04. Modular Arithmetic (1)

Number theory is a branch of mathematics devoted primarily to the study of the integers. Modular arithmetic is heavily used in cryptography.

Divisibility

Definition. Let $a, b, c \in \mathbb{Z}$ such that $a = bc$. Then,

1. $b$ and $c$ are said to divide $a$, and are called factors of $a$.
2. $a$ is said to be a multiple of $b$ and $c$.

Notation. For $a, b \in \mathbb{Z}$, we write $a \mid b$ if $a$ divides $b$. If not, we write $a \nmid b$.

These are simple lemmas for checking divisibility.

Lemma. Let $a, b, c \in \mathbb{Z}$.

1. If $a \mid b$ and $a \mid c$, then $a \mid (b + c)$.
2. If $a \mid b$, then $a \mid bc$.
3. If $a \mid b$ and $b \mid c$, then $a \mid c$.

Prime Numbers

Definition. Integer $n \geq 2$ is prime if it is only divisible by $1$ and itself. If it is not prime, then it is composite.

Note that $1$ is neither prime nor composite.

Primality Tests

It is hard to verify if some given number is prime. Many encryption schemes heavily rely on this fact.

The following is a simple algorithm to check if a given integer is prime.

bool naive_prime_test(int n) {
    if (n < 2) {
        return false;
    }

    for (int i = 2; i < sqrt(n); ++i) {
        if (n % i == 0) {
            return false;
        }
    }
    return true;
}

However, this algorithm has complexity $\mathcal{O}(\sqrt{n})$, which is slow. We have better algorithms like Fermat’s test, Miller-Rabin test, Pollard’s rho algorithm… (Not covered in this lecture)

Division Algorithm

Theorem. (Euclidean Division) For $a, b \in \mathbb{Z}$ with $b \neq 0$, there exist unique integers $q, r$ with $0 \leq r < \left\lvert b \right\rvert$ such that $a = bq + r$.

Proof. By induction.

Other proofs use the well-ordering principle.

Modulo Operation

There are two ways to think about ‘mod’: as a function, and as a congruence.

Modulo as a Function

As a function, $a \bmod b$ return the remainder of $a$ divided by $b$. This operation is commonly denoted % in many programming languages.¹

Modulo as a Congruence

As a congruence, it means that $a, b$ are in the same equivalence class.²

Definition. For $a, b, n \in \mathbb{Z}$ and $n \neq 0$, $a \equiv b \pmod n$ if and only if $n \mid (a - b)$.

Properties of modulo operation.

Lemma. Suppose that $a \equiv b \pmod n$ and $c \equiv d \pmod n$. Then, the following hold.

1. $a + c \equiv (b + d) \pmod n$.
2. $ac \equiv bd \pmod n$.
3. $a^k \equiv b^k \pmod n$.
4. $a \equiv (a \bmod n) \pmod n$.

Proof. Trivial. :)

The last one is very useful in computing. For example, if $a, b$ are very large integers, using the identity

\[(a + b)^k \equiv ((a + b) \bmod n)^k \pmod n\]

allows us to reduce the size of the numbers before exponentiation.

Modular Arithmetic

For modulus $n$, modular arithmetic is operation on $\mathbb{Z}_n$.

Residue Classes

For each positive integer $n$, we can partition $\mathbb{Z}$ into $n$ cells according to whether the remainder is $0, 1, 2, \dots, n - 1$ when the integer is divided by $n$. These cells are the residue classes modulo $n$ in $\mathbb{Z}$.

We write each residue class as follows.

\[\overline{k} = [k] = \left\lbrace m \in \mathbb{Z} : m \bmod n = k\right\rbrace\]

Consider the relation

\[R = \left\lbrace (a, b) : a \equiv b \pmod m \right\rbrace \subset \mathbb{Z} \times \mathbb{Z}\]

then $R$ has the following properties.

• Reflexive: $\forall a \in \mathbb{Z}$, $(a, a) \in R$.
• Symmetric: $\forall a, b \in \mathbb{Z}$, if $(a, b) \in R$, then $(b, a) \in R$.
• Transitive: $\forall a, b, c \in \mathbb{Z}$, if $(a, b), (b, c) \in R$ then $(a, c) \in R$.

Thus, $R$ is an equivalence relation and each residue class $[k]$ is an equivalence class.

We write the set of residue classes modulo $n$ as

\[\mathbb{Z}_n = \left\lbrace \overline{0}, \overline{1}, \overline{2}, \dots, \overline{n-1} \right\rbrace.\]

Note that $\mathbb{Z}_n$ is closed under addition and multiplication.

Identity

Definition. For a binary operation $\ast$ defined on a set $S$, $e$ is the identity if

\[\forall a \in S,\, a * e = e * a = a.\]

In $\mathbb{Z}_n$, the additive identity is $0$, the multiplicative identity is $1$.

Inverse

Definition. For a binary operation $\ast$ defined on a set $S$, let $e$ be the identity. $x$ is the inverse of $a$ if

\[x * a = a * x = e.\]

We write $x = a^{-1}$.

In the language of modular arithmetic, $x$ is the inverse of $a$ if

\[ax \equiv 1 \pmod n.\]

The inverse exists if and only if $\gcd(a, n) = 1$.

Lemma. For $n \geq 2$ and $a \in \mathbb{Z}$, its inverse $a^{-1} \in \mathbb{Z}_n$ exists if and only if $\gcd(a, n) = 1$.

Proof. We use the Extended Euclidean Algorithm. There exists $u, v \in \mathbb{Z}$ such that

\[au + nv = \gcd(a, n).\]

($\impliedby$) If $\gcd(a, n) = 1$, then $au + nv = 1$, so $au = 1 - nv \equiv 1 \pmod n$. Thus $a^{-1} = u$.

($\implies$) Suppose that $x = a^{-1}$ exists. Then $ax \equiv 1 \pmod n$, so $ax = 1 + kn$ for some $n \in \mathbb{Z}$. Then $ax - nk = 1$. $\gcd(a, n)$ must divide the LHS, so $\gcd(a, n) = 1$.

Euclidean Algorithm

Greatest Common Divisor

Definition. Let $a, b \in \mathbb{Z} \setminus \left\lbrace 0 \right\rbrace$ . The greatest common divisor of $a$ and $b$ is the largest integer $d$ such that $d \mid a$ and $d \mid b$. We write $d = \gcd(a, b)$.

Definition. If $\gcd(a, b) = 1$, we say that $a$ and $b$ are relatively prime.

Euclidean Algorithm

Euclidean Algorithm is an efficient way to find $\gcd(a, b)$. It relies on the following lemma.

Lemma. For $a, b \in \mathbb{Z}$ and $b \neq 0$, $\gcd(a, b) = \gcd(b, a \bmod b)$.

Proof. By the division algorithm, there exists $q, r \in \mathbb{Z}$ such that $a = bq + r$. Here, $r = a \bmod b$.

Let $d = \gcd(a, b)$. Then $d \mid a$ and $d \mid b$, so $d \mid (a - bq)$. Thus $d \leq \gcd(b, r)$. Conversely, let $d’ = \gcd(b, r)$. Then $d’ \mid b$ and $d’ \mid (a - bq)$, so $d’ \mid a$. Thus $d’ \leq \gcd(a, b)$. Thus $d = d’$.

The following code computes the greatest common divisor.

int gcd(int a, int b) {
    if (b == 0) {
        return a;
    } else {
        return gcd(b, a % b);
    }
}

Extended Euclidean Algorithm

We can extend the Euclidean algorithm to compute $u, v \in \mathbb{Z}$ such that

\[ua + vb = \gcd(a, b).\]

Basically, we use the Euclidean algorithm and solve for the remainder (which is the $\gcd$).

Calculating Modular Multiplicative Inverse

We can use the extended Euclidean algorithm to find modular inverses. Suppose we want to calculate $a^{-1}$ in $\mathbb{Z}_n$. We assume that the inverse exist, so $\gcd(a, n) = 1$.

Therefore, we use the extended Euclidean algorithm and find $x, y \in \mathbb{Z}$ such that

\[ax + ny = 1.\]

Then $ax \equiv 1 - ny \equiv 1 \pmod n$, thus $x$ is the inverse of $a$ in $\mathbb{Z}_n$.

1. Note that in C standards, (a / b) * b + (a % b) == a. ↩

2. $a$ and $b$ are in the same coset of $\mathbb{Z}/n\mathbb{Z}$. ↩