Hykilpikonna
120 lines
2.8 KiB
TeX
120 lines
2.8 KiB
TeX
\documentclass[fontsize=11pt]{article}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{amsthm}
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\usepackage[utf8]{inputenc}
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\usepackage[margin=0.75in]{geometry}
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\title{CSC110 Assignment 4: Number Theory, Cryptography, and Algorithm Running Time}
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\author{Azalea Gui \& Peter Lin}
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\date{\today}
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% Some useful LaTeX commands. You are free to use these or not, and also add your own.
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\newcommand{\N}{\mathbb{N}}
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\newcommand{\Z}{\mathbb{Z}}
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\newcommand{\R}{\mathbb{R}}
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\newcommand{\cO}{\mathcal{O}}
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\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
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\begin{document}
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\maketitle
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\section*{Part 1: Practice with Proofs}
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\begin{enumerate}
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\item[1.] Statement to prove:
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$$\forall a, k, n \in \Z,~ \gcd(a, n) = 1 \Rightarrow \gcd(a + kn, n) = 1$$
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\item[2.] Statement to prove (we've expanded the definition of Omega for you!):
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$$\exists c, n_0 \in \R^+,~ \forall n \in \N,~ n \geq n_0 \Rightarrow \log_{3} n - \log_{11} n \geq c \cdot \log_{14} n$$
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\item[3.] Statement to prove (we haven't expanded the definition of Big-O for you, but we encourage you to do so yourself):
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$$\forall f, g: \N \to \R^{\geq 0},~ g \in \cO(f) \land \big(\forall m \in \N,~ f(m) \geq 1 \big) \Rightarrow g \in \cO(\floor{f})$$
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\end{enumerate}
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\newpage
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\section*{Part 2: Generating Coprime Numbers}
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\begin{enumerate}
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\item[1.]
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Not to be handed in.
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\item[2.]
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Complete this part in the provided \texttt{a4\_part2.py} starter file.
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Do \textbf{not} include your solution in this file.
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\item[3.]
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Prove that each loop invariant holds.
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\begin{enumerate}
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\item[a.] Loop Invariant 1
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\item[b.] Loop Invariant 2
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\item[c.] Loop Invariant 3
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\item[d.] Loop Invariant 4
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\begin{proof}
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TODO: Your proof goes here.
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\end{proof}
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\end{enumerate}
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\item[4.]
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Complete this part in the provided \texttt{a4\_part2.py} starter file.
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Do \textbf{not} include your solution in this file.
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\item[5.]
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Complete this part in the provided \texttt{a4\_part2.py} starter file.
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Do \textbf{not} include your solution in this file.
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\end{enumerate}
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\newpage
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\section*{Part 3: Running-Time Analysis}
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\begin{enumerate}
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\item[1.]
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TODO: Running-time analysis of \texttt{coprime\_to\_2\_and\_3}.
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\item[2.]
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TODO: Running-time analysis of \texttt{starting\_coprime\_numbers}.
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\item[3.]
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TODO: Running-time analysis of \texttt{coprime\_to\_all}.
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\end{enumerate}
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\section*{Part 4: Two New Cryptosystems}
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Complete this part in the provided \texttt{a4\_part4.py} starter file.
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Do \textbf{not} include your solution in this file.
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\end{document}
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