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	On exponentiation - libzahl - big integer library
	git clone git://git.suckless.org/libzahl
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	---
	commit fba103dca7a472eb58159d45ff8700736f89a136
	parent a70f79dfb22e8ea00231f5739f89ecc8d552643f
	Author: Mattias Andrée <[email protected]>
	Date: Thu, 12 May 2016 00:30:53 +0200

	On exponentiation

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/arithmetic.tex \| 66 +++++++++++++++++++++++++++++…
	M doc/libzahl.tex \| 2 +-

	2 files changed, 66 insertions(+), 2 deletions(-)
	---
	diff --git a/doc/arithmetic.tex b/doc/arithmetic.tex
	@@ -130,7 +130,71 @@ TODO % zdiv zmod zdivmod
	\section{Exponentiation}
	\label{sec:Exponentiation}

	-TODO % zpow zpowu zmodpow zmodpowu
	+Exponentiation refers to raising a number to
	+a power. libzahl provides two functions for
	+regular exponentiation, and two functions for
	+modular exponentiation. libzahl also provides
	+a function for raising a number to the second
	+power, see \secref{sec:Multiplication} for
	+more details on this. The functions for regular
	+exponentiation are
	+
	+\begin{alltt}
	+ void zpow(z_t power, z_t base, z_t exponent);
	+ void zpowu(z_t, z_t, unsigned long long int);
	+\end{alltt}
	+
	+\noindent
	+They are identical, except {\tt zpowu} expects
	+and intrinsic type as the exponent. Both functions
	+calculate
	+
	+\vspace{1em}
	+$power \gets base^{exponent}$
	+\vspace{1em}
	+
	+\noindent
	+The functions for modular exponentiation are
	+\begin{alltt}
	+ void zmodpow(z_t, z_t, z_t, z_t modulator);
	+ void zmodpowu(z_t, z_t, unsigned long long int, z_t);
	+\end{alltt}
	+
	+\noindent
	+They are identical, except {\tt zmodpowu} expects
	+and intrinsic type as the exponent. Both functions
	+calculate
	+
	+\vspace{1em}
	+$power \gets base^{exponent} \mod modulator$
	+\vspace{1em}
	+
	+The sign of {\tt modulator} does not affect the
	+result, {\tt power} will be negative if and only
	+if {\tt base} is negative and {\tt exponent} is
	+odd, that is, under the same circumstances as for
	+{\tt zpow} and {\tt zpowu}.
	+
	+These four functions are implemented using
	+exponentiation by squaring. {\tt zmodpow} and
	+{\tt zmodpowu} are optimised, they modulate
	+results for multiplication and squaring at
	+every multiplication and squaring, rather than
	+modulating every at the end. Exponentiation
	+by modulation is a very simple algorithm which
	+can be expressed as a simple formula
	+
	+\vspace{-1em}
	+\[ \hspace*{-0.4cm}
	+ a^b = \prod_{i = 0}^{\lceil \log_2 b \rceil}
	+ a^{2^i} \left \lfloor {b \over 2^i} \hspace*{-1ex} \mod 2 \right \rfloor
	+\]
	+
	+{\tt zmodpow} does \emph{not} calculate the
	+modular inverse if the exponent is negative,
	+rather, you should expect the result to be
	+1 and 0 depending of whether the base is 1
	+or not 1.


	\newpage
	diff --git a/doc/libzahl.tex b/doc/libzahl.tex
	@@ -1,4 +1,4 @@
	-\documentclass[11pt,b5paper,openright]{book}
	+\documentclass[11pt,b5paper,openright,fleqn]{book}
	\special{papersize=176mm,250mm}
	\usepackage[utf8]{inputenc}
	\usepackage[T1]{fontenc}