MR素数测试在密码学库中应用广泛，通常作为BSPW的一部分来进行素数测试，由于在其算法中，有随机数的使用（选择一个随机的base），若一个MR伪素数$n$，已知其在某一个强伪证$a$（随机base）下表现出伪素性，那么我们可以逆向其算法过程，构造一个伪随机数生成器，使其通过MR素数测试。这通常是绕过BSPW必不可少的一部分。

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1.MR素数测试

设$n$是一个素数，且$n>2$，则$n-1$为偶数，$n-1$可以表示为$2^{s}d$，$s$和$d$为正整数，且$d$为奇数。对任意在$(Z/nZ{)}^{\ast }$范围内的$a$，必须满足以下两种形式的一种：

$$\begin{gathered} a^d\equiv 1(modn)\stackrel{◯}{1} \\ {a}^{2^{r}d}\equiv -1(modn)\stackrel{◯}{2} \end{gathered}$$

其中$r$是满足$,0\le r\le s-1$的整数。
由费马定理可得，对于一个素数$n$，有
$$a^{n-1}\equiv 1(modn)$$

不断对$a^{n-1}$取平方根后，总会得到$1$或$-1$。如果得到了$-1$，意味着$\stackrel{◯}{2}$成立。如果从未得到-1，那么这个过程已经取遍了所有$2$的幂次，意味着$\stackrel{◯}{1}$成立。

Miller–Rabin素数测试基于上述原理的逆否，如果能找到一个$a$，使得对于任意$0\le r\le s-1$以下两个式子均满足：

$$\begin{gathered} a^d\not\equiv 1(modn) \\ {a}^{2^{r}d}\not\equiv -1(modn) \end{gathered}$$
那么$n$是一个合数。这样的$a$称为$n$是合数的一个凭证（witness）。否则$a$可能是一个证明$n$是素数的“强伪证”（strong liar），即当$n$确实是一个合数，但是对于当前选取的$a$来说上述两个式子均不满足，这时我们认为$n$是基于$a$的大概率素数。

详情参考维基百科：https://zh.wikipedia.org/wiki/%E7%B1%B3%E5%8B%92-%E6%8B%89%E5%AE%BE%E6%A3%80%E9%AA%8C

2.pycryptodome 下MR素数测试源码分析

2.1 相关版本

• python 3.9.0
• pycryptodome 3.18.0

2.2 源码分析

直接取自Crypto.Math.Primality下的miller_rabin_test方法：

def miller_rabin_test(candidate, iterations, randfunc=None):
    """Perform a Miller-Rabin primality test on an integer.

    The test is specified in Section C.3.1 of `FIPS PUB 186-4`__.

    :Parameters:
      candidate : integer
        The number to test for primality.
      iterations : integer
        The maximum number of iterations to perform before
        declaring a candidate a probable prime.
      randfunc : callable
        An RNG function where bases are taken from.

    :Returns:
      ``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.

    .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
    """

    if not isinstance(candidate, Integer):
        candidate = Integer(candidate)

    if candidate in (1, 2, 3, 5):
        return PROBABLY_PRIME

    if candidate.is_even():
        return COMPOSITE

    one = Integer(1)
    minus_one = Integer(candidate - 1)

    if randfunc is None:
        randfunc = Random.new().read

    # Step 1 and 2
    m = Integer(minus_one)
    a = 0
    while m.is_even():
        m >>= 1
        a += 1

    # Skip step 3

    # Step 4
    for i in iter_range(iterations):

        # Step 4.1-2
        base = 1
        while base in (one, minus_one):
            base = Integer.random_range(min_inclusive=2,
                    max_inclusive=candidate - 2,
                    randfunc=randfunc)
            assert(2 <= base <= candidate - 2)

        # Step 4.3-4.4
        z = pow(base, m, candidate)
        if z in (one, minus_one):
            continue

        # Step 4.5
        for j in iter_range(1, a):
            z = pow(z, 2, candidate)
            if z == minus_one:
                break
            if z == one:
                return COMPOSITE
        else:
            return COMPOSITE

    # Step 5
    return PROBABLY_PRIME

三个参数分别是待测数、迭代次数、伪随机数生成器，同时注释也表明了是遵循nist规范（http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf）的一个实现。

按照第1章的符号规范的话，其中的minus_one是$n-1$，m是$d$，a是$s$，base是随机选择的$a$。

整体流程是：

• 1.先将$n-1$表示为$2^{s}d$。
• 2.在指定迭代次数下，每次随机选取一个$a$，满足$1<a<n-1$，分别测试$\stackrel{◯}{1}$和$\stackrel{◯}{2}$的逆否。

再来看一下默认迭代次数是如何选择的，在test_probable_prime方法中：

选择的依据是保证伪素数的概率是$10^{-30}$，按照220比特内，迭代30次，220-280比特内，迭代20次这样的方式预设迭代次数的值。

3.已知MR伪素数以及强伪证 生成指定伪随机数生成器

如何构造一个MR伪素数及强伪证不是本文的重点，可以自行参考相应论文（Fran ̧cois Arnault. Constructing Carmichael numbers which are strong pseudoprimes to several bases. Journal of Symbolic Computation, 20(2):151–161, 1995.）

假设已知一个MR伪素数$n$，一个强伪证$a$，若要让第2章中的miller_rabin_test通过测试，我们需要确定其随机数的选择机理，同时将$a$转换为一系列伪随机生成器的输出，同时逆向该伪随机数生成器，在调用miller_rabin_test的时候指定为该伪随机数生成器，或覆盖系统的os.urandom为该伪随机数生成器。

3.1 base生成机理

base的生成，调用的是Integer.random_range方法：

    @classmethod
    def random_range(cls, **kwargs):
        """Generate a random integer within a given internal.

        :Keywords:
          min_inclusive : integer
            The lower end of the interval (inclusive).
          max_inclusive : integer
            The higher end of the interval (inclusive).
          max_exclusive : integer
            The higher end of the interval (exclusive).
          randfunc : callable
            A function that returns a random byte string. The length of the
            byte string is passed as parameter. Optional.
            If not provided (or ``None``), randomness is read from the system RNG.
        :Returns:
            An Integer randomly taken in the given interval.
        """

        min_inclusive = kwargs.pop("min_inclusive", None)
        max_inclusive = kwargs.pop("max_inclusive", None)
        max_exclusive = kwargs.pop("max_exclusive", None)
        randfunc = kwargs.pop("randfunc", None)

        if kwargs:
            raise ValueError("Unknown keywords: " + str(kwargs.keys))
        if None not in (max_inclusive, max_exclusive):
            raise ValueError("max_inclusive and max_exclusive cannot be both"
                         " specified")
        if max_exclusive is not None:
            max_inclusive = max_exclusive - 1
        if None in (min_inclusive, max_inclusive):
            raise ValueError("Missing keyword to identify the interval")

        if randfunc is None:
            randfunc = Random.new().read

        norm_maximum = max_inclusive - min_inclusive
        bits_needed = cls(norm_maximum).size_in_bits()

        norm_candidate = -1
        while not 0 <= norm_candidate <= norm_maximum:
            norm_candidate = cls.random(
                                    max_bits=bits_needed,
                                    randfunc=randfunc
                                    )
        return norm_candidate + min_inclusive

在该方法中，将其分为了norm_candidate + min_inclusive（最小值保证），随后调用了cls.random获取norm_candidate：

    @classmethod
    def random(cls, **kwargs):
        """Generate a random natural integer of a certain size.

        :Keywords:
          exact_bits : positive integer
            The length in bits of the resulting random Integer number.
            The number is guaranteed to fulfil the relation:

                2^bits > result >= 2^(bits - 1)

          max_bits : positive integer
            The maximum length in bits of the resulting random Integer number.
            The number is guaranteed to fulfil the relation:

                2^bits > result >=0

          randfunc : callable
            A function that returns a random byte string. The length of the
            byte string is passed as parameter. Optional.
            If not provided (or ``None``), randomness is read from the system RNG.

        :Return: a Integer object
        """

        exact_bits = kwargs.pop("exact_bits", None)
        max_bits = kwargs.pop("max_bits", None)
        randfunc = kwargs.pop("randfunc", None)

        if randfunc is None:
            randfunc = Random.new().read

        if exact_bits is None and max_bits is None:
            raise ValueError("Either 'exact_bits' or 'max_bits' must be specified")

        if exact_bits is not None and max_bits is not None:
            raise ValueError("'exact_bits' and 'max_bits' are mutually exclusive")

        bits = exact_bits or max_bits
        bytes_needed = ((bits - 1) // 8) + 1
        significant_bits_msb = 8 - (bytes_needed * 8 - bits)
        msb = bord(randfunc(1)[0])
        if exact_bits is not None:
            msb |= 1 << (significant_bits_msb - 1)
        msb &= (1 << significant_bits_msb) - 1

        return cls.from_bytes(bchr(msb) + randfunc(bytes_needed - 1))

流程如下：

• 1.先产生一个随机字节，用于确定最高位。
• 2.然后产生剩余的字节，一起组合成一个整数。

3.2 转换为伪随机数生成器的输出

我们使用python的random模块，其使用的是MT19937。若要逆向MT19937得到一个完全一致的伪随机数生成器，我们需要明确在整个调用伪随机数生成器期间，生成了哪些32位数。

MT19937默认生成的随机数是32字节，若调用random.randbytes(1)生成一个字节，那么从源码可以看出：

是将一个32位的数右移24位。

剩余的字节数，按每4个字节是一个完整的MT19937输出计算，好需要特殊处理末尾的几个字节，可能为0-3个字节。

具体转换过程见代码：

def get_mr_test_rand_list(p, base):
    def get_iter_count(x):
        x = Integer(x)
        mr_ranges = ((220, 30), (280, 20), (390, 15), (512, 10),
                     (620, 7), (740, 6), (890, 5), (1200, 4),
                     (1700, 3), (3700, 2))

        bit_size = x.size_in_bits()
        try:
            mr_iterations = list(filter(lambda x: bit_size < x[0],
                                        mr_ranges))[0][1]
        except IndexError:
            mr_iterations = 1
        return mr_iterations

    res_rands = []
    base_bytes = long_to_bytes(base - 2)
    iter_count = get_iter_count(p)
    for _ in range(iter_count):
        # randbytes(1)
        res_rands += [base_bytes[0] << 24]
        # randbytes(bytes_needed - 1)
        base_byte_size = len(base_bytes) - 1
        base_int_size_r = base_byte_size % 4
        for i in range(1, base_byte_size - base_int_size_r + 1, 4):
            res_rands += [base_bytes[i] + base_bytes[i + 1] * 2**8 +
                          base_bytes[i + 2] * 2**16 + base_bytes[i + 3] * 2**24]
        last_int = 0
        for i in range(0, base_int_size_r):
            last_int += base_bytes[base_byte_size - base_int_size_r + 1 + i] * 2**(i*8)
        res_rands += [last_int << ((4 - base_int_size_r) * 8)]
    return res_rands

3.3 生成指定伪随机数生成器

参考之前逆向MT19937的文章：

MT19937在连续输出存在截断的情况下利用z3符号执行推导内部状态以及等价种子: 点我前往

只需要将get_mr_test_rand_list函数的输出，传入recover_seed，即可得到一个等价种子，从而得到一个指定的伪随机数生成器。

3.4 测试

给定MR伪素数及强伪证如下：

p1 = 142445387161415482404826365418175962266689133006163
p2 = 5840260873618034778597880982145214452934254453252643
p3 = 14386984103302963722887462907235772188935602433622363
n = p1 * p2 * p3

a = 29

在控制其伪随机数生成器后，成功通过MR素数测试。

import random

from Crypto.Math.Primality import *
from Crypto import Random
from Crypto.Math.Numbers import Integer
from Crypto.Util.number import *

def mr_test(x, randfunc=None):
    if randfunc is None:
        randfunc = Random.new().read
    x = Integer(x)
    mr_ranges = ((220, 30), (280, 20), (390, 15), (512, 10),
                 (620, 7), (740, 6), (890, 5), (1200, 4),
                 (1700, 3), (3700, 2))
    bit_size = x.size_in_bits()
    try:
        mr_iterations = list(filter(lambda x: bit_size < x[0],
                                        mr_ranges))[0][1]
    except IndexError:
        mr_iterations = 1
    if miller_rabin_test(x, mr_iterations,
                         randfunc=randfunc) == COMPOSITE:
        return COMPOSITE
    return PROBABLY_PRIME
def get_mr_test_rand_list(p, base):
    def get_iter_count(x):
        x = Integer(x)
        mr_ranges = ((220, 30), (280, 20), (390, 15), (512, 10),
                     (620, 7), (740, 6), (890, 5), (1200, 4),
                     (1700, 3), (3700, 2))

        bit_size = x.size_in_bits()
        try:
            mr_iterations = list(filter(lambda x: bit_size < x[0],
                                        mr_ranges))[0][1]
        except IndexError:
            mr_iterations = 1
        return mr_iterations

    res_rands = []
    base_bytes = long_to_bytes(base - 2)
    iter_count = get_iter_count(p)
    for _ in range(iter_count):
        # randbytes(1)
        res_rands += [base_bytes[0] << 24]
        # randbytes(bytes_needed - 1)
        base_byte_size = len(base_bytes) - 1
        base_int_size_r = base_byte_size % 4
        for i in range(1, base_byte_size - base_int_size_r + 1, 4):
            res_rands += [base_bytes[i] + base_bytes[i + 1] * 2**8 +
                          base_bytes[i + 2] * 2**16 + base_bytes[i + 3] * 2**24]
        last_int = 0
        for i in range(0, base_int_size_r):
            last_int += base_bytes[base_byte_size - base_int_size_r + 1 + i] * 2**(i*8)
        res_rands += [last_int << ((4 - base_int_size_r) * 8)]
    return res_rands
p1 = 142445387161415482404826365418175962266689133006163
p2 = 5840260873618034778597880982145214452934254453252643
p3 = 14386984103302963722887462907235772188935602433622363
q = p1 * p2 * p3
a = 29

res_rands = get_mr_test_rand_list(q, a)

#from find_seed_u import find_seed
#seed_int = find_seed(res_rands)
#print(hex(seed_int))
seed_int = 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

random.seed(seed_int)
print(mr_test(q, randfunc=random.randbytes))

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ATFWUS 2023-11-20