CSE 311 Lecture 13: Primes and GCD

Topics

Modular arithmetic applications

A quick wrap-up of Lecture 12.

Primes

Fundamental theorem of arithmetic, Euclid’s theorem, factoring.

Greatest Common Divisors (GCD)

GCD definition and properties.

Euclidean algorithm

Computing GCDs with the Euclidean algorithm.

Extended Euclidean algorithm

Bézout’s theorem and the extended Euclidean algorithm.

Applications of modular arithmetic

Modular arithmetic is the basis of modern computing, with many applications.

Examples include

• hashing,
• pseudo-random numbers, and
• simple ciphers.

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 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

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

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

$a^{-1}$ is the multiplicative inverse of $a$ modulo 26, and we’ll soon see how to compute these inverses.

Primality

Prime number

An integer $p > 1$ is called prime if its only positive factors are $1$ and $p$.

Composite number

An integer $c > 1$ is called composite if it is not prime.

A key theorem about all positive integers

Fundamental theorem of arithmetic

Every positive integer greater than 1 has a unique prime factorization.

A key theorem about primes

Euclid’s theorem

There are infinitely many primes.

Important algorithmic problems

Primality testing

Given an integer $n$, determine if $n$ is prime.

Factoring

Given an integer $n$, determine the prime factorization of $n$.

• We don’t know of an efficient algorithm for factoring large numbers.
• The security of commonly used cryptographic protocols (e.g., RSA) hinges on this fact.
• For example, it took two years and thousands of machine-hours to factor a 232-digit (768-bit) number known as RSA-768.
• But factoring is easy for quantum computers!

Definition of greatest common divisor (GCD)

Greatest common divisor (GCD)

The greatest common divisor of integers $a$ and $b$, written as $\gcd{a}{b}$, is the largest integer $d$ such that $d\vert a$ and $d\vert b$.

How can we compute $\gcd{a}{b}$?

A naive approach is to first factor both $a$ and $b$:

$a = 2^3\cdot 3\cdot 5^2\cdot 7\cdot 11 = 46,20$

$b = 2\cdot 3^2\cdot 5^3\cdot 7\cdot 13 = 204,750$

And then compute $\gcd{a}{b}$ as follows:

$\gcd{a}{b} = 2^{\min(3,1)}\cdot 3^{\min(1,2)}\cdot 5^{\min(2,3)}\cdot 7^{\min(1,1)}\cdot 11^{\min(1,0)} \cdot 13^{\min(0,1)}$

But factoring is expensive! Can we compute $\gcd{a}{b}$ without factoring?

Euclidean algorithm is based on two useful facts

$\gcd{a}{0}$

If $a$ is a positive integer, then $\gcd{a}{0} = a$.

Proof follows straightforwardly from the definition of GCD and divisibility.

GCD and modulo

If $a$ and $b$ are positive integers, then $\gcd{a}{b} = \gcd{b}{\mod{a}{b}}$.

Euclidean algorithm

Apply $\gcd{a}{b} = \gcd{b}{\mod{a}{b}}$ until you get $\gcd{a}{0} = a$.

Example implementation:

// Assumes a >= b >= 0.
public static int gcd(int a, int b) {
  if (b == 0)
    return a;             // GCD(a, 0) = a
  else  
    return gcd(b, a % b); // GCD(a, b) = GCD(b, a mod b)
}

Bézout’s theorem about GCDs

Bézout’s theorem

If $a$ and $b$ are positive integers, then there exist integers $s$ and $t$ such that $\gcd{a}{b} = sa + tb$.

We can extend Euclidean algorithm to find $s$ and $t$ in addition to computing $\gcd{a}{b}$.

Extended Euclidean algorithm

Extended Euclidean algorithm

Extended Euclidean algorithm

Multiplicative inverse $\mod{}{m}$

Suppose $\gcd{a}{m} = 1$.

By Bézout’s Theorem, there exist integers $s$ and $t$ such that $sa + tm = 1$.

$\mod{s}{m}$ is the multiplicative inverse of $a$

$1 = \mod{(sa+tm)}{m} = \mod{sa}{m}$

Using multiplicative inverses to solve modular equations

Solve: $\congruent{7x}{1}{26}$