pfranusic/why-RSA-works

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 %%%% why-RSA-works/multiple-plus-one.tex %%%% Copyright 2012 Peter Franusic. %%%% All rights reserved. %%%% RSA uses two integers as exponents. One is the encryptor $e$ and the other is the decryptor $d$. In order for RSA to work, the product $ed$ must satisfy a strict condition. The condition is that the product $ed$ must have a \emph{multiple-plus-one} form. The product must be able to be written in the form $k\lambda+1$. The reason for this condition will become apparent later. For now, however, we need to understand what the expression $k\lambda+1$ means. %% This paragraph shall introduce Table \ref{mult-plus-one} below. Table \ref{mult-plus-one} contains some examples of multiple-plus-one products. Each product ends in 001. Each product is a multiple of 1000, plus one. In the first example, the product 174001 is equal to $174 \cdot 1000 + 1$. \vspace{2ex} %%%% multiple-plus-one table \begin{table}[!ht] \begin{small} \input{mult-plus-one.tex} \end{small} \caption{Multiples of 1000, plus one} \label{mult-plus-one} \end{table} \vspace{2ex} %% This paragraph shall introduce $\lambda$. The Greek letter $\lambda$ (pronounced \textsf{LAM duh}) is specified in the RSA literature.\cite{RSA-standard} We use $\lambda$ here as an integer constant. It typically has a huge value, almost as large as modulus $n$. In the context of Table \ref{mult-plus-one} it has a small value, $\lambda=1000$. The products can therefore be written like this: \begin{eqnarray*} 911 \cdot 191 &=& 174 \lambda + 1 \\ 913 \cdot 977 &=& 892 \lambda + 1 \\ 917 \cdot 253 &=& 232 \lambda + 1 \\ & \vdots & \end{eqnarray*} %% This paragraph shall explain that $k$ is some unspecified positive integer. The products in the table can be written in the form $k\lambda+1$. The symbol $k$ signifies some positive integer. Its value is not important. The term $k\lambda$ simply means \emph{some integer multiple of $\lambda$}. This meaning of $k$ allows the multiple-plus-one condition to be stated succinctly. %% This paragraph shall formally state the condition. \paragraph{Multiple-plus-one condition:} Given positive integers $e$, $d$, and $\lambda$, the product $ed$ shall be an integer multiple of $\lambda$, plus one. \begin{equation} \label{eq:inv-pair} ed = k\lambda + 1 \end{equation}