\documentclass[12pt,ifthen]{article} \usepackage{url} \newcommand{\lf}{\left\lfloor} \newcommand{\rf}{\right\rfloor} \newcommand{\lc}{\left\lceil} \newcommand{\rc}{\right\rceil} \newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} \newcommand{\abit }{\hat{a}} \newcommand{\bbit }{\hat{b}} \newcommand{\bits}[1]{\{0,1\}^{{#1}}} \newcommand{\nth}{n^{th}} \newif{\ifshowsoln} \showsolntrue \newcommand{\und}{\_\_\_\_\_\_\_\_\_} \newcommand{\Z}{\mathbb{Z}} \usepackage{amsmath} \usepackage{amssymb} % for \nmid \begin{document} \centerline{\textbf{HW 10 CMSC 456. MORALLY DUE Nov 26}} {\textbf{NOTE- THE HW IS ONE PAGE LONG!!!!!!!}} \begin{enumerate} \item (0 points) READ the syllabus- Content and Policy. What is your name? Write it clearly. What is the day of the final? READ the slides and notes on Secret Sharing. \item (30 points) Let $1\le t\le L$. Show that there CANNOT be a $(t,L)$ VSS scheme if all the players are all powerful and they want information-theoretic security. The players shares can be of any finite length. (WARNING- DO NOT prove that the VSS scheme WE gave in class would not work. You need to show that NO VSS scheme works.) \item (30 points) \begin{enumerate} \item (20 points) In class we showed how to use the Paillier Public Key Crypto System and Secret Sharing to hold an election where there are TWO candidates. Find a way to hold an election with THREE candidates and $V$ voters. You are GIVEN $V$ and need to put conditions on $N$ so that your scheme works. \item (10 points) If 1,000,000 people want to vote then how large does $N$ have to be? \end{enumerate} \item (20 points) Zelda wants to do $(3,3)$ secret sharing with polynomials. The secret is $1001$ which is 9 in base 2, so she uses mod 11. Zelda picks out $r_2=3$ and $r_1=7$. What shares does she give out? Give the ACTUAL NUMBER, do not just say, for example $f(1)$. (NOTE- this was an issue on the midterm when some people for Diffie Helman wrote that Alice sends $2^{4} \pmod {11}$. I am asking this question now so that you DO NOT make the same MISTAKE on the FINAL.) \item (20 points) In the last problem Zelda had secret 9 and used mod 11. The players DO know the length of the secret (that is not considered a leak of info). The players DO know that they work mod 11. Does the choice of 11 leak any information? Explain your answer. \end{enumerate} \end{document}