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\lecture{5 and 6: Chinese Remaindering}{CS681}{Piyush P Kurur}{Ramprasad Saptharishi}

\section*{Overview}

In the next two lectures, we shall see a very important techinique
used in computer science. The technique is called {\em Chinese
Remaindering.} This comes in extremely handy in various arithmetic
problems. We shall be looking at the problem of evaluating the
determinant as a motivation.

\section{Motivation for CRT: The Determinant}

We are given an integer square matrix $A$ and we are to find the
determinant of the matrix. Before we talk about solving the problem,
we need to understand the input as such. How big is the input?

The size is not just $n^2$ since the entries of the matrix also need
to be represented. Hence the input size also depends on the size of
the entries in the matrix. Let us call $\lambda = \max_{i,j} \abs{a_{ij}}.$ Then each entry in the matrix requires atmost
$\log\lambda$ bits and there are $n^2$ entries. Thus, the input size
is $n^2 \log \lambda.$ We are looking for an algorithm that
runs in time polynomial in the input size. \\

The naive approach is to do Gaussian Elimination, or the elementary
row-operation method done in high-school: pick the first non-zero
element in the first row, divide that row by the number (making it
1), and use this row to clear all other entries in that row.

This however has two problems:
\begin{enumerate}
\item Involves division and hence manipulating rational numbers.
\item Numbers in the matrix can become huge during gaussian
elimination.
\end{enumerate}

Firstly we need to understand why the first point is really a
problem. We are given a matrix with just integer entries. We shall see
now that that such a matrix will have an integer determinant. Thus, it
may not be efficient to have rational number manipulation.

\subsection{Integer Matrices have Integer Determinants}

The group of permutations over $n$ indices is denoted by $S_n.$ Given
any permutation in $S_n$, we can talk about the sign of the
permutation. The definition is based on the fact that every
permutation can be written as a product of cycles of length $2.$

\begin{lemma}
Every permutation $\sigma$ can be written as a product of disjoint cycles.
\end{lemma}
\begin{proof}
Start with the index $1$. Look at the image of $1$ which is
$\sigma(1)$ and its image $\sigma(\sigma(1))$ etc. Eventually some $\sigma^i(1) = 1$ since the number of elements is finite. Hence this corresponds to
the cycle $\inparen{1\ \sigma(1)\ \sigma^2(1)\ \cdots \ \sigma^i(1)}.$ Now look at the next smallest index that has
not been covered in this cycle and do the same. Our permutation
$\sigma$ is the product of these cycles and they are clearly disjoint.
\end{proof}

\begin{lemma}
Every permutation $\sigma$ can be written as a product of cycles of
length $2.$
\end{lemma}
\begin{proof}
By the previous lemma, it is enough to show that every cycle can be
written as a product of $2$-cycles. And this is very easy to
see. Consider any cycle of the form $(a_1\ a_2\ \cdots\ a_k).$ Easy to
check that this is equal to the product $(a_1\ a_2)(a_1\ a_3)\cdots (a_1\ a_k).$
\end{proof}

Now, if a permutation $\sigma$ can be represented as a product of $m$
$2$-cycles, then we define the sign of $\sigma$ to be $(-1)^m.$ An
immediate question is whether this is well defined. That is, suppose a
permutation can be represented as a product of $7$ such $2$-cycles and
also as a product of $24$ $2$-cycles in a different way, won't it give
two conflicting values for the sign of the permutation. The answer is
that such a thing won't happen. It is not too hard to check but we
leave this to the interested reader.

(Hint: Every permutation of $n$ indices can be thought of as a tuple
$(x_1,\cdots,x_n)$ where the $i$-th index corresponds to the image of
$i$ under the permutation. Now look at the sign of the expression
$$\prod_{i>j}(x_i - x_j)$$ The sign of this expression is an
equivalent definition of the sign of the permutation.

Now can you see why a permutation cannot be expressed as a product of
$s$ transpositions and $t$ transpositions where $s$ and $t$ are of
different parity?)
\\

This is another formula to evaluate the determinant.
$$det A = \sum_{\sigma \in S_n} sign(\sigma)\prod_{i=1}^n a_{i\sigma(i)}$$

As an example, any $2\times 2$ matrix has its determinant as
$a_{11}a_{22} - a_{12}a_{21}$ where the first term corresponds to the
permutation $(1)(2)$ and the second to the permutation $(1\ 2).$

It is clear from the above formula that a determinant of an integer
matrix has to be an integer.

\subsection{First Attempt: An Euclidian Approach}

Since we are assured that the answer is going to be an integer, it
doesn't make much of sense to use rational numbers during our
computation. And besides, the denominators can grow really huge
through successive row operations.

But the problem of division can be sorted out using a Euclid's
Algorithm sort of approach. Consider the first row of the
matrix. Suppose each element is a multiple of the least entry, then we
are in good shape. There would be no need to divide at all. How do we
make sure that this happens? Somehow get the gcd of the numbers as one
of the entries!

Pick up the least element in the row, say $a_{11}.$ Now every other
$a_{i1} = qa_{11} + r.$ Now subtract $q$ times the first row from the
$i$-th row. This essentially reduces every entry to the remainder when
divided by $a_{11}.$ Now continue this procedure by picking up the least
element until you get the gcd of the numbers and then use it to kill
every other entry in the row. \\

But, this still causes numbers to blow up. While we do operations to
work on the first column, the other entries can grow to become too
large.

\subsection{Second Attempt: The Big Primes Method}

One clever trick is to do all computations modulo a prime large
enough. Since we know that the determinant  is equal to $\sum sign(\sigma)\prod a_{i\sigma(i)}$, this value is atmost $n!\lambda^n$
since there are $n!$ terms and each term can be atmost $M = \lambda^n.$
Now choose a prime $P$ larger than this bound $M.$

Now division reduces to multiplying by the inverse modulo $P$ and this
can be done efficiently (this is the reason we want our $P$ to be a
prime. We can't choose any arbitrary number since inverses may not
exist). Now the gaussian elimination works and numbers will be bounded
by $P.$ Gaussian elimination modulo this prime $P$ will give us a
final answer $D$ that is in the range $[0,P-1].$ But this could still
mean that there are two choices for the integer determinant. The
determinant could either be $D$ or $D-P.$ So to get around this small
catch, we choose $P$ to be a prime larger than $2M.$ In this way, if
the value we get is less than $P/2$, we know it is the
determinant. Else, it will be $D-P.$ \\

Thus we can solve the determinant problem by doing all computations
modulo a large prime. But how do we get a large prime? How do we find
a prime larger than the bound $M$ quickly?

By a theorem on the density of primes, a random number between $m$
and $2m$ is a prime with reasonably good probability. Thus we can just
pick a random number, test if it is prime, if not pick again. We will
hit a prime soon enough. \\

This however introduces randomness in our algorithm. We would like to
have a deterministic polynomial time algorithm. This is where Chinese
Remaindering comes in.

\section{Chinese Remainder Theorem: Over Integers}

\begin{theorem}
Let $N = a_1a_2\cdots a_k$ such that each pair $a_i,a_j$ are
coprime. Then we have the following isomorphism between the two
rings.
$$\Z/(N\Z) \iso \Z/(a_1\Z) \times \Z/(a_2\Z) \times \cdots \Z/(a_k\Z)$$
And more so, the isomorphism and the inverse map are computable easily.
\end{theorem}
\begin{proof}
First we look at the following homomorphism
\bea{
\phi:\Z & \longrightarrow & \Z/(a_1\Z) \times \Z/(a_2\Z) \times
\cdots \Z/(a_k\Z)\\
x & \mapsto & \inparen{x\bmod{a_1},x\bmod{a_2},\cdots, x\bmod{a_k}}
}
It is easy to check that this is indeed a homomorphism of rings. What
is the kernel of the map? We are looking at the inverse image of
$(0,0,\cdots, 0).$ This just means that any $x$ in the kernel must be
$0\bmod{a_i}$ for each $i.$ And since the $a_i$'s are coprime, this
inturn means that $x$ must be divisible by $N$. Thus the kernel of
this map is $(N\Z).$

Hence, by the first isomorphism theorem, we have that the induced
quotient map is an injective homomorphism:
$$\hat{\phi}:\Z/(N\Z) \hookrightarrow \Z/(a_1\Z) \times \Z/(a_2\Z) \times \cdots \Z/(a_k\Z)$$

It's just left to show that the map is not only injective but also
surjective; that would establish that it is indeed a
homomorphism. Since the rings in the picture are finite rings, we can
use a cardinality argument here. The cardinality of the ring on the
left is $N$ and so is the cardinality of the ring on the right $N$
(since it is equal to $a_1a_2\cdots a_n$). Hence, since the map is
injective between two sets of the same finite cardinality, it has to
be an isomorphism.

This however will now help when the rings are infinite. For example
$\R$ happily sits injectively inside $\C$ but they are clearly not
isomorphic. We shall soon be getting to a general setting when such a
cardinality argument won't work. Thus we need a more algebraic
proof. \\

Here we shall use a small lemma.
\begin{lemma}
We can easily compute elements $x_i$ such that $x_i = 1\bmod{a_i}$ and
$x_i = 0 \bmod{a_j}$ for all $i\neq j.$ In other words, the image of
$x_i$ is the tuple that has $1$ on the $i$-th coordinate and $0$
everywhere else.
\end{lemma}

{\em Pf:} Since all the $a_i$'s are pairwise coprime, $a_i$ is coprime
to $\bar{a}_i = \prod_{j\neq i} a_j.$ Thus, by euclid's lemma, there exists
elements $x$ and $y$ such that
$$xa_i + y\bar{a}_i = 1$$

Going modulo $a_i$, we get $y\bar{a}_i = 1\bmod{a_i}.$ And since
$y\bar{a}_i$ is divisible by each other $a_j$, it is $0\bmod{a_j}.$
Thus this number $y\bar{a}_i$ is our required $x_i$ and hence can be
computed easily by the extended euclid's algorithm. \qed

Now that we have these $x_i$'s, computing the inverse map is very
simple. Given a tuple $(z_1,z_2,\cdots, z_k)$, the inverse image is
just $\sum_{i=1}^k z_ix_i.$

Thus the map $\hat{\phi}$ is indeed an isomorphism and its image and
inverse images can be computed easily.
\end{proof}

\subsection{Solving Determinant through CRT}

We are going to pick up small primes ${p_1, p_2, \cdots, p_m}$ such
that $\prod p_i = N > 2M$ and then use chinese remaindering. How many
primes do we need to pick? Since each prime is greater than or equal
to two, the product of $m$ distinct primes is clearly greater than
$2^m.$ Thus in order to go larger than $2M$, we just need to pick
$\log 2M$ primes, which is $O(n\log n + n\log \lambda)$ and is clearly
polynomial in the input size.

How do we go about picking them? Just keep testing numbers from $2$
onwards, check if it is prime, and do this until we have enough
primes. How long do we have to go before we get enough primes?

The prime number theorem tells us that the number of primes less than
$n$ is  $O\inparen{\frac{n}{\log n}}.$ With a little bit of calculations, it
is easy to see that we would have found our $m$ primes if we go just
up till $m^2\log^2 m.$ And since $m$ is polynomial in the input size,
so is $m^2\log^2m.$

So we just need to look at all numbers up till $m^2\log^2m$ where $m = \log (2n!\lambda^n)$, and pick up all the primes. Note all these
primes are extremely small, even their magnitude smaller than $m$
which is about $\log M.$ In the big prime method, we were picking a
prime that required $\log M$ bits to even represent it in binary; its
magnitude was about $2^{\log M} = M.$ These primes are logarithmically
smaller. Hence, to check if the numbers are really primes, you can use
even the extremely inefficient exponential time sieve method or
something. It also makes sense to store these small primes in the
library, precompute them and keep it.

Now that we have these primes, we compute the $x_i$'s as indicated in
the lemma using the extended euclid's algorithm. Now we compute the
determinant of the matrix $A$ modulo each of these primes $p_i$ using
gaussian elimination. Let us say we get our value of the determinant
mod $p_i$ as $d_i.$ Once you have done this calculation for each
$p_i$, we get the tuple $(d_1, d_2, \cdots, d_m).$ Now using the
$x_i$'s, find the inverse map to get the value of the determinant $D$ mod
$N.$ If this value is less than $N/2$, return it. Else, return $D - N.$

\begin{algorithm}
\caption{\sc{Determinant: Using CRT}}
\begin{algorithmic}[1]
\STATE Let $M = n!\lambda^n$ and $m > 2\log M.$
\STATE Enumerate the first $m^2\log^2m$ numbers and check for
primes. Pick the first $m$ primes.
\STATE Let $N = p_1p_2\cdots p_m.$
\FOR{$i=1$ to $m$}
\STATE Evaluate, using gaussian elimination, $det(A)\bmod{p_i}.$
\STATE Let $d_i = det(A)\bmod{p_i}.$
\STATE Evaluate, using extended euclid's algorithm, the $x_i$ as in
the lemma.
\ENDFOR
\STATE Let $D = \sum_{i=1}^m x_id_i.$
\IF{$D < N/2$}
\STATE {\bf return} $D.$
\ELSE
\STATE {\bf return} $D - N.$
\ENDIF
\end{algorithmic}
\end{algorithm}

\section{Chinese Remainder Theorem for Arbitrary Rings}

In order to state the theorem for arbitrary rings, we need analogues
of divisibility and coprimeness in terms of rings. This can be done
using ideals. We say $m\mid n$, or $m$ divides $n$, if every multiple
of $n$ is also a multiple of $m.$ This interms of ideals translates to
$m\Z$ containing the ideal $n\Z.$

As for coprimes, we know that two numbers $a,b$ are coprime if there
exist $x,y$ such that $xa+yb = 1.$ For this, we need a notion of an
ideal sum.

Given ideals $\mathfrak{a},\mathfrak{b}$ of a ring $R$, we define the
sum-ideal as
$$\mathfrak{a}+\mathfrak{b} = \setdef{a+b}{a\in \mathfrak{a},b\in \mathfrak{b}}$$
This is also the ideal generated by the union of the two ideals.

And now we can say that two ideals $\mathfrak{a}$ and $\mathfrak{b}$
are coprime if the ideal $\mathfrak{a} + \mathfrak{b} = R.$

Using these definitions, we have the Chinese Remainder Theorem for
arbitrary rings. We state and prove the theorem for two ideals, but
the general case is similar.

\begin{theorem}
Let $R$ be any commutative ring with identity. Let $\mathfrak{a}$ and
$\mathfrak{b}$ be two ideals of $R$ that are coprime to each
other. Then we have the following ring isomorphism:
$$R/(\mathfrak{a}\cap \mathfrak{b}) \iso R/\mathfrak{a} \times R/\mathfrak{b}$$
\end{theorem}
\begin{proof}
The proof is almost exactly the same as the proof of CRT over
integers. Consider the following homomorphism.
\bea{
\phi: R & \longrightarrow & R/\mathfrak{a} \times
R/\mathfrak{b}\\
x& \mapsto & (x\bmod{\mathfrak{a}},x\bmod{\mathfrak{b}})
}

Here, mod corresponds to the coset that contains $x.$ This again is
clearly a homomorphism. And the kernel is the set of all elements that
ar $0\bmod{\mathfrak{a}}$ and $0\bmod{\mathfrak{b}}$ which means that
the element is present in both $\mathfrak{a}$ and $\mathfrak{b}$.
Thus the kernel of the homomorphism is $\mathfrak{a}\cap \mathfrak{b}.$

Thus all that's left to do is to show that the quotient map
$$\hat{\phi}: R/(\mathfrak{a}\cap \mathfrak{b}) \hookrightarrow R/\mathfrak{a}\times R/\mathfrak{b}$$
is also surjective. To show that, we construct the inverse map.

We need to find the inverse image of any given tuple $(x,y).$ Since we
know that the two ideals are coprime, $\mathfrak{a} + \mathfrak{b} = R$ and in particular contains the identity element $1.$

Hence there exists two elements $a\in\mathfrak{a},b\in\mathfrak{b}$
such that $a+b = 1.$ Just as in the integer case, if we go modulo
$\mathfrak{b}$ we see that $a$ is an element that is
$0\bmod{\mathfrak{a}}$ but $1 \bmod{\mathfrak{b}}$ and similarly the
element $b$. Now consider the element $xb+ ya.$ This is easily seen to
be $x\bmod{\mathfrak{a}}$ and $y\bmod{\mathfrak{b}}.$ Hence $xb + ya$
is the inverse image of $(x,y).$

Thus the map is also surjective and hence is indeed an isomorphism.
\end{proof}

Thus any ring that lets us compute the elements $a,b$ such that $a+b = 1$ will allow CRT to go through. One such example is the ring of
polynomials over integers. In this ring, the irreducible polynomial
will act as the primes and we will be able to compute the determinant
of a matrix with polynomial entries as well.

Another important property is that the units go to the units. That is,
if you take any element $x$ in the ring
$R/(\mathfrak{a}+\mathfrak{b})$ whose inverse exists, corresponding
tuple $(a,b)$ also have the property that $a$ and $b$ are
invertible. The proof of this is pretty easy.

Suppose an invertible element $x$ was mapped to $(a,b).$ Then, since
the inverse exists, look at the image of the inverse of $x.$ Lets call
it $(a',b').$ By definition, $1 = xx^{-1} \mapsto (aa',bb') = (1,1).$
And hence $a'$ must be the inverse of $a$ and $b'$ the inverse of
$b$. And thus if $x$ is invertible, so is  $a,b$ and viceversa. \\

Chinese Remainder Theorem is a really powerful tool used in numerous
occasions in computer science. We shall see more in the days to come.

\end{document}