什麼是隨機亂數

import random
import numpy as np

# 設定種子值以確保可重現性，42 是宇宙的終極答案
random.seed(42)
np.random.seed(42)

# 產生0-1之間的浮點數
random_float = random.random()
print(f"Random float: {random_float}")

# 產生指定範圍的整數
random_int = random.randint(1, 100)
print(f"Random integer between 1 and 100: {random_int}")

# 從序列中隨機選擇
sequence = ['A', 'B', 'C', 'D', 'E']
random_choice = random.choice(sequence)
print(f"Random choice from sequence: {random_choice}")

# 隨機打亂序列
random.shuffle(sequence)
print(f"Shuffled sequence: {sequence}")

class LCG:
    def __init__(self, seed, a=1664525, c=1013904223, m=2 ** 32):
        self.seed = seed
        self.a = a  # 乘數
        self.c = c  # 增量
        self.m = m  # 模數
        self.current = seed

    def next(self):
        self.current = (self.a * self.current + self.c) % self.m
        return self.current

    def random(self):
        # 轉換為0-1之間的浮點數
        return self.next() / self.m


# 使用範例
lcg = LCG(seed=42)
for _ in range(5):
    print(f"Random number: {lcg.random():.6f}")

class MersenneTwister:
    def __init__(self, seed=5489):
        # MT19937 constants
        self.w, self.n, self.m, self.r = 32, 624, 397, 31
        self.a = 0x9908B0DF
        self.u, self.d = 11, 0xFFFFFFFF
        self.s, self.b = 7, 0x9D2C5680
        self.t, self.c = 15, 0xEFC60000
        self.l = 18
        self.f = 1812433253

        # Create a length n array to store the state of the generator
        self.MT = [0] * self.n
        self.index = self.n + 1
        self.lower_mask = (1 << self.r) - 1
        self.upper_mask = (~self.lower_mask) & self.d

        # Initialize the generator from a seed
        self.seed_mt(seed)

    def seed_mt(self, seed):
        """Initialize the generator from a seed"""
        self.index = self.n
        self.MT[0] = seed
        for i in range(1, self.n):
            self.MT[i] = self.d & (
                    self.f *
                    (self.MT[i - 1] ^ (self.MT[i - 1] >> (self.w - 2))) + i
            )

    def twist(self):
        """Generate the next n values from the series"""
        for i in range(self.n):
            x = (self.MT[i] & self.upper_mask) + \
                (self.MT[(i + 1) % self.n] & self.lower_mask)
            xA = x >> 1
            if x % 2 != 0:
                xA = xA ^ self.a
            self.MT[i] = self.MT[(i + self.m) % self.n] ^ xA
        self.index = 0

    def extract_number(self):
        """Extract a tempered value based on MT[index]"""
        if self.index >= self.n:
            if self.index > self.n:
                self.seed_mt(5489)  # Default seed
            self.twist()

        y = self.MT[self.index]
        y = y ^ ((y >> self.u) & self.d)
        y = y ^ ((y << self.s) & self.b)
        y = y ^ ((y << self.t) & self.c)
        y = y ^ (y >> self.l)

        self.index += 1
        return self.d & y

    def random(self):
        """Generate a random float in [0, 1)"""
        return self.extract_number() / ((1 << self.w) - 1)


if __name__ == '__main__':
    mt = MersenneTwister()
    for _ in range(5):
        print(mt.random())

import secrets
import os

# 產生密碼學安全的隨機位元組
random_bytes = secrets.token_bytes(16)
print(f"Random bytes: {random_bytes.hex()}")

# 產生密碼學安全的隨機整數
random_int = secrets.randbelow(100)
print(f"Random integer below 100: {random_int}")

# 產生密碼學安全的Token
token = secrets.token_urlsafe(32)
print(f"Secure URL-safe token: {token}")

# 使用os.urandom()
os_random_bytes = os.urandom(16)
print(f"OS random bytes: {os_random_bytes.hex()}")

import numpy as np
from scipy import stats


def chi_square_uniformity_test(numbers, bins=10, confidence_level=0.05):
    """
    使用卡方檢定評估亂數序列的均勻性
    
    參數：
    numbers: 要檢測的亂數序列
    bins: 分箱數量
    confidence_level: 顯著水準
    
    返回：
    is_uniform: 是否通過均勻性檢定
    p_value: p值
    """
    # 計算每個區間的觀察頻率
    observed_freq, _ = np.histogram(numbers, bins=bins, range=(0, 1))

    # 計算期望頻率（均勻分布下，每個區間的期望頻率應相等）
    expected_freq = len(numbers) / bins

    # 進行卡方檢定
    chi_square_stat, p_value = stats.chisquare(observed_freq)

    # 判斷是否通過檢定
    is_uniform = p_value > confidence_level

    print(f"卡方檢定結果：")
    print(f"p-value: {p_value:.4f}")
    print(f"檢定結果: {'通過' if is_uniform else '未通過'} (α = {confidence_level})")
    print(f"自由度: {bins - 1}")

    return is_uniform, p_value


# 測試範例
# 1. 測試內建的隨機數產生器
np.random.seed(42)
random_numbers = np.random.random(1000)
print("測試 NumPy 隨機數產生器：")
chi_square_uniformity_test(random_numbers)


# 2. 測試線性同餘法產生的隨機數
def lcg_sequence(n, seed=42, a=1664525, c=1013904223, m=2 ** 32):
    numbers = []
    current = seed
    for _ in range(n):
        current = (a * current + c) % m
        numbers.append(current / m)
    return numbers


print("\n測試線性同餘法：")
lcg_numbers = lcg_sequence(1000)
chi_square_uniformity_test(lcg_numbers)