_bases, bob_bases)

# verify
print("Alice Bit of:", alice_bits)
print("Alice Basis for:", alice_bases)
print("Bob Basis for:", bob_bases)
print("Shared key:", shared_key)

The above code demonstrates the key distribution process of the BB84 protocol and verifies the generation of the same shared key when Alice and Bob base are the same.

summary

The above technical documents provide a detailed description of the mathematical formulas, algorithm steps, and actual implementation of distributed ledgers, PoW and PoS consensus mechanisms, BB84 protocol, and LWE algorithm. Through these detailed descriptions and verifications, the correctness and security of the implementation can be ensured.

Project Structure

quantum-crypto
├── README.md
├── LICENSE
├── .gitignore
├── blockchain
│ ├── blockchain.py
│ ├── consensus.py
│ ├── transaction.py
│ └── wallet.py
├── quantum
│ ├── qkd.py
│ ├── lwe.py
│ └── utils.py
├── tests
│ ├── test_blockchain.py
│ ├── test_qkd.py
│ ├── test_lwe.py
│ ├── test_consensus.py
│ └── test_wallet.py
└── smart_contracts
├── contract.py
└── vm.py

README.md

# Quantum Crypto
A quantum-secure cryptocurrency project implementing blockchain, quantum key distribution (BB84), and lattice-based cryptography (LWE).
## Features
- **Blockchain**: A simple blockchain with PoW and PoS consensus mechanisms.
- **Quantum Key Distribution (QKD)**: Implementation of BB84 protocol.
- **Lattice-based Encryption**: Implementation of Learning With Errors (LWE) encryption scheme.
- **Wallet**: Basic wallet functionality to manage keys and transactions.
## Installation
1. Clone the repository:
```sh
git clone https://github.com/YourProjectName/quantum-crypto.git
cd quantum-crypto

Install dependencies:

pip install -r requirements.txt

Usage

To start the blockchain:

python blockchain/blockchain.py
To run tests:

pytest

License

This project is licensed under the MIT License.

### `blockchain/blockchain.py`

```python

import hashlib

import json

import time

from consensus import proof_of_work, proof_of_stake

class Block:

def __init__(self, index, previous_hash, timestamp, data, proof):

self.index = index

self.previous_hash = previous_hash

self.timestamp = timestamp

self.data = data

self.proof = proof

self.hash = self.calculate_hash()

def calculate_hash(self):

block_string = f"{self.index}{self.previous_hash}{self.timestamp}{self.data}{self.proof}"

return hashlib.sha256(block_string.encode()).hexdigest()

class Blockchain:

def __init__(self):

self.chain = [self.create_genesis_block()]

self.difficulty = 2

self.stakeholders = {}

def create_genesis_block(self):

return Block(0, "0", time.time(), "Genesis Block", 0)

def get_latest_block(self):

return self.chain[-1]

def add_block(self, new_block):

new_block.previous_hash = self.get_latest_block().hash

new_block.hash = new_block.calculate_hash()

self.chain.append(new_block)

def proof_of_work(self, block):

return proof_of_work(block, self.difficulty)

def proof_of_stake(self, validator, block):

return proof_of_stake(validator, self.stakeholders, block)

def create_block(self, data, validator=None):

latest_block = self.get_latest_block()

new_block = Block(len(self.chain), latest_block.hash, time.time(), data, 0)

if validator:

if self.proof_of_stake(validator, new_block):

new_block.proof = validator

self.add_block(new_block)

else:

new_block.proof = self.proof_of_work(new_block)

self.add_block(new_block)

if __name__ == "__main__":

blockchain = Blockchain()

blockchain.create_block("First Block")

blockchain.create_block("Second Block", validator="validator_address")

for block in blockchain.chain:

print(vars(block))

blockchain/consensus.py

import hashlib

def proof_of_work(block, difficulty):
block.proof = 0
computed_hash = block.calculate_hash()
while not computed_hash.startswith('0' * difficulty):
block.proof += 1
computed_hash = block.calculate_hash()
return block.proof

def proof_of_stake(validator, stakeholders, block):
if validator not in stakeholders:
return False
# Simplified PoS mechanism for illustration
return True

blockchain/transaction.py

import hashlib
import json

class Transaction:
def __init__(self, sender, receiver, amount, signature):
self.sender = sender
self.receiver = receiver
self.amount = amount
self.signature = signature

def to_dict(self):
return {
"sender": self.sender,
"receiver": self.receiver,
"amount": self.amount,
"signature": self.signature
}

def calculate_hash(self):
transaction_string = json.dumps(self.to_dict(), sort_keys=True).encode()
return hashlib.sha256(transaction_string).hexdigest()

def __repr__(self):
return str(self.to_dict())

blockchain/wallet.py

import os
import binascii
import hashlib
from ecdsa import SigningKey, VerifyingKey, SECP256k1

class Wallet:
def __init__(self):
self.private_key = SigningKey.generate(curve=SECP256k1)
self.public_key = self.private_key.get_verifying_key()

def sign_transaction(self, transaction_hash):
return binascii.hexlify(self.private_key.sign(transaction_hash.encode())).decode()

def verify_signature(self, transaction_hash, signature, public_key):
vk = VerifyingKey.from_string(binascii.unhexlify(public_key.encode()), curve=SECP256k1)
return vk.verify(binascii.unhexlify(signature.encode()), transaction_hash.encode())

def get_address(self):
pub_key_bytes = self.public_key.to_string()
return hashlib.sha256(pub_key_bytes).hexdigest()

def export_keys(self):
return {
"private_key": self.private_key.to_pem().decode(),
"public_key": self.public_key.to_pem().decode()
}

# Example usage
if __name__ == "__main__":
wallet = Wallet()
keys = wallet.export_keys()
print(f"Private Key:\n{keys['private_key']}")
print(f"Public Key:\n{keys['public_key']}")
print(f"Address: {wallet.get_address()}")

quantum/qkd.py

import numpy as np

def bb84_key_exchange(alice_bits, alice_bases, bob_bases):
key = []
for ab, bb, bit in zip(alice_bases, bob_bases, alice_bits):
if ab == bb:
key.append(bit)
return key

# Example usage
if __name__ == "__main__":
alice_bits = np.random.randint(2, size=10)
alice_bases = np.random.randint(2, size=10)
bob_bases = np.random.randint(2, size=10)
shared_key = bb84_key_exchange(alice_bits, alice_bases, bob_bases)
print("Alice's bits:", alice_bits)
print("Alice's bases:", alice_bases)
print("Bob's bases:", bob_bases)
print("Shared Key:", shared_key)

quantum/lwe.py

import numpy as np

def generate_lwe_keys(n, m, q):
A = np.random.randint(q, size=(m, n))
s = np.random.randint(q, size=n)
e = np.random.randint(q, size=m)
b = (A @ s + e) % q
return A, b, s

def encrypt_lwe(A, b, m, q):
r = np.random.randint(q, size=A.shape[0])
u = (A.T @ r) % q
v = (b.T @ r + m) % q
return u, v

def decrypt_lwe(u, v, s, q):
return (v - (u.T @ s) % q) % q

# Example usage
if __name__ == "__main__":
n, m, q = 8, 12, 256
A, b, s = generate_lwe_keys(n, m, q)
message = 42
u, v = encrypt_lwe(A, b, message, q)
decrypted_message = decrypt_lwe(u, v, s, q)
print("Original Message:", message)
print("Decrypted Message:", decrypted_message)

quantum/utils.py

import hashlib

def hash_data(data):
return hashlib.sha256(data.encode()).hexdigest()

# Example usage
if __name__ == "__main__":
data = "Hello, Quantum World!"
hashed_data = hash_data(data)
print("Original Data:", data)
print("Hashed Data:", hashed_data)

tests/test_blockchain.py

import unittest
from blockchain.blockchain import Blockchain, Block

class TestBlockchain(unittest.TestCase):
def setUp(self):
self.blockchain = Blockchain()

def test_genesis_block(self):
self.assertEqual(len(self.blockchain.chain), 1)
self.assertEqual(self.blockchain.chain[0].data, "Genesis Block")

def test_create_block(self):
self.blockchain.create_block("Test Block")
self.assertEqual(len(self.blockchain.chain), 2)
self.assertEqual(self.blockchain.chain[-1].data, "Test Block")

def test_proof_of_work(self):
new_block = Block(1, self.blockchain.get_latest_block().hash, time.time(), "Test Block", 0)
proof = self.blockchain.proof_of_work(new_block)
self.assertTrue(str(proof).startswith('0' * self.blockchain.difficulty))

def test_proof_of_stake(self):
self.blockchain.stakeholders["validator_address"] = 100
new_block = Block(1, self.blockchain.get_latest_block().hash, time.time(), "Test Block", 0)
self.assertTrue(self.blockchain.proof_of_stake("validator_address", new_block))

if __name__ == '__main__':
unittest.main()

tests/test_qkd.py

import unittest
import numpy as np
from quantum.qkd import bb84_key_exchange

class TestQKD(unittest.TestCase):
def test_bb84_key_exchange(self):
alice_bits = np.random.randint(2, size=10)
alice_bases = np.random.randint(2, size=10)
bob_bases = alice_bases # for simplicity, use same bases
key = bb84_key_exchange(alice_bits, alice_bases, bob_bases)
self.assertEqual(len(key), 10)

if __name__ == '__main__':
unittest.main()

tests/test_lwe.py

import unittest
from quantum.lwe import generate_lwe_keys, encrypt_lwe, decrypt_lwe

class TestLWE(unittest.TestCase):
def test_lwe_encryption_decryption(self):
n, m, q = 8, 12, 256
A, b, s = generate_lwe_keys(n, m, q)
message = 42
u, v = encrypt_lwe(A, b, message, q)
decrypted_message = decrypt_lwe(u, v, s, q)
self.assertEqual(message, decrypted_message)

if __name__ == '__main__':
unittest.main()

tests/test_consensus.py

import unittest
from blockchain.consensus import proof_of_work
from blockchain.blockchain import Block

class TestConsensus(unittest.TestCase):
def test_proof_of_work(self):
block = Block(0, "0", time.time(), "Genesis Block", 0)
proof = proof_of_work(block, 2)
self.assertTrue(str(proof).startswith("00"))

if __name__ == '__main__':
unittest.main()

tests/test_wallet.py

import unittest
from blockchain.wallet import Wallet

class TestWallet(unittest.TestCase):
def setUp(self):
self.wallet = Wallet()

def test_wallet_keys(self):
keys = self.wallet.export_keys()
self.assertIn("private_key", keys)
self.assertIn("public_key", keys)

def test_wallet_address(self):
address = self.wallet.get_address()
self.assertEqual(len(address), 64)

if __name__ == '__main__':
unittest.main()

This code template shows the main functional modules of the quantum cryptocurrency project in more detail and provides test cases. Each module includes detailed comments and example usage for better understanding.

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