帮我写一段带输出的NTRU数字签名的python代码

好的，以下是一段带输出的NTRU数字签名的python代码：

import random

# Define NTRU parameters
N = 503
q = 2048
p = 3
K = 4
d = 79

# Generate key pair
def key_gen():
    f = [random.randint(-1,1) for i in range(N)]
    while sum(f) == 0:
        f = [random.randint(-1,1) for i in range(N)]
    g = [random.randint(-1,p) for i in range(N)]
    h = [((p+1)//3)*fi%p for fi in f]
    while sum(h) == 0:
        f = [random.randint(-1,1) for i in range(N)]
        h = [((p+1)//3)*fi%p for fi in f]
    return (f,g,h)

sk = key_gen()[0]
pk = key_gen()[1:]

# Generate random message
m = random.randint(0,1)

# Hash message to polynomial
def hash_to_poly(m):
    pm = [int(mi) for mi in bin(m)[2:].zfill(K*d)]
    while len(pm) < N:
        pm.append(0)
    return pm

# Sign message
def sign(m, sk):
    f = sk
    pm = hash_to_poly(m)
    r = [random.randint(-1,1) for i in range(N)]
    a = [fi*pi%p for fi,pi in zip(f,pk)]
    e = [pi*ri%p for pi,ri in zip(pk,r)]
    b = [(pi+ei+pmi)%p for pi,ei,pmi in zip(pk,e,pm)]
    u = [(bi-fi*ri)%p for bi,ri in zip(b,r)]
    v = [ui*inv_fi%p for ui,inv_fi in zip(u,sk)]
    return (u,v)

s = sign(m, sk)

# Verify signature
def verify(m, s, pk):
    pm = hash_to_poly(m)
    u,v = s
    a = [fi*pi%p for fi,pi in zip(sk,pk)]
    b = [(pi+ei+pmi)%p for pi,ei,pmi in zip(pk,[0]*N,pm)]
    r = [(fi*ui+vi)%p for fi,ui,vi in zip(sk,u,v)]
    if sum([(bi-ai)%p for bi,ai in zip(b,a)]) == 0 and sum([(ri-fi*ui)%p for ri,fi,ui in zip(r,sk,u)]) == 0:
        return True
    else:
        return False

print("Message: ", m)
print("Public key: ", pk)
print("Signature: ", s)
print("Verification: ", verify(m, s, pk))

这段代码实现了NTRU数字签名的生成、签名和验证，其中包括生成公钥私钥、哈希消息为多项式、用私钥签署消息、用公钥验证签名。输入的m为消息，输出的s为签名。如果验证成功，将返回True，否则返回False。