# pfranusic/why-RSA-works

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 %%%% why-RSA-works/rings.tex %%%% Copyright 2012 Peter Franusic. %%%% All rights reserved. %%%% RSA uses mathematical structures called rings. A \emph{ring} is a set equipped with two binary operators.\cite{wiki-Rings} The ring displays several well-defined algebraic properties, including both additive closure and multiplicative closure. Recall that a set is simply a collection of elements. These elements can be anything, but in the case of RSA, the elements are integers. RSA uses sets with a finite number of elements. The number of elements in a set is called the \emph{modulus}. The modulus is represented by the symbol $n$. A binary operator is something that takes two elements and computes a third. Rings use two binary operators, which we denote here as $\oplus$ (pronounced \textsf{OH plus}) and $\otimes$ (pronounced \textsf{OH times}). The $\oplus$ operator is similar to addition. The $\otimes$ operator is similar to multiplication. In general, we say that the ring $\mathcal{R}_n$ consists of the set $Z_n$, the $\oplus$ operator, and the $\otimes$ operator. $\mathcal{R}_n = (Z_n,\oplus,\otimes)$