CSE 311 Lecture 12: Modular Arithmetic and Applications

Topics

Modular arithmetic basics

Review of Lecture 11.

Modular arithmetic properties

Congruence, addition, multiplication, proofs.

Modular arithmetic and integer representations

Unsigned, sign-magnitude, and two’s complement representation.

Applications of modular arithmetic

Hashing, pseudo-random numbers, ciphers.

Key definition: divisibility

Definition: $a$ divides $b$, written as $a \vert b$.

For $a\in\Z, b\in\Z$, $a \vert b \leftrightarrow \exists k\in\Z. b = ka$.

We also say that $b$ is divisible by $a$ when $a \vert b$.

Key theorem: division theorem

Division theorem

For $a\in\Z, d\in\Z$ with $d > 0$,

there exist unique integers $q, r$ with $0 \leq r < d$

such that $a = dq+r$.

That is, if we divide $a$ by $d$, we get a unique

• quotient $q = \div{a}{d}$ and
• non-negative remainder $r = \mod{a}{d}$.

So, $a = d(\div{a}{d}) + (\mod{a}{d})$.

Congruence modulo a positive integer

Definition: $a$ is congruent to $b$ modulo $m$, written as $\congruent{a}{b}{m}$

For $a,b,m\in\Z$ with $m>0$, $\congruent{a}{b}{m} \leftrightarrow m \vert (a-b)$

We read “$\congruent{a}{b}{m}$” as “$a$ is congruent to $b$ modulo $m$”, which means $m \vert (a-b)$.

So, “congruence modulo $m$” is a predicate on integers, written using the notation “$\congruent{}{}{m}$”.

Congruence and equality

Congruence property

Let $a, b, m \in \Z$ with $m>0$.

Then, $\congruent{a}{b}{m}$ if and only if $\mod{a}{m} = \mod{b}{m}$.

The $\mod{}{m}\ $ function vs the $\congruent{}{\!}{m}$ predicate

The $\mod{}{m}$ function takes any $a\in\Z$ and maps it to a remainder $\mod{a}{m}\in\{0, 1, \ldots, m-1\}$.

In other words, $\mod{}{m}$ places all integers that have the same remainder modulo $m$ into the same “group” (a.k.a. “congruence class”).

The $\congruent{}{\!}{m}$ predicate compares $a,b\in\Z$ and returns true if and only if $a$ and $b$ are in the same group according to the $\mod{}{m}$ function.

Modular addition property

Modular addition property

Let $m$ be a positive integer ($m \in \Z$ with $m>0$).

If $\congruent{a}{b}{m}$ and $\congruent{c}{d}{m}$, then $\congruent{a+c}{b+d}{m}$.

Modular multiplication property

Modular multiplication property

Let $m$ be a positive integer ($m \in \Z$ with $m>0$).

If $\congruent{a}{b}{m}$ and $\congruent{c}{d}{m}$, then $\congruent{ac}{bd}{m}$.

Example: a proof using modular arithmetic

Let $n\in\Z$, and prove that $\congruent{n^2}{0}{4}$ or $\congruent{n^2}{1}{4}$.

Unsigned integer representation

Represent integer $x$ as a sum of $n$ powers of 2:

If $x = \sum_{i=0}^{n-1} b_i2^i$ where each $b_i\in\{0,1\}$,

then the representation is $b_{n-1}\ldots b_2 b_1 b_0$.

Sign-magnitude integer representation

If $-2^{n-1} < x < 2^{n-1}$, represent $x$ with $n$ bits as follows:

Use the first bit as the sign (0 for positive and 1 for negative), and

the remaining $n-1$ bits as the (unsigned) value.

Two’s complement integer representation

Represent $x$ with $n$ bits as follows:

If $0 \leq x < 2^{n-1}$, use the $n$-bit unsigned representation of $x$.

If $-2^{n-1} \leq x < 0$, use the $n$-bit unsigned representation of $2^n - |x|$.

Computing the two’s complement representation

For $-2^{n-1} \leq x < 0$, $x$ is represented using the $n$-bit unsigned representation of $2^n - |x|$. To compute this value:

• Compute the $n$-bit unsigned representation of $|x|$.
• Flip the bits of $|x|$ to get the representation of $2^n-1-|x|$.
• Add 1 to get $2^n - |x|$.
• This works because $x+\overline{x}$ is all 1s, which represents $2^n-1$. So $\overline{x} = 2^n - 1 - x$ and $\overline{x} + 1 = 2^n - x$.

Example: -18 in 8-bit two’s complement

18 in 8-bit unsigned: $0001\ 0010$

Flip the bits: $1110\ 1101$

Add 1: $1110\ 1110$

Hashing

Problem:

We want to map a small number of data values from a large domain $\{0, 1, \ldots, M-1\}$ into a small set of locations $\{0, 1, \ldots, n-1\}$ to be able to quickly check if a value is present.

Solution:

Compute $\text{hash}(x) = \mod{x}{p}$ for a prime $p$ close to $n$.

Or, compute $\text{hash}(x) = \mod{ax+b}{p}$ for a prime $p$ close to $n$.

This approach depends on all of the bits of data the data.

Helps avoid collisions due to similar values.

But need to manage them if they occur.

Pseudo-random number generation

Linear Congruential method

$x_{n+1} = \mod{(ax_n + c)}{m}$

Choose $x_0$ randomly and $a, c, m$ carefully to produce a sequence of $x_n$’s.

Example

$a = 1103515245, c = 12345, m = 2^{31}$ from BSD

$x_0 = 311$

$x_1 = 1743353508, x_2 = 1197845517, x_3 = 1069836226, \ldots$

Simple ciphers

Ceasar or shift cipher

Treat letters as numbers: A = 0, B = 1, …

$f(p) = \mod{(p + k)}{26}$

$f^{-1}(p) = \mod{(p - k)}{26}$

More general version

$f(p) = \mod{(ap + b)}{26}$

$f^{-1}(p) = \mod{(a^{-1}(p - b))}{26}$