# pfranusic/why-RSA-works

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 9e6d37e Added actual files to repo. pfranusic authored Jun 25, 2012 1 %%%% why-RSA-works/exponential-notation.tex 2 %%%% Copyright 2012 Peter Franusic. 3 %%%% All rights reserved. 4 %%%% 5 6 Let's say we're given three elements $a,b,c$ which are members of the set $Z_n$. 7 We're also given the expression $a \otimes b \otimes c$. 8 The question is this: How do we compute this expression? 9 Do we first multiply $a$ and $b$ and then multiply $c$? 10 Or do we multiply $b$ and $c$ and then multiply $a$? 11 The answer is that either way is correct. 12 It doesn't matter what order we multiply the elements. 13 This is because the ring $\mathcal{R}_n$ has the property of \emph{multiplicative association}. 14 The multiplicative association property says that 15 when we have a series of $\otimes$ operations, 16 we can do the operations in whatever order we want. 17 The answer will be the same. 18 \begin{eqnarray*} 19 a \otimes b \otimes c &=& (a \otimes b) \otimes c \\ 20 &=& a \otimes (b \otimes c) 21 \end{eqnarray*} 22 23 The modex function is represented mathematically using \emph{exponential notation}. 24 Exponential notation is an efficient way to describe a series of multiplications of the same value. 25 For example, the value $m$ can be multiplied by itself any number of times. 26 We use exponential notation to describe this. 27 Remember that it doesn't matter in what order the $m$'s are multiplied together. 28 \begin{eqnarray*} 29 \overbrace{m}^1 &=& m^1 \\ 30 \overbrace{m \otimes m}^2 &=& m^2 \\ 31 \overbrace{m \otimes m \otimes m}^3 &=& m^3 \\ 32 \overbrace{m \otimes m \otimes m \otimes m}^4 &=& m^4 \\ 33 &\vdots& 34 \end{eqnarray*} 35 36 RSA uses the exponential notation $m^e$. 37 The value $m$ is the \emph{message} integer. 38 The value $e$ is the \emph{encryptor} exponent. 39 The exponential notation $m^e$ means that $e$ copies of $m$ are multiplied together 40 using the $\otimes$ operator in the ring $\mathcal{R}_n$. 41 $m^e \quad = \quad \overbrace{m \otimes m \otimes m \, \cdots \otimes m \otimes m}^e$ 42 43 RSA also uses the exponential notation $c^d$. 44 The value $c$ is the \emph{ciphertext} integer. 45 The value $d$ is the \emph{decryptor} exponent. 46 The exponential notation $c^d$ means that $d$ copies of $c$ are multiplied together 47 using the $\otimes$ operator in the ring $\mathcal{R}_n$. 48 $c^d \quad = \quad \overbrace{c \otimes c \otimes c \, \cdots \otimes c \otimes c}^d$ 49