The Chinese Remainder Theorem

Suppose we are asked to solve

\[ x = 2 \pmod{5} \] \[ x = 3 \pmod{7} \]

for $x$. If we have a solution $y$, then $y + 35$ is also a solution, and more generally $y$ plus any mulitple of $35$. So we only need to look for solutions modulo $35$. By brute force, we find the only solution is $x = 17 \pmod{35}$.

For any system of equations like this, the Chinese Remainder Theorem tells us that there is always a unique solution up to a certain modulus, and describes how to find the solution efficiently.

Theorem: Let $p, q$ be coprime. Then the system of equations

\[ x = a \pmod{p} \] \[ x = b \pmod{q} \]

has a unique solution for $x$ modulo $p q$.

The reverse direction is trivial: given $x \in \mathbb{Z}_{p q}$, we can reduce $x$ modulo $p$ and $x$ modulo $q$ to obtain two equations of the above form.

Proof: Let $p_1 = p^{-1} \pmod{q}$ and $q_1 = q^{-1} \pmod{p}$. These must exist since $p, q$ are coprime. Then we claim that if $y$ is an integer such that

\[ y = a q q_1 + b p p_1 \pmod{p q} \]

then $y$ satisfies both equations:

Modulo $p$, we have $y = a q q_1 = a \pmod{p}$ since $q q_1 = 1 \pmod{p}$. Similarly $y = b \pmod{q}$. Thus $y$ is a solution for $x$.

It remains to show no other solutions exist modulo $p q$. If $z = a \pmod{p}$ then $z - y$ is a multiple of $p$. If $z = b \pmod{q}$ as well, then $z - y$ is also a multiple of $q$. Since $p$ and $q$ are coprime, this implies $z - y$ is a multiple of $p q$, hence $z = y \pmod {p q}$. ∎

This theorem implies we can represent an element of $\mathbb{Z}_{p q}$ by one element of $\mathbb{Z}_p$ and one element of $\mathbb{Z}_q$, and vice versa. In other words, we have a bijection between $\mathbb{Z}_{p q}$ and $\mathbb{Z}_p \times \mathbb{Z}_q$.

Examples: We can write $17 \in \mathbb{Z}_{35}$ as $(2,3) \in \mathbb{Z}_5 \times \mathbb{Z}_7$. We can write $1 \in \mathbb{Z}_{p q}$ as $(1,1) \in \mathbb{Z}_p \times \mathbb{Z}_q$.

In fact, this correspondence goes further than a simple relabelling. Suppose $x, y \in \mathbb{Z}_{p q}$ correspond to $(a,b), (c,d) \in \mathbb{Z}_p \times \mathbb{Z}_q$ respectively. Then a little thought shows $x + y$ corresponds to $(a + c, b + d)$, and similarly $x y$ corresponds to $(a c, b d)$.

A practical application: if we have many computations to perform on $x \in \mathbb{Z}_{p q}$ (e.g. RSA signing and decryption), we can convert $x$ to $(a,b) \in \mathbb{Z}_p \times \mathbb{Z}_q$ and do all the computations on $a$ and $b$ instead before converting back. This is often cheaper because for many algorithms, doubling the size of the input more than doubles the running time.

Example: To compute $17 \times 17 \pmod{35}$, we can compute $(2\times 2,3\times 3) = (4,2)$ in $\mathbb{Z}_5\times\mathbb{Z}_7$, and then apply the Chinese Remainder Theorem to find that $(4,2)$ is $9 \pmod{35}$.

Let us restate the Chinese Remainder Theorem in the form it is usually presented.

For Several Equations

Theorem: Let $m_1,...,m_n$ be pairwise coprime (that is $\gcd(m_i, m_j) = 1$ whenever $i \ne j$). Then the system of $n$ equations

\[ x = a_1 \pmod{m_1} \] \[ ... \] \[ x = a_n \pmod{m_n} \]

has a unique solution for $x$ modulo $M$ where $M = m_1...m_n$.

Proof: This is an easy induction from the previous form of the theorem, or we can write down the solution directly.

Define $b_i = M / m_i$ (the product of all the moduli except for $m_i$) and $b'_i = b_i^{-1} \pmod{m_i}$. Then by a similar argument to before,

\[ x = \sum_{i=1}^n a_i b_i b'_i \pmod{M} \]

is the unique solution.∎

Prime Powers First

An important consequence of the theorem is that when studying modular arithmetic in general, we can first study modular arithmetic a prime power and then appeal to the Chinese Remainder Theorem to generalize any results. For any integer $n$, we factorize $n$ into primes $n = p_1^{k_1} ... p_m^{k_m}$ and then use the Chinese Remainder Theorem to get

\[ \mathbb{Z}_n = \mathbb{Z}_{p_1^{k_1}} \times ... \times \mathbb{Z}_{p_m^{k_m}} \]

To prove statements in $\mathbb{Z}_{p^k}$, one starts from $\mathbb{Z}_p$, and inductively works up to $\mathbb{Z}_{p^k}$. Thus the most important case to study is $\mathbb{Z}_p$.