Chapter 3, Some Elementary Number Theory Video Solutions, Modern Cryptography and Elliptic Curves: A Beginner’s Guide

Problem 1

Exercise. In contrast to the proof we give below, which has certain pedagogical motivations, construct a proof using the idea in the remark above. For example, consider the intervals of the form $[q|b|,(q+1)|b|)$ where $|b|$ is the absolute value of $b$ and $q$ ranges over all the integers.

Norman Atentar

Numerade Educator

Problem 2

Exercise.
- Show that $3 \mid 0$, but $0 \nmid 3$.
- Show that if $a \mid b$ and $b \mid c$, then $a \mid c$.
- Show that if $a \mid b$ and $c \mid d$, then $a c \mid b d$.
- Show that if $m \neq 0$, then $a \mid b$ if and only if $a m \mid b m$.

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Problem 3

Exercise. Let $p>1$ be a prime. Show that
- For any integer $n, \operatorname{gcd}(p, n)=1$ or $p$.
- For integers $m, n$ if $p \mid m n$, then either $p \mid m$ or $p \mid n$.

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Problem 4

Can you find integers $x, y, z$ so that $987654319=x^2+$ $y^2+z^2$ ? Hint: Determine the possible values of $x^2+y^2+z^2(\bmod 8)$.

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05:40

Problem 5

(A precursor to the Chinese Remainder Theorem). Find the smallest number of marbles in a jar so that one remains if taken out $2,3,5$ at a time, but none remain if taken out 11 at a time.

James Chok

Numerade Educator

Problem 6

To get more of a feel for congruences and how to move between congruences and equalities, consider the following exercises:
- Show (by example) that the congruence $a x \equiv a y(\bmod n)$ does not necessarily imply that $x \equiv y(\bmod n)$.
- On the other hand, show that if $x \equiv y(\bmod n)$, then $a x \equiv a y(\bmod n)$ for any integer $a$.
- Show that there exist integers $u, v$ so that $a u+n v=b$ if and only if $a x \equiv b(\bmod n)$ is solvable.

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Problem 7

Explore an encryption scheme known as ROT13; it is a shift cipher. What can you say about the encryption and decryption functions $E$ and $D$ ?

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04:59

Problem 8

Exercise. Before reading the propositions that follow, can you determine the values of $a$ for which a decryption algorithm to $C \equiv$ $a P+b(\bmod 26)$ be produced?

Prathan Jarupoonphol

Numerade Educator

01:49

Problem 9

Let $m, n>1$ be coprime integers, and let $a, b$ be arbitrary integers. Then the system of congruences
$$
\begin{aligned}
& x \equiv a \quad(\bmod m), \\
& x \equiv b \quad(\bmod n)
\end{aligned}
$$
has a unique solution modulo mn . A generous hint: Note that since $\operatorname{gcd}(m, n)=1$, Bézout's identity says there exists $u, v \in \mathbb{Z}$ so that $m u+n v=1$. Show that the number $b m u+a n v$ is a solution to the system, and then prove it is unique modulo mn .

Trang Hoang

Numerade Educator

Problem 10

Explain how to use the above version of the CRT to solve a system
$$
\begin{aligned}
& x \equiv a \quad(\bmod \ell), \\
& x \equiv b \quad(\bmod m), \\
& x \equiv c \quad(\bmod n),
\end{aligned}
$$
where $\ell, m, n>1$ are integers which are coprime in pairs.

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