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libtommath/libtommath-tex.patch

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diff -Naur libtommath-1.0.old/bn.tex libtommath-1.0/bn.tex
--- libtommath-1.0.old/bn.tex 2016-02-23 11:24:13.632735129 +0100
+++ libtommath-1.0/bn.tex 2016-02-23 11:43:13.580826898 +0100
@@ -258,7 +258,7 @@
So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think
are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|l|c|c|l|}
@@ -300,7 +300,7 @@
There are three possible return codes a function may return.
\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM}
-\begin{figure}[here!]
+\begin{figure}[h!]
\begin{center}
\begin{small}
\begin{tabular}{|l|l|}
@@ -823,7 +823,7 @@
for any comparison.
\index{MP\_GT} \index{MP\_EQ} \index{MP\_LT}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|c|c|}
\hline \textbf{Result Code} & \textbf{Meaning} \\
@@ -1290,7 +1290,7 @@
\end{alltt}
Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|l|l|}
@@ -1739,7 +1739,7 @@
(see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in
mp\_prime\_random().
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|r|l|}
diff -Naur libtommath-1.0.old/tommath.src libtommath-1.0/tommath.src
--- libtommath-1.0.old/tommath.src 2016-02-03 19:07:27.000000000 +0100
+++ libtommath-1.0/tommath.src 2016-02-23 11:46:48.428479620 +0100
@@ -175,7 +175,7 @@
typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
@@ -366,7 +366,7 @@
exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the
scoring system used.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
@@ -573,7 +573,7 @@
used within LibTomMath.
\index{mp\_int}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
%\begin{verbatim}
@@ -670,7 +670,7 @@
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).
\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
@@ -719,7 +719,7 @@
structure are set to valid values. The mp\_init algorithm will perform such an action.
\index{mp\_init}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init}. \\
@@ -793,7 +793,7 @@
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be
returned to the application's memory pool with the mp\_clear algorithm.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
@@ -857,7 +857,7 @@
is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it
must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
@@ -911,7 +911,7 @@
of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it
will allocate \textit{at least} a specified number of digits.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -963,7 +963,7 @@
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement. It is essentially a shortcut to multiple initializations.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\
@@ -1022,7 +1022,7 @@
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to
\textbf{MP\_ZPOS}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
@@ -1090,7 +1090,7 @@
a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
@@ -1205,7 +1205,7 @@
useful within functions that need to modify an argument but do not wish to actually modify the original copy. The
mp\_init\_copy algorithm has been designed to help perform this task.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
@@ -1235,7 +1235,7 @@
Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to
perform this task.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_zero}. \\
@@ -1265,7 +1265,7 @@
With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
the absolute value of an mp\_int.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_abs}. \\
@@ -1297,7 +1297,7 @@
With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
the negative of an mp\_int input.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_neg}. \\
@@ -1334,7 +1334,7 @@
\subsection{Setting Small Constants}
Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set}. \\
@@ -1375,7 +1375,7 @@
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set\_int}. \\
@@ -1425,7 +1425,7 @@
To facilitate working with the results of the comparison functions three constants are required.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
@@ -1438,7 +1438,7 @@
\caption{Comparison Return Codes}
\end{figure}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp\_mag}. \\
@@ -1479,7 +1479,7 @@
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
comparison a trivial signed comparison algorithm can be written.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
@@ -1560,7 +1560,7 @@
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
\newpage
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{center}
\begin{small}
\begin{tabular}{l}
@@ -1663,7 +1663,7 @@
For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{center}
\begin{small}
\begin{tabular}{l}
@@ -1754,7 +1754,7 @@
Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
@@ -1783,7 +1783,7 @@
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly
straightforward but restricted since subtraction can only produce positive results.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
@@ -1833,7 +1833,7 @@
\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
@@ -1865,7 +1865,7 @@
\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
@@ -1911,7 +1911,7 @@
In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -1967,7 +1967,7 @@
\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2024,7 +2024,7 @@
degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2080,7 +2080,7 @@
Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2136,7 +2136,7 @@
\subsection{Multiplication by Power of Two}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2199,7 +2199,7 @@
\subsection{Division by Power of Two}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2252,7 +2252,7 @@
The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2363,7 +2363,7 @@
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}).
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2422,7 +2422,7 @@
For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized in the following table.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|l|}
\hline && & 5 & 7 & 6 & \\
@@ -2500,7 +2500,7 @@
Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication
of $576$ and $241$.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
@@ -2521,7 +2521,7 @@
Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is
congruent to adding a leading zero digit.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2577,7 +2577,7 @@
the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2783,7 +2783,7 @@
of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2888,7 +2888,7 @@
the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2925,7 +2925,7 @@
\caption{Algorithm mp\_toom\_mul}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2997,7 +2997,7 @@
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3051,7 +3051,7 @@
For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
required for multiplication. The following diagram gives an example of the operations required.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
@@ -3080,7 +3080,7 @@
The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines
will not handle.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3159,7 +3159,7 @@
mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3249,7 +3249,7 @@
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3342,7 +3342,7 @@
derive their own Toom-Cook squaring algorithm.
\subsection{High Level Squaring}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3560,7 +3560,7 @@
is considerably faster than the straightforward $3m^2$ method.
\subsection{The Barrett Algorithm}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3634,7 +3634,7 @@
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3680,7 +3680,7 @@
From these two simple facts the following simple algorithm can be derived.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3706,7 +3706,7 @@
final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|}
@@ -3736,7 +3736,7 @@
and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
Fortunately there exists an alternative representation of the algorithm.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3758,7 +3758,7 @@
This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single
precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|r|}
@@ -3791,7 +3791,7 @@
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3849,7 +3849,7 @@
The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for
Montgomery reductions.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3933,7 +3933,7 @@
With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4014,7 +4014,7 @@
\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4070,7 +4070,7 @@
The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4183,7 +4183,7 @@
of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
exponentiations are performed.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4249,7 +4249,7 @@
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for
completeness.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4271,7 +4271,7 @@
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4304,7 +4304,7 @@
In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new
algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4346,7 +4346,7 @@
\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4385,7 +4385,7 @@
that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most
significant bit. The resulting sum will be a power of two.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4487,7 +4487,7 @@
While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to
be computed in an auxilary variable. Consider the following equivalent algorithm.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4539,7 +4539,7 @@
to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of
$b$ that are greater than three.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4590,7 +4590,7 @@
computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a
portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4626,7 +4626,7 @@
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
@@ -4655,7 +4655,7 @@
Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm {\ref{fig:KARY}}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
@@ -4677,7 +4677,7 @@
\label{fig:OPTK2}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4728,7 +4728,7 @@
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm
terminates with an error.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4780,7 +4780,7 @@
\subsection{Barrett Modular Exponentiation}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4830,7 +4830,7 @@
\caption{Algorithm s\_mp\_exptmod}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4944,7 +4944,7 @@
Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is
equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4986,7 +4986,7 @@
will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5082,7 +5082,7 @@
At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.
\subsection{Radix-$\beta$ Division with Remainder}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5125,7 +5125,7 @@
\caption{Algorithm mp\_div}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5248,7 +5248,7 @@
Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction
algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5281,7 +5281,7 @@
multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands
only has one digit.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5323,7 +5323,7 @@
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5388,7 +5388,7 @@
such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5441,7 +5441,7 @@
factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented
is solely for simulations and not intended for cryptographic use.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5483,7 +5483,7 @@
such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
mediums.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{cc|cc|cc|cc}
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\
@@ -5511,7 +5511,7 @@
\label{fig:ASC}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5551,7 +5551,7 @@
\subsection{Generating Radix-$n$ Output}
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5620,7 +5620,7 @@
The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5646,7 +5646,7 @@
greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5680,7 +5680,7 @@
However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5725,7 +5725,7 @@
The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly
and will produce the greatest common divisor.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5812,7 +5812,7 @@
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5935,7 +5935,7 @@
factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6040,7 +6040,7 @@
equation.
\subsection{General Case}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6150,7 +6150,7 @@
approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6196,7 +6196,7 @@
integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows
in size.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6228,7 +6228,7 @@
value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
diff -Naur libtommath-1.0.old/tommath.tex libtommath-1.0/tommath.tex
--- libtommath-1.0.old/tommath.tex 2016-02-23 11:24:13.648735326 +0100
+++ libtommath-1.0/tommath.tex 2016-02-23 11:43:13.591827034 +0100
@@ -175,7 +175,7 @@
typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
@@ -366,7 +366,7 @@
exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the
scoring system used.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
@@ -534,7 +534,7 @@
for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease.
\begin{center}
-\begin{figure}[here]
+\begin{figure}[h]
\includegraphics{pics/design_process.ps}
\caption{Design Flow of the First Few Original LibTomMath Functions.}
\label{pic:design_process}
@@ -579,7 +579,7 @@
used within LibTomMath.
\index{mp\_int}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
%\begin{verbatim}
@@ -676,7 +676,7 @@
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).
\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
@@ -725,7 +725,7 @@
structure are set to valid values. The mp\_init algorithm will perform such an action.
\index{mp\_init}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init}. \\
@@ -831,7 +831,7 @@
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be
returned to the application's memory pool with the mp\_clear algorithm.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
@@ -925,7 +925,7 @@
is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it
must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
@@ -1022,7 +1022,7 @@
of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it
will allocate \textit{at least} a specified number of digits.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -1108,7 +1108,7 @@
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement. It is essentially a shortcut to multiple initializations.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\
@@ -1212,7 +1212,7 @@
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to
\textbf{MP\_ZPOS}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
@@ -1310,7 +1310,7 @@
a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
@@ -1479,7 +1479,7 @@
useful within functions that need to modify an argument but do not wish to actually modify the original copy. The
mp\_init\_copy algorithm has been designed to help perform this task.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
@@ -1527,7 +1527,7 @@
Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to
perform this task.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_zero}. \\
@@ -1579,7 +1579,7 @@
With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
the absolute value of an mp\_int.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_abs}. \\
@@ -1640,7 +1640,7 @@
With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
the negative of an mp\_int input.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_neg}. \\
@@ -1703,7 +1703,7 @@
\subsection{Setting Small Constants}
Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set}. \\
@@ -1759,7 +1759,7 @@
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set\_int}. \\
@@ -1843,7 +1843,7 @@
To facilitate working with the results of the comparison functions three constants are required.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
@@ -1856,7 +1856,7 @@
\caption{Comparison Return Codes}
\end{figure}
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp\_mag}. \\
@@ -1938,7 +1938,7 @@
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
comparison a trivial signed comparison algorithm can be written.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
@@ -2047,7 +2047,7 @@
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
\newpage
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{center}
\begin{small}
\begin{tabular}{l}
@@ -2244,7 +2244,7 @@
For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{center}
\begin{small}
\begin{tabular}{l}
@@ -2411,7 +2411,7 @@
Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
@@ -2440,7 +2440,7 @@
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly
straightforward but restricted since subtraction can only produce positive results.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
@@ -2529,7 +2529,7 @@
\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
@@ -2561,7 +2561,7 @@
\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
@@ -2651,7 +2651,7 @@
In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2775,7 +2775,7 @@
\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2886,7 +2886,7 @@
degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -2929,7 +2929,7 @@
\newpage
\begin{center}
-\begin{figure}[here]
+\begin{figure}[h]
\includegraphics{pics/sliding_window.ps}
\caption{Sliding Window Movement}
\label{pic:sliding_window}
@@ -3001,7 +3001,7 @@
Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3115,7 +3115,7 @@
\subsection{Multiplication by Power of Two}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3249,7 +3249,7 @@
\subsection{Division by Power of Two}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3387,7 +3387,7 @@
The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3541,7 +3541,7 @@
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}).
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3600,7 +3600,7 @@
For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized in the following table.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|l|}
\hline && & 5 & 7 & 6 & \\
@@ -3753,7 +3753,7 @@
Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication
of $576$ and $241$.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
@@ -3774,7 +3774,7 @@
Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is
congruent to adding a leading zero digit.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -3830,7 +3830,7 @@
the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4129,7 +4129,7 @@
of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4387,7 +4387,7 @@
the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4424,7 +4424,7 @@
\caption{Algorithm mp\_toom\_mul}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4768,7 +4768,7 @@
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -4875,7 +4875,7 @@
For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
required for multiplication. The following diagram gives an example of the operations required.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
@@ -4903,7 +4903,7 @@
The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines
will not handle.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5052,7 +5052,7 @@
mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5242,7 +5242,7 @@
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5442,7 +5442,7 @@
derive their own Toom-Cook squaring algorithm.
\subsection{High Level Squaring}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5705,7 +5705,7 @@
is considerably faster than the straightforward $3m^2$ method.
\subsection{The Barrett Algorithm}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5865,7 +5865,7 @@
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5931,7 +5931,7 @@
From these two simple facts the following simple algorithm can be derived.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -5957,7 +5957,7 @@
final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|}
@@ -5987,7 +5987,7 @@
and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
Fortunately there exists an alternative representation of the algorithm.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6009,7 +6009,7 @@
This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single
precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{small}
\begin{center}
\begin{tabular}{|c|l|r|}
@@ -6042,7 +6042,7 @@
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6100,7 +6100,7 @@
The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for
Montgomery reductions.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6287,7 +6287,7 @@
With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6526,7 +6526,7 @@
\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6627,7 +6627,7 @@
The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6740,7 +6740,7 @@
of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
exponentiations are performed.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6888,7 +6888,7 @@
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for
completeness.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6928,7 +6928,7 @@
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -6990,7 +6990,7 @@
In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new
algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7081,7 +7081,7 @@
\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7153,7 +7153,7 @@
that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most
significant bit. The resulting sum will be a power of two.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7293,7 +7293,7 @@
While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to
be computed in an auxilary variable. Consider the following equivalent algorithm.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7345,7 +7345,7 @@
to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of
$b$ that are greater than three.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7465,7 +7465,7 @@
computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a
portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7501,7 +7501,7 @@
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
@@ -7530,7 +7530,7 @@
Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm {\ref{fig:KARY}}.
-\begin{figure}[here]
+\begin{figure}[h]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
@@ -7552,7 +7552,7 @@
\label{fig:OPTK2}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7603,7 +7603,7 @@
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm
terminates with an error.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7754,7 +7754,7 @@
\subsection{Barrett Modular Exponentiation}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7804,7 +7804,7 @@
\caption{Algorithm s\_mp\_exptmod}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -7896,7 +7896,7 @@
the two cases of $mode = 1$ and $mode = 2$ respectively.
\begin{center}
-\begin{figure}[here]
+\begin{figure}[h]
\includegraphics{pics/expt_state.ps}
\caption{Sliding Window State Diagram}
\label{pic:expt_state}
@@ -8163,7 +8163,7 @@
Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is
equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -8239,7 +8239,7 @@
will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -8335,7 +8335,7 @@
At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.
\subsection{Radix-$\beta$ Division with Remainder}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -8378,7 +8378,7 @@
\caption{Algorithm mp\_div}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -8782,7 +8782,7 @@
Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction
algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -8913,7 +8913,7 @@
multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands
only has one digit.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9020,7 +9020,7 @@
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9186,7 +9186,7 @@
such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9255,7 +9255,7 @@
factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented
is solely for simulations and not intended for cryptographic use.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9339,7 +9339,7 @@
such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
mediums.
-\newpage\begin{figure}[here]
+\newpage\begin{figure}[h]
\begin{center}
\begin{tabular}{cc|cc|cc|cc}
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\
@@ -9367,7 +9367,7 @@
\label{fig:ASC}
\end{figure}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9478,7 +9478,7 @@
\subsection{Generating Radix-$n$ Output}
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9608,7 +9608,7 @@
The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9634,7 +9634,7 @@
greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9668,7 +9668,7 @@
However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9713,7 +9713,7 @@
The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly
and will produce the greatest common divisor.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -9891,7 +9891,7 @@
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -10060,7 +10060,7 @@
factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -10268,7 +10268,7 @@
equation.
\subsection{General Case}
-\newpage\begin{figure}[!here]
+\newpage\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -10407,7 +10407,7 @@
approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -10536,7 +10536,7 @@
integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows
in size.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
@@ -10616,7 +10616,7 @@
value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.
-\begin{figure}[!here]
+\begin{figure}[!h]
\begin{small}
\begin{center}
\begin{tabular}{l}
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