SPHINCS+ 算法实现

根据提供的引用内容，SPHINCS+是一种数字签名方案，其实现基于Hash安全假设。SPHINCS+具有短的公钥和私钥size，但前签名要远长于Dilithium签名和Falcon签名。SPHINCS+数字签名方案分为三个级别：SPHINCS+-128 (Level 1)、SPHINCS+-192 (Level 3)和SPHINCS+-256 (Level 5)。下面是SPHINCS+算法的一个简单实现：

# 导入所需的库
import hashlib
import random

# 定义SPHINCS+算法的参数
n = 32
w = 16
d = 16
k = 32
L = 64
m = 32
tau = 16
h = hashlib.shake_256

# 定义SPHINCS+算法的签名函数
def sign(sk, message):
    # 从私钥中提取种子和掩码
    seed = sk[:n]
    masks = [sk[n+(i*w):n+((i+1)*w)] for i in range(d)]
    
    # 计算公钥
    pk = h(seed).digest(L)
    
    # 计算消息的哈希值
    message_hash = h(message).digest(m)
    
    # 初始化签名
    signature = b''
    
    # 计算每个层级的签名
    for i in range(d):
        # 计算层级的哈希种子
        layer_seed = h(seed + bytes([i])).digest(n)
        
        # 计算层级的掩码
        layer_masks = [h(layer_seed + bytes([j])).digest(w) for j in range(2**w)]
        
        # 计算层级的哈希值
        layer_hash = h(layer_seed + signature + message_hash).digest(L)
        
        # 计算层级的签名
        layer_signature = b''
        for j in range(2**w):
            if masks[i][j]:
                layer_signature += layer_masks[j]
                layer_hash = h(layer_hash + layer_masks[j]).digest(L)
            else:
                layer_signature += h(layer_hash + layer_masks[j]).digest(n)
        
        # 将层级签名添加到总签名中
        signature += layer_signature
    
    # 返回签名和公钥
    return signature, pk

# 定义SPHINCS+算法的验证函数
def verify(pk, message, signature):
    # 计算消息的哈希值
    message_hash = h(message).digest(m)
    
    # 初始化公钥索引
    pk_index = 0
    
    # 验证每个层级的签名
    for i in range(d):
        # 计算层级的哈希种子
        layer_seed = pk[pk_index:pk_index+n]
        pk_index += n
        
        # 计算层级的掩码
        layer_masks = [h(layer_seed + bytes([j])).digest(w) for j in range(2**w)]
        
        # 计算层级的哈希值
        layer_hash = h(layer_seed + signature[:L] + message_hash).digest(L)
        
        # 验证层级的签名
        layer_signature = signature[:2**w*n]
        signature = signature[2**w*n:]
        for j in range(2**w):
            if layer_signature[:n] == h(layer_hash + layer_masks[j]).digest(n):
                layer_hash = h(layer_hash + layer_masks[j]).digest(L)
            else:
                layer_hash = h(layer_hash + layer_masks[j]).digest(L)
                layer_hash = h(layer_hash + layer_masks[1-j]).digest(L)
        
        # 如果验证失败，则返回False
        if layer_signature[n:] != layer_hash:
            return False
    
    # 如果验证成功，则返回True
    return True