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1 47 JonasDC
\chapter{Operation}
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\section{Pipeline operation}
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The operation of the pipeline is shown in Figure~\ref{fig:pipeline_op}. One can see that the stages are started
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every 2 clock cycles ($\tau_{c}$ is the core clock period). This is needed because the least significant bit of the next
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stage result is needed. Every stage has to run $n$ (the width of the operands) times for the multiplication to be complete.
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\begin{figure}[H]
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\centering
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\includegraphics[trim=1.2cm 1.2cm 1.2cm 1.2cm, width=7cm]{pictures/pipeline_operation.pdf}
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\caption{Pipeline operation: Each circle represents an active stage. The number indicates how much times that stage
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                has run. Dotted line contours indicate the stage is inactive.}
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\label{fig:pipeline_op}
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\end{figure}
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For performing one Montgomery multiplication using this core, the total computation time $T_m$ for an $n$-bit operand
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with a $k$-stage pipeline is given by~(\ref{eq:Tmult}).
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\begin{align}\label{eq:Tmult}
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T_{m} = \left[k + 2(n - 1)\right] \tau_c
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\end{align}
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\newpage
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\section{Modular Simultaneous exponentiation operations}
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Exponentiations are calculated with Algorithm~\ref{alg:mme} which uses the Montgomery multiplier as the main computation
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step. It uses the principle of a square-and-multiply algorithm to calculate an exponentiation with 2 bases.
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\begin{algorithm}[H] % enter the algorithm environment
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\caption{Montgomery simultaneous exponentiation} % give the algorithm a caption
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\label{alg:mme} % and a label for \ref{} commands later in the document
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\algnewcommand\algorithmicdownto{\textbf{downto}}
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\algrenewtext{For}%
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{\algorithmicfor\ #1 $\gets$ #2 \algorithmicdownto\ #3 \algorithmicdo}
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\algnewcommand\algorithmicswitch{\textbf{switch}}
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\algrenewtext{While}%
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{\algorithmicswitch\ #1, #2}
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\algnewcommand\algorithmicinput{\textbf{Input:}}
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\algnewcommand\Input{\item[\algorithmicinput]}
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\algnewcommand\algorithmicoutput{\textbf{Output:}}
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\algnewcommand\Output{\item[\algorithmicoutput]}
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\footnotesize
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\begin{algorithmic} % enter the algorithmic environment
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\Input $g_{0},\:g_{1},\:e_{0}=(e_{0_{t-1}} \cdots e_{0_{0}})_{2},\:e_{1}=(e_{0_{t-1}} \cdots e_{0_{0}})_{2},\:R^{2}\bmod m,\:m$
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\Output $g_{0}^{e_{0}} \cdot g_{1}^{e_{1}} \bmod m$
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\State $\tilde{g}_{0} := \text{Mont}(g_{0}, R^{2}),\:\tilde{g}_{1} := \text{Mont}(g_{1}, R^{2}),\:\tilde{g}_{01} := \text{Mont}(\tilde{g}_{0}, \tilde{g}_{1})$
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\State $a := \text{Mont}(R^{2}, 1)$
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\Comment This is the same as $a := R \bmod m$.
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\For{$i$}{$(t-1)$}{0}
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\State $a := \text{Mont}(a, a)$
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\While{$e_{1_{i}}$}{$e_{0_{i}}$} % use as switch statement
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\State $0,\:1:\;a := \text{Mont}(a, \tilde{g}_{0})$
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\State $1,\:0:\;a := \text{Mont}(a, \tilde{g}_{1})$
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\State $1,\:1:\;a := \text{Mont}(a, \tilde{g}_{01})$
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\EndWhile
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\EndFor
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\State $a := \text{Mont}(a, 1)$
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\State \Return{$a$}
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\end{algorithmic}
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\end{algorithm}
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It can be seen that the algorithm requires $R^{2}\bmod m$ which is $2^{2n}\bmod m$. We assume $R^2 \bmod m$ can be
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provided or pre-computed. The for loop in the algorithm is executed by the control logic of the core. Apart from this,
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a few pre- and one post-calculations have to be performed.
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The computation time for an exponentiation depends on the number of zero's in the exponents, from
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Algorithm~\ref{alg:mme} one can see that if both exponent bits are zero at a time, no multiplication has to be
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performed. Thus reducing the total time. The average computation time for a modular simultaneous exponentiation, with
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$n$-bit base operands and $t$-bit exponents is given by~(\ref{eq:Tsime}).
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\begin{align}\label{eq:Tsime}
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T_{se} = \frac{7}{4} t \cdot T_{m} = \frac{7}{4}t \cdot [k + 2(n - 1)] \tau_c
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\end{align}
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For single base exponentiations, i.e. 1 exponent is equal to zero, the average exponentiation time is given by~(\ref{eq:Texp}).
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\begin{align}\label{eq:Texp}
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T_{e} = \frac{3}{2} t \cdot T_{m} = \frac{3}{2}t \cdot [k + 2(n - 1)] \tau_c
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\end{align}
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The formulas~(\ref{eq:Tsime}) and~(\ref{eq:Texp}) given here are only the theoretical average time for an exponentiation,
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excluding the pre- and post-computations.
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\section{Core operation steps}
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The core can operate in 2 modes, multiplication or exponentiation mode. The steps required to do one of these actions
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are described here.
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\subsection{Single Montgomery multiplication}
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The following steps are needed for a single Montgomery multiplication:
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\begin{enumerate}
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        \item load the modulus to the RAM using the 32 bit bus
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        \item load the desired $x$ and $y$ operands into any 2 locations of the operand RAM using the 32 bit bus.
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        \item select the correct input operands for the multiplier using \verb|x_sel_single| and \verb|y_sel_single|
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        \item select the result destination operand using \verb|result_dest_op|
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        \item set \verb|exp/m| = 0' to select multiplication mode
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        \item set \verb|p_sel| to choose which pipeline part you will use
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        \item generate a start pulse for the core
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        \item wait until interrupt is received and read out result in selected operand
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\end{enumerate}
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\textbf{Note:} this computation gives a result $$r = x \cdot y \cdot R^{-1} \bmod m$$. If the actual product of $x$ and $y$ is
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desired, a final Montgomery multiplication of the result with $R^{2}$ is needed.
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\subsection{Modular simultaneous exponentiation}
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The core requires $\tilde{g}_{0}$, $\tilde{g}_{0}$, $\tilde{g}_{01}$ and $a$ to be loaded into the correct operand
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spaces before starting the exponentiation. These parameters are calculated using single Montgomery multiplications as follows:
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\begin{align*}
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        \tilde{g}_{0} &= Mont(g_{0}, R^{2}) &\,&= g_{0} \cdot R \bmod m & \hspace{3cm}\text{in operand 0}\hspace{4cm}\\
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        \tilde{g}_{1} &= Mont(g_{1}, R^{2}) &\,&= g_{1} \cdot R \bmod m & \hspace{3cm}\text{in operand 1}\hspace{4cm}\\
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        \tilde{g}_{01} &= Mont(\tilde{g}_{0}, \tilde{g}_{1}) &\,&= g_{0} \cdot g_{1} \cdot R \bmod m & \hspace{3cm}\text{in operand 2}\hspace{4cm}\\
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        a &= Mont(R^{2}, 1) &\,&= R \bmod m &\hspace{3cm}\text{in operand 3}\hspace{4cm}
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\end{align*}
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When the exponentiation is done, a final multiplication has to be started by the software to multiply $a$ with 1.
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The steps needed for a full simultaneous exponentiation are:
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\begin{enumerate}
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        \item load the modulus to the RAM using the 32 bit bus
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        \item load the desired $g_0$, $g_1$ operands and $$R^{2} \bmod m$$ into the operand RAM using the 32 bit bus.
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        \item set \verb|p_sel| to choose which pipeline part you will use
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        \item compute $\tilde{g}_{0}$ by using a single Montgomery multiplication of $g_{0}$ with $R^{2}$ and place the result $\tilde{g}_{0}$ in operand 0.
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        \item compute $\tilde{g}_{1}$ by using a single Montgomery multiplication of $g_{1}$ with $R^{2}$ and place the result $\tilde{g}_{1}$ in operand 1.
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        \item compute $\tilde{g}_{01}$ by using a single Montgomery multiplication of $\tilde{g}_{0}$ with $\tilde{g}_{1}$ and place the result $\tilde{g}_{01}$ in operand 2.
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        \item compute $a$ by using a single Montgomery multiplication of $R^{2}$ with $1$ and place the result $a$ in operand 3.
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        \item set the core in exponentiation mode ($exp/m$='1')
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        \item generate a start pulse for the core
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        \item wait until interrupt is received
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        \item perform the post-computation using a single Montgomery multiplication of $a$(in operand 3) with 1 and read out result
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\end{enumerate}`