# publicpfranusic/why-RSA-works

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 %%%% why-RSA-works/set-Zn.tex%%%% Copyright 2012 Peter Franusic.%%%% All rights reserved.%%%%A finite set $Z_n$ can be specified in several different ways.When a set has just a few elements, they can be explicitly enumerated, listed within curly brackets.For example, the set $Z_{15}$ consists of the 15 integers starting with 0 and ending with 14.$Z_{15} = \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14\}$When a set has a huge number of elements, they cannot be enumerated.But if a set consists entirely of sequential elements, it can be specifiedby listing the first few elements, an ellipsis, and the last few elements.For example, the set $Z_n$ consists of a sequence of $n$ integers,starting with 0 and ending with $(n-1)$.$Z_n = \{0,1,2,3,\ldots,(n-2),(n-1)\}$When RSA generates a pair of keys, it selects some modulus $n$ that is the product of two distinct primes $p$ and $q$.The term \emph{product} means that we multiply $p$ times $q$.Instead of writing $p \times q$ we use the abbreviation $pq$.$n = pq$The term \emph{distinct} means that $p$ and $q$ are different from each other.That is, $p \ne q$.Recall that a \emph{prime} is any integer greater than 1that cannot be divided evenly by any other integer except 1 and itself.The first five primes are 2, 3, 5, 7, and 11.In the example of $Z_{15}$ above, the modulus $15$ is the product of the two distinct primes $3$ and $5$.
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