Deep Analysis of the Chinese Remainder Theorem (CRT) in Cryptanalysis of Subgroups of Prime Orders (implemented on Google Colab)

Abstract. This paper presents a detailed analysis of the application of the Chinese Remainder Theorem (CRT) to cryptanalysis problems, specifically to solving the discrete logarithm in groups of composite order. The Pohlig–Hellman attack, which allows reducing a complex problem to a set of subproblems in subgroups of prime order, is considered, after which the results are combined using the CRT. Special attention is paid to the Bitcoin cryptocurrency and the elliptic curve secp256k1: it is shown that due to the prime order of the base point, a direct attack is impossible; however, vulnerabilities in nonce generation can lead to partial application of these methods. The paper contains ready-made Python scripts, adapted for execution in Google Colab, demonstrating all the key stages. In modern public-key cryptography, the security of systems is based on the computational complexity of problems such as integer factorization or discrete logarithm (DLP). Elliptic curves, used in Bitcoin, Ethereum, and many other blockchain platforms, occupy a special place. The secp256k1 curve is chosen so that the order of its base point G is a large prime, guaranteeing resistance to the Pohlig-Hellman attack, which is based on factorizing the order and then applying the Chinese Remainder Theorem.

Nevertheless, understanding the mechanisms of CRT and its role in cryptanalysis remains critical for information security professionals. First, many legacy or alternative protocols can use groups with composite ordering. Second, even in systems with prime ordering, implementation vulnerabilities are possible (for example, leaking the low-order bits of a nonce during an ECDSA signature), leading to problems where CRT can be useful for secret key recovery. In this article, we systematically present the mathematical apparatus of CRT and the Pohlig–Hellman algorithm, demonstrate their relationship to elliptic curves, and provide practical Google Colab scripts to clearly illustrate each step. This article is devoted to an in-depth analysis of the application of the Chinese Remainder Theorem (CRT) in the context of cryptanalysis of subgroups of prime orders q _i . We discuss a methodology for solving a series of subproblems in individual subgroups and then combining the results using CRT. Particular attention is paid to interaction with the cryptography of the Bitcoin cryptocurrency, including vulnerability analysis and mathematical foundations.

In modern cryptography, particularly in elliptic curve-based systems (such as secp256k1, used in Bitcoin), the discrete logarithm problem (DLP) is fundamental. The security of the system depends on the difficulty of computing the private key from the public key. However, if the order of the group factorizes into relatively small prime factors, the problem can be broken down into smaller subproblems using the Pohlig-Hellman algorithm, and the results can be combined using the Chinese Remainder Theorem (CRT).

Historical example: In 1978, Steven Pohlig and Martin Hellman published an algorithm that showed that if the order of a group N has only small prime divisors, then the DLP can be solved in polynomial time. This discovery led to a strict requirement: in cryptosystems (e.g., Bitcoin), the order of the base point must be prime.

2. Mathematical foundations of CRT

The Chinese remainder theorem states that if the modules m ₁ , m ₂ , …, m _k are pairwise relatively prime, then the system of congruences:

$x \equiv a_{1} (\mod m_{1})$ $x \equiv a_{2} (\mod m_{2})$ $\dots$ $x \equiv a_{k} (\mod m_{k})$

has a unique solution modulo M = m ₁ \cdot m ₂ \cdot \dots \cdot m _k .

Algorithm for finding a solution:

Calculate M = m ₁ \cdot m ₂ \dots m _k .
For each i, calculate M _i = M / m _i .
Find the inverse elements y _i = M _i^-1 \pmod{m_i} .
Calculate the final value: x = \sum_{i=1}^{k} a_i \cdot M_i \cdot y_i \pmod{M} .

3. Application to subgroups of prime orders q _i

Let the order of the elliptic curve group N not be prime, but decompose into factors: N = q ₁ \cdot q ₂ \dots q _k , where q _i are prime numbers.

If we want to solve the DLP: Q = xP , we project the problem into subgroups of order q _i by multiplying both sides by N / q _i :

$(N / q_{i}) Q = x_{i} ((N / q_{i}) P)$

By solving the DLP in each of these small subgroups, we find x _i \equiv x \pmod{q_i} . We then use the CRT to find x \pmod{N} itself .

Connection with Bitcoin:

Bitcoin uses the secp256k1 curve . The order of the base point G in this curve is a very large prime number n (close to ²²⁵⁶ ). Because n is prime, the Pohlig-Hellman attack using a CRT does not work directly against Bitcoin. However, cryptanalysis using a CRT is applicable in situations where random numbers (nonces) in ECDSA signatures are generated with vulnerabilities (for example, by leaking some bits or using weak pseudorandom number generators whose periods can decompose).

4. Implementation examples

Python (SymPy)

from sympy.ntheory.modular import crt
m = [3, 5, 7]
a = [2, 3, 2]
x, M = crt(m, a)
print(f"Решение: x ≡ {x} (mod {M})") # Вывод: x ≡ 23 (mod 105)

SageMath

m = [3, 5, 7]
a = [2, 3, 2]
x = crt(a, m)
print(x) # Вывод: 23

Magma

m := [3, 5, 7];
a := [2, 3, 2];
x := CRT(a, m);
x; // Вывод: 23

PARI/GP

a = [2, 3, 2];
m = [3, 5, 7];
x = chinese(vector(#m, i, Mod(a[i], m[i])));
print(x); \ Вывод: Mod(23, 105)

2. Chinese Remainder Theorem: Basics

The Chinese remainder theorem is a classical result in number theory that provides a solution to a system of linear congruences. Let $m_{1}, m_{2}, \dots, m_{k}$ — pairwise coprime positive integers. Then for any integers $a_{1}, a_{2}, \dots, a_{k}$ system

${\begin{cases} x \equiv a_{1} (\mod m_{1}), \\ x \equiv a_{2} (\mod m_{2}), \\ ⋮ \\ x \equiv a_{k} (\mod m_{k}) \end{cases}$

has a unique solution modulo $M = m_{1} m_{2} \dots m_{k}$ Moreover, this solution can be found explicitly:

$x = \sum_{i = 1}^{k} a_{i} \cdot M_{i} \cdot y_{i} (\mod M),$

Where $M_{i} = M / m_{i}$ , A $y_{i}$ — reverse to $M_{i}$ modulo $m_{i}$ , i.e. $M_{i} y_{i} \equiv 1 (\mod m_{i})$ .

This theorem is the cornerstone of many algorithms in cryptography, including fast exponentiation, secret sharing schemes, and, most importantly for our topic, the Pohlig–Hellman attack.

3. The Pohlig–Hellman algorithm and its connection with CRT

Let us have a cyclic group G of order N , where N is a composite number with decomposition $N = q_{1}^{e_{1}} q_{2}^{e_{2}} \dots q_{t}^{e_{t}}$ For simplicity, let us consider the case where N is square-free, i.e. all $e_{i} = 1$ , And $N = q_{1} q_{2} \dots q_{t}$ , Where $q_{i}$ — different prime numbers.

Discrete logarithm problem: given $P, Q \in G$ , it is necessary to find $x \in Z_{N}$ such that $Q = x P$ If N is large, a direct solution (for example, by the “large and small step” method) requires $O (\sqrt{N})$ operations, which is often infeasible for cryptographic sizes. However, if N is factorable into small prime factors, the problem can be solved significantly faster.

The idea behind the Pohlig–Hellman algorithm:

For every prime divisor $q_{i}$ of order N solve DLP in a subgroup of order $q_{i}$ To do this, we multiply both parts of the equality $Q = x P$ on $N / q_{i}$ : $(\frac{N}{q_{i}}) Q = x (\frac{N}{q_{i}}) P .$ Let us denote $P_{i} = (N / q_{i}) P$ , $Q_{i} = (N / q_{i}) Q$ . Then $Q_{i} = x P_{i}$ , and the order $P_{i}$ equal $q_{i}$ (If $x$ multiple $q_{i}$ , then the order may be smaller, but we can solve DLP in a subgroup of prime order using methods like Pollard’s rho or exhaustive search if $q_{i}$ few).
Solving the DLP in each subgroup, we obtain $x \equiv x_{i} (\mod q_{i})$ .
Now we have a system of comparisons $x \equiv x_{i} (\mod q_{i})$ . Because everything $q_{i}$ pairwise relatively prime, CRT gives a unique solution $x (\mod N)$ .

Thus, the complexity of the algorithm is determined by the sum of the complexities of solving DLP in subgroups of prime orders, which turns out to be significantly less than $O (\sqrt{N})$ , if the prime divisors are small. This is a classical attack, which requires choosing cryptosystems with a prime group order.

4. Application to Elliptic Curves and Bitcoin

The Bitcoin system uses an elliptic curve secp256k1 , defined by the equation $y^{2} = x^{3} + 7$ over the field $F_{p}$ , Where $p = 2^{256} - 2^{32} - 977$ The base point G has order n equal to

n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
(this is a prime number ≈ 2²⁵⁶)

Since n is prime, there is no factorization, and the Pohlig-Hellman algorithm cannot be applied directly to find the secret key from the public key. This is one of Bitcoin’s fundamental security requirements.

However, in practice, vulnerabilities related to ECDSA implementations are possible. For example, if a non-random or predictable value k (nonce) is used in signature generation, an attacker can recover some of the bits of k . This, combined with known public keys and signatures, leads to comparison systems where a CRT can be useful for ultimately recovering the secret key. Such attacks are considered in the literature as attacks on incomplete nonces (e.g., lattice attacks ), but they are sometimes simplified to use a CRT when the number of leaks is small.

Additionally, there are alternative cryptocurrencies or protocols where the group order is not simple, and where a CRT attack is a real threat.

5. Practical Scripts for Google Colab

Below are three Python scripts that illustrate the key steps:

The CRT implementation is a demonstration of how the theorem works.
The Pohlig–Hellman algorithm for DLP in a multiplicative group — using the example of a finite field where the order of the group is composite.
A secp256k1 primality check confirms that the CRT attack is not applicable.

All scripts are adapted for execution in the Google Colab environment and use only standard libraries sympy. random

5.1 Basic CRT Implementation

import sympy
from sympy.ntheory.modular import crt

# Пример системы
m = [3, 5, 7]   # модули (попарно взаимно простые)
a = [2, 3, 2]   # остатки

x, M = crt(m, a)
print(f"Решение: x ≡ {x} (mod {M})")  # 23 mod 105

# Проверка
for mi, ai in zip(m, a):
    print(f"x mod {mi} = {x % mi} (ожидалось {ai})")

5.2. The Pohlig–Hellman Algorithm for DLP in the Multiplicative Group Modulo a Composite Number

Let us work in a group $Z_{p}^{*}$ order $p - 1$ , Where $p$ – simple, but $p - 1$ has small prime factors. We will set a small $p = 41$ , Then $p - 1 = 40 = 2^{3} \cdot 5$ For simplicity, we’ll only consider prime factors 2 and 5 (ignoring powers to keep things simple). We’ll implement the DLP solution using subgroups of prime orders and combining the results via a CRT.

import sympy
from sympy.ntheory.modular import crt
import random

def dlog_prime_subgroup(g, h, p, q):
    """
    Решает DLP: g^x = h mod p, где порядок g равен q (простое).
    Используем полный перебор (для малых q).
    """
    for x in range(q):
        if pow(g, x, p) == h:
            return x
    return None

def pohlig_hellman(g, h, p):
    """
    Решает DLP: g^x = h mod p, где p-1 = prod q_i (простые).
    Возвращает x mod (p-1).
    """
    N = p - 1
    factors = sympy.factorint(N)
    # Берём простые делители (без учёта степеней)
    primes = list(factors.keys())
    residues = []
    moduli = []
    for q in primes:
        # Вычисляем элементы подгруппы порядка q
        g_q = pow(g, N // q, p)
        h_q = pow(h, N // q, p)
        x_q = dlog_prime_subgroup(g_q, h_q, p, q)
        if x_q is None:
            raise ValueError("Не удалось решить DLP в подгруппе")
        residues.append(x_q)
        moduli.append(q)
    # Объединяем через CRT
    x, M = crt(moduli, residues)
    return x % N

# Пример: p=41, g=3 (первообразный корень), h = g^x
p = 41
g = 3   # первообразный корень по модулю 41
x_secret = random.randint(1, p-2)
h = pow(g, x_secret, p)

print(f"Исходные данные: p={p}, g={g}, h={h}")
print(f"Секретный x = {x_secret}")

x_found = pohlig_hellman(g, h, p)
print(f"Восстановленный x = {x_found}")
assert x_found == x_secret, "Ошибка!"
print("Успешно!")

# Вывод: факторизация p-1 и восстановленный x
factors = sympy.factorint(p-1)
print(f"p-1 = {p-1} = {factors}")

5.3 Checking the primality of the secp256k1 base point order

In this example we use the library ecdsa (if installed) or simply check if a number n is prime using sympy.isprime.

import sympy

# Порядок базовой точки G кривой secp256k1
n_hex = "0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141"
n = int(n_hex, 16)

print(f"n = {n}")
print(f"n - простое? {sympy.isprime(n)}")

# Попытка разложения на множители (займёт много времени для такого большого числа, поэтому не делаем)
# print(sympy.factorint(n))  # закомментировано, т.к. слишком медленно

Running this code, we see that n is prime, hence the CRT attack is not applicable.

Note: To run all scripts in Google Colab, you only need to install the library sympy (it’s pre-installed in most environments). If not, run it !pip install sympy in a cell.

6. Discussion of vulnerabilities and applicability limits

Although secp256k1 is secure against a direct Pohlig-Hellman attack, real-world systems may encounter situations where the problem reduces to solving a system of comparisons. For example, in multi-signature protocols, where nonces are partially known, equations of the form $s_{i} k_{i} \equiv z_{i} + r_{i} d (\mod n)$ , Where $d$ — a secret key. If several bits of each are known $k_{i}$ , you can construct a lattice and use LLL methods, but in some simplified cases (for example, if $k_{i}$ are small or linearly related) CRT can be used for recovery $d$ modulo the product of small divisors. However, in practice, for secp256k1, such attacks require a very large number of signatures and are impractical if the nonce is correctly generated.

Nevertheless, studying CRT and its role in cryptanalysis remains important for understanding the fundamental principles of constructing secure cryptosystems. In particular, when choosing elliptic curve parameters, it is necessary to ensure that the order of the group has no small prime divisors (or is prime at all).

Conclusion

In this paper, we examined in detail the Chinese Remainder Theorem and its application to cryptanalysis via the Pohlig–Hellman algorithm. We demonstrated that this technique effectively reduces the complexity of solving the discrete logarithm in groups of composite order, making it a powerful tool for cracking systems with poorly chosen parameters. Using Bitcoin as an example, we demonstrated that choosing a prime base point order is essential for security. The provided Google Colab scripts allow the reader to independently reproduce all stages: from a basic CRT to a full attack on a composite group.

In the future, researchers may extend the methods discussed to more complex scenarios, including attacks against incomplete nonces and the use of CRTs in combination with lattice algorithms. Understanding these mechanisms is critical for developers of cryptographic protocols and security auditors.

The Pohlig-Hellman attack, strengthened by the Chinese Remainder Theorem, demonstrates a fundamental principle of elliptic curve cryptography: the order of the group must be a large prime number. Although secp256k1 (Bitcoin) is secure against this type of direct attack by choosing a curve with prime order n , understanding the mechanics of the CRT is vital for cryptanalysts analyzing other protocols or vulnerable PRNG implementations for signature generation.

Bibliography

Polig, S., Hellman, M. (1978). An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. Inf. Theory, 24(1), 106–110.
Menezes, A., van Oorschot, P., Vanstone, S. (1996). Handbook of Applied Cryptography. CRC Press.
Hankerson, D., Menezes, A., Vanstone, S. (2004). Guide to Elliptic Curve Cryptography. Springer.
Bitcoin Wiki: Secp256k1. https://en.bitcoin.it/wiki/Secp256k1
Nguyen, P. Q., Shparlinski, I. E. (2002). The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces. Designs, Codes and Cryptography, 30(2), 201–217.