利用math包用python语言写出带输出NTRU数字签名的代码

以下是利用Python的math包实现NTRU数字签名的代码，其中包括生成公钥和私钥、签名和验证。

import math
import random

# Generate public and private keys
def generate_key_pair(n, p, q):
    # Choose random polynomial f such that all coefficients are either -1, 0, or 1
    f = [random.choice([-1,0,1]) for _ in range(n)]

    # Choose random polynomial g such that all coefficients are either 0 or 1
    g = [random.choice([0,1]) for _ in range(n)]

    # Calculate h
    h = poly_mod(poly_mult(poly_mult(f,g),2),p,q)

    # Return public key (h) and private key (f,g)
    return (h, f, g)


# Sign a message using the private key
def sign(message, f, g, p, q):
    # Convert message to polynomial
    m = [int(c) for c in bin(int.from_bytes(message.encode(), 'big'))[2:]]

    # Pad message with zeros
    m += [0] * (len(f) - len(m))

    # Calculate e
    e = poly_mod(poly_mult(poly_mult(g,m),2),p,q)

    # Calculate s
    s = poly_mod(poly_mod(poly_mult(e,f),p,q),2)

    # Convert s to string and return
    return ''.join([str(c) for c in s])


# Verify a message using the public key and signature
def verify(message, signature, h, p, q):
    # Convert message to polynomial
    m = [int(c) for c in bin(int.from_bytes(message.encode(), 'big'))[2:]]

    # Pad message with zeros
    m += [0] * (len(h) - len(m))

    # Convert signature to polynomial
    s = [int(c) for c in signature]

    # Calculate e
    e = poly_mod(poly_mult(poly_mult(h,s),2),p,q)

    # Calculate m'
    m_prime = poly_mod(poly_mod(poly_mult(e,f),p,q),2)

    # Compare m and m'
    return m == m_prime


# Helper function to perform polynomial multiplication modulo p and q
def poly_mult(f,g):
    n = len(f)
    m = len(g)
    res = [0] * (n+m-1)
    for i in range(n):
        for j in range(m):
            res[i+j] += f[i] * g[j]
    return poly_mod(res,2,3)


# Helper function to perform polynomial modulo p and q
def poly_mod(f,p,q):
    res = [0] * len(f)
    for i in range(len(f)):
        res[i] = f[i] % q
        if res[i] > p/2:
            res[i] -= p
        elif res[i] < -p/2:
            res[i] += p
    return res

使用示例：

# Generate key pair
p = 3
q = 32
n = 10
h, f, g = generate_key_pair(n, p, q)
print('Public key: ', h)
print('Private key: ', f, g)

# Sign message
message = 'Hello, world!'
signature = sign(message, f, g, p, q)
print('Signature: ', signature)

# Verify message
print('Message verified: ', verify(message, signature, h, p, q))

注意：NTRU数字签名算法的安全性并未得到广泛认可，因此不建议在实际应用中使用。