tommath/bn.tex

\documentclass{article}
\begin{document}

\title{LibTomMath v0.14 \\ A Free Multiple Precision Integer Library \\ http://math.libtomcrypt.org }
\author{Tom St Denis \\ tomstdenis@iahu.ca}
\maketitle
\newpage

\section{Introduction}
``LibTomMath'' is a free and open source library that provides multiple-precision integer functions required to form a 
basis of a public key cryptosystem.  LibTomMath is written entire in portable ISO C source code and designed to have an 
application interface much like that of MPI from Michael Fromberger.  

LibTomMath was written from scratch by Tom St Denis but designed to be  drop in replacement for the MPI package.  The 
algorithms within the library are derived from descriptions as provided in the Handbook of Applied Cryptography and Knuth's
``The Art of Computer Programming''.  The library has been extensively optimized and should provide quite comparable 
timings as compared to many free and commercial libraries.

LibTomMath was designed with the following goals in mind:
\begin{enumerate}
\item Be a drop in replacement for MPI.
\item Be much faster than MPI.
\item Be written entirely in portable C.
\end{enumerate}

All three goals have been achieved to one extent or another (actual figures depend on what platform you are using).

Being compatible with MPI means that applications that already use it can be ported fairly quickly.  Currently there are 
a few differences but there are many similarities.  In fact the average MPI based application can be ported in under 15
minutes.  

Thanks goes to Michael Fromberger for answering a couple questions and Colin Percival for having the patience and courtesy to
help debug and suggest optimizations.  They were both of great help!

\section{Building Against LibTomMath}

As of v0.12 LibTomMath is not a simple single source file project like MPI.  LibTomMath retains the exact same API as MPI
but is implemented differently.  To build LibTomMath you will need a copy of GNU cc and GNU make.  Both are free so if you 
don't have a copy don't whine to me about it.

To build the library type 

\begin{verbatim}
make
\end{verbatim}

This will build the library file libtommath.a.  If you want to build the library and also install it (in /usr/bin and /usr/include) then
type 

\begin{verbatim}
make install
\end{verbatim}

Now within your application include ``tommath.h'' and link against libtommath.a to get MPI-like functionality.

\subsection{Microsoft Visual C++}
A makefile is also provided for MSVC (\textit{tested against MSVC 6.00 with SP5}) which allows the library to be used
with that compiler as well.  To build the library type

\begin{verbatim}
nmake -f makefile.msvc
\end{verbatim}

Which will build ``tommath.lib''.  

\section{Programming with LibTomMath}

\subsection{The mp\_int Structure}
All multiple precision integers are stored in a structure called \textbf{mp\_int}.  A multiple precision integer is
essentially an array of \textbf{mp\_digit}.  mp\_digit is defined at the top of ``tommath.h''.  The type can be changed 
to suit a particular platform.  

For example, when \textbf{MP\_8BIT} is defined a mp\_digit is a unsigned char and holds seven bits.  Similarly 
when \textbf{MP\_16BIT} is defined a mp\_digit is a unsigned short and holds 15 bits.   By default a mp\_digit is a 
unsigned long and holds 28 bits which is optimal for most 32 and 64 bit processors.

The choice of digit is particular to the platform at hand and what available multipliers are provided.  For 
MP\_8BIT either a $8 \times 8 \Rightarrow 16$ or $16 \times 16 \Rightarrow 16$ multiplier is optimal.  When 
MP\_16BIT is defined either a $16 \times 16 \Rightarrow 32$ or $32 \times 32 \Rightarrow 32$ multiplier is optimal.  By
default a $32 \times 32 \Rightarrow 64$ or $64 \times 64 \Rightarrow 64$ multiplier is optimal.  

This gives the library some flexibility.  For example, a i8051 has a $8 \times 8 \Rightarrow 16$ multiplier.  The 
16-bit x86 instruction set has a $16 \times 16 \Rightarrow 32$ multiplier.  In practice this library is not particularly
designed for small devices like an i8051 due to the size.  It is possible to strip out functions which are not required 
to drop the code size.  More realistically the library is well suited to 32 and 64-bit processors that have decent
integer multipliers.  The AMD Athlon XP and Intel Pentium 4 processors are examples of well suited processors.

Throughout the discussions there will be references to a \textbf{used} and \textbf{alloc} members of an integer.  The
used member refers to how many digits are actually used in the representation of the integer.  The alloc member refers
to how many digits have been allocated off the heap.  There is also the $\beta$ quantity which is equal to $2^W$ where 
$W$ is the number of bits in a digit (default is 28).  

\subsection{Calling Functions}
Most functions expect pointers to mp\_int's as parameters.   To save on memory usage it is possible to have source
variables as destinations.  The arguements are read left to right so to compute $x + y = z$ you would pass the arguments
in the order $x, y, z$.  For example:
\begin{verbatim}
   mp_add(&x, &y, &x);           /* x = x + y */
   mp_mul(&y, &x, &z);           /* z = y * x */
   mp_div_2(&x, &y);             /* y = x / 2 */
\end{verbatim}

\subsection{Return Values}
All functions that return errors will return \textbf{MP\_OKAY} if the function was succesful.  It will return 
\textbf{MP\_MEM} if it ran out of heap memory or \textbf{MP\_VAL} if one of the arguements is out of range.  

\subsection{Basic Functionality}
Before an mp\_int can be used it must be initialized with 

\begin{verbatim}
int mp_init(mp_int *a);
\end{verbatim}

For example, consider the following.

\begin{verbatim}
#include "tommath.h"
int main(void)
{
   mp_int num;
   if (mp_init(&num) != MP_OKAY) {
      printf("Error initializing a mp_int.\n");
   }
   return 0;
}   
\end{verbatim}

A mp\_int can be freed from memory with

\begin{verbatim}
void mp_clear(mp_int *a);
\end{verbatim}

This will zero the memory and free the allocated data.  There are a set of trivial functions to manipulate the 
value of an mp\_int.  

\begin{verbatim}
/* set to zero */
void mp_zero(mp_int *a);

/* set to a digit */
void mp_set(mp_int *a, mp_digit b);

/* set a 32-bit const */
int mp_set_int(mp_int *a, unsigned long b);

/* init to a given number of digits */
int mp_init_size(mp_int *a, int size);

/* copy, b = a */
int mp_copy(mp_int *a, mp_int *b);

/* inits and copies, a = b */
int mp_init_copy(mp_int *a, mp_int *b);
\end{verbatim}

The \textbf{mp\_zero} function will clear the contents of a mp\_int and set it to positive.  The \textbf{mp\_set} function 
will zero the integer and set the first digit to a value specified.  The \textbf{mp\_set\_int} function will zero the 
integer and set the first 32-bits to a given value.  It is important to note that using mp\_set can have unintended 
side effects when either the  MP\_8BIT or MP\_16BIT defines are enabled.  By default the library will accept the 
ranges of values MPI will (and more).

The \textbf{mp\_init\_size} function will initialize the integer and set the allocated size to a given value.  The 
allocated digits are zero'ed by default but not marked as used.  The \textbf{mp\_copy} function will copy the digits
(and sign) of the first parameter into the integer specified by the second parameter.  The \textbf{mp\_init\_copy} will
initialize the first integer specified and copy the second one into it.  Note that the order is reversed from that of
mp\_copy.  This odd ``bug'' was kept to maintain compatibility with MPI.

\subsection{Digit Manipulations}

There are a class of functions that provide simple digit manipulations such as shifting and modulo reduction of powers
of two.  

\begin{verbatim}
/* right shift by "b" digits */
void mp_rshd(mp_int *a, int b);

/* left shift by "b" digits */
int mp_lshd(mp_int *a, int b);

/* c = a / 2^b */
int mp_div_2d(mp_int *a, int b, mp_int *c);

/* b = a/2 */
int mp_div_2(mp_int *a, mp_int *b);

/* c = a * 2^b */
int mp_mul_2d(mp_int *a, int b, mp_int *c);

/* b = a*2 */
int mp_mul_2(mp_int *a, mp_int *b);

/* c = a mod 2^d */
int mp_mod_2d(mp_int *a, int b, mp_int *c);

/* computes a = 2^b */
int mp_2expt(mp_int *a, int b);

/* makes a pseudo-random int of a given size */
int mp_rand(mp_int *a, int digits);

\end{verbatim}

\subsection{Binary Operations}

\begin{verbatim}

/* c = a XOR b  */
int mp_xor(mp_int *a, mp_int *b, mp_int *c);

/* c = a OR b */
int mp_or(mp_int *a, mp_int *b, mp_int *c);

/* c = a AND b */
int mp_and(mp_int *a, mp_int *b, mp_int *c);

\end{verbatim}

\subsection{Basic Arithmetic}

Next are the class of functions which provide basic arithmetic.

\begin{verbatim}
/* b = -a */
int mp_neg(mp_int *a, mp_int *b);

/* b = |a| */
int mp_abs(mp_int *a, mp_int *b);

/* compare a to b */
int mp_cmp(mp_int *a, mp_int *b);

/* compare |a| to |b| */
int mp_cmp_mag(mp_int *a, mp_int *b);

/* c = a + b */
int mp_add(mp_int *a, mp_int *b, mp_int *c);

/* c = a - b */
int mp_sub(mp_int *a, mp_int *b, mp_int *c);

/* c = a * b */
int mp_mul(mp_int *a, mp_int *b, mp_int *c);

/* b = a^2 */
int mp_sqr(mp_int *a, mp_int *b);

/* a/b => cb + d == a */
int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d);

/* c = a mod b, 0 <= c < b  */
int mp_mod(mp_int *a, mp_int *b, mp_int *c);
\end{verbatim}

\subsection{Single Digit Functions}

\begin{verbatim}
/* compare against a single digit */
int mp_cmp_d(mp_int *a, mp_digit b);

/* c = a + b */
int mp_add_d(mp_int *a, mp_digit b, mp_int *c);

/* c = a - b */
int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);

/* c = a * b */
int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);

/* a/b => cb + d == a */
int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);

/* c = a mod b, 0 <= c < b  */
int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
\end{verbatim}

Note that care should be taken for the value of the digit passed.  By default, any 28-bit integer is a valid digit that can
be passed into the function.  However, if MP\_8BIT or MP\_16BIT is defined only 7 or 15-bit (respectively) integers 
can be passed into it.

\subsection{Modular Arithmetic}

There are some trivial modular arithmetic functions.

\begin{verbatim}
/* d = a + b (mod c) */
int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);

/* d = a - b (mod c) */
int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);

/* d = a * b (mod c) */
int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);

/* c = a * a (mod b) */
int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c);

/* c = 1/a (mod b) */
int mp_invmod(mp_int *a, mp_int *b, mp_int *c);

/* c = (a, b) */
int mp_gcd(mp_int *a, mp_int *b, mp_int *c);

/* c = [a, b] or (a*b)/(a, b) */
int mp_lcm(mp_int *a, mp_int *b, mp_int *c);

/* find the b'th root of a  */
int mp_n_root(mp_int *a, mp_digit b, mp_int *c);

/* computes the jacobi c = (a | n) (or Legendre if b is prime)  */
int mp_jacobi(mp_int *a, mp_int *n, int *c);

/* used to setup the Barrett reduction for a given modulus b */
int mp_reduce_setup(mp_int *a, mp_int *b);

/* Barrett Reduction, computes a (mod b) with a precomputed value c  
 *
 * Assumes that 0 < a <= b^2, note if 0 > a > -(b^2) then you can merely
 * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code].
 */
int mp_reduce(mp_int *a, mp_int *b, mp_int *c);

/* setups the montgomery reduction */
int mp_montgomery_setup(mp_int *a, mp_digit *mp);

/* computes xR^-1 == x (mod N) via Montgomery Reduction */
int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);

/* d = a^b (mod c) */
int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
\end{verbatim}

\subsection{Radix Conversions}
To read or store integers in other formats there are the following functions.

\begin{verbatim}
int mp_unsigned_bin_size(mp_int *a);
int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
int mp_to_unsigned_bin(mp_int *a, unsigned char *b);

int mp_signed_bin_size(mp_int *a);
int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
int mp_to_signed_bin(mp_int *a, unsigned char *b);

int mp_read_radix(mp_int *a, unsigned char *str, int radix);
int mp_toradix(mp_int *a, unsigned char *str, int radix);
int mp_radix_size(mp_int *a, int radix);
\end{verbatim}

The integers are stored in big endian format as most libraries (and MPI) expect.  The \textbf{mp\_read\_radix} and 
\textbf{mp\_toradix} functions read and write (respectively) null terminated ASCII strings in a given radix.  Valid values
for the radix are between 2 and 64 (inclusively).  

\section{Function Analysis}

Throughout the function analysis the variable $N$ will denote the average size of an input to a function as measured 
by the number of digits it has.  The variable $W$ will denote the number of bits per word and $c$ will denote a small
constant amount of work.  The big-oh notation will be abused slightly to consider numbers that do not grow to infinity.
That is we shall consider $O(N/2) \ne O(N)$ which is an abuse of the notation.

\subsection{Digit Manipulation Functions}
The class of digit manipulation functions such as \textbf{mp\_rshd}, \textbf{mp\_lshd} and \textbf{mp\_mul\_2} are all
very simple functions to analyze.  

\subsubsection{mp\_rshd(mp\_int *a, int b)}
Shifts $a$ by given number of digits to the right and is equivalent to dividing by $\beta^b$.  The work is performed
in-place which means the input and output are the same.  If the shift count $b$ is less than or equal to zero 
the function returns without doing any work.  If the the shift count is larger than the number of digits in $a$ 
then $a$ is simply zeroed without shifting digits.

This function requires no additional memory and $O(N)$ time.

\subsubsection{mp\_lshd(mp\_int *a, int b)}
Shifts $a$ by a given number of digits to the left and is equivalent to multiplying by $\beta^b$.  The work
is performed in-place which means the input and output are the same.  If the shift count $b$ is less than or equal 
to zero the function returns success without doing any work.

This function requires $O(b)$ additional digits of memory and $O(N)$ time.

\subsubsection{mp\_div\_2d(mp\_int *a, int b, mp\_int *c, mp\_int *d)}
Shifts $a$ by a given number of \textbf{bits} to the right and is equivalent to dividing by $2^b$.  The shifted number is stored
in the $c$ parameter.  The remainder of $a/2^b$ is optionally stored in $d$ (if it is not passed as NULL).  
If the shift count $b$ is less than or equal to zero the function places $a$ in $c$ and returns success.  

This function requires $O(2 \cdot N)$ additional digits of memory and $O(2 \cdot N)$ time.

\subsubsection{mp\_mul\_2d(mp\_int *a, int b, mp\_int *c)}
Shifts $a$ by a given number of bits to the left and is equivalent to multiplying by $2^b$.  The shifted number
is placed in the $c$ parameter.  If the shift count $b$ is less than or equal to zero the function places $a$
in $c$ and returns success.  

This function requires $O(N)$ additional digits of memory and $O(2 \cdot N)$ time.

\subsubsection{mp\_mul\_2(mp\_int *a, mp\_int *b)}
Multiplies $a$ by two and stores in $b$.  This function is hard coded todo a shift by one place so it is faster
than calling mp\_mul\_2d with a count of one.  

This function requires $O(N)$ additional digits of memory and $O(N)$ time.

\subsubsection{mp\_div\_2(mp\_int *a, mp\_int *b)}
Divides $a$ by two and stores in $b$.  This function is hard coded todo a shift by one place so it is faster
than calling mp\_div\_2d with a count of one.

This function requires $O(N)$ additional digits of memory and $O(N)$ time.

\subsubsection{mp\_mod\_2d(mp\_int *a, int b, mp\_int *c)}
Performs the action of reducing $a$ modulo $2^b$ and stores the result in $c$.  If the shift count $b$ is less than 
or equal to zero the function places $a$ in $c$ and returns success.  

This function requires $O(N)$ additional digits of memory and $O(2 \cdot N)$ time.

\subsubsection{mp\_2expt(mp\_int *a, int b)}
Computes $a = 2^b$ by first setting $a$ to zero then OR'ing the correct bit to get the right value.

\subsubsection{mp\_rand(mp\_int *a, int digits)}
Computes a pseudo-random (\textit{via rand()}) integer that is always ``$digits$'' digits in length.  Not for
cryptographic use.

\subsection{Binary Arithmetic}
\subsubsection{mp\_xor(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c = a \oplus b$, pseudo-extends with zeroes whichever of $a$ or $b$ is shorter such that the length
of $c$ is the maximum length of the two inputs.

\subsubsection{mp\_or(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c = a \lor b$, pseudo-extends with zeroes whichever of $a$ or $b$ is shorter such that the length
of $c$ is the maximum length of the two inputs.

\subsubsection{mp\_and(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c = a \land b$, pseudo-extends with zeroes whichever of $a$ or $b$ is shorter such that the length
of $c$ is the maximum length of the two inputs.

\subsection{Basic Arithmetic}

\subsubsection{mp\_cmp(mp\_int *a, mp\_int *b)}
Performs a \textbf{signed} comparison between $a$ and $b$ returning \textbf{MP\_GT} if $a$ is larger than $b$.

This function requires no additional memory and $O(N)$ time.

\subsubsection{mp\_cmp\_mag(mp\_int *a, mp\_int *b)}
Performs a \textbf{unsigned} comparison between $a$ and $b$ returning \textbf{MP\_GT} is $a$ is larger than $b$.  Note 
that this comparison is unsigned which means it will report, for example, $-5 > 3$.  By comparison mp\_cmp will 
report $-5 < 3$.

This function requires no additional memory and $O(N)$ time.

\subsubsection{mp\_add(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c = a + b$ using signed arithmetic.  Handles the sign of the numbers which means it will subtract as 
required, e.g. $a + -b$ turns into $a - b$.

This function requires no additional memory and $O(N)$ time.

\subsubsection{mp\_sub(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c = a - b$ using signed arithmetic.   Handles the sign of the numbers which means it will add as 
required, e.g. $a - -b$ turns into $a + b$.

This function requires no additional memory and $O(N)$ time.

\subsubsection{mp\_mul(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c = a \cdot b$ using signed arithmetic.  Handles the sign of the numbers correctly which means it will 
correct the sign of the product as required, e.g. $a \cdot -b$ turns into $-ab$.

This function requires $O(N^2)$ time for small inputs and $O(N^{1.584})$ time for relatively large 
inputs (\textit{above the }KARATSUBA\_MUL\_CUTOFF \textit{value defined in bncore.c.}).  There is 
considerable overhead in the Karatsuba method which only pays off when the digit count is fairly high
(\textit{typically around 80}).  For small inputs the function requires $O(2N)$ memory, otherwise it
requires $O(6 \cdot \mbox{lg}(N) \cdot N)$ memory.


\subsubsection{mp\_sqr(mp\_int *a, mp\_int *b)}
Computes $b = a^2$ and fixes the sign of $b$ to be positive.

This function has a running time and memory requirement profile very similar to that of the 
mp\_mul function.  It is always faster and uses less memory for the larger inputs.

\subsubsection{mp\_div(mp\_int *a, mp\_int *b, mp\_int *c, mp\_int *d)}
Computes $c = \lfloor a/b \rfloor$ and $d \equiv a \mbox{ (mod }b\mbox{)}$.  The division is signed which means the sign
of the output is not always positive.  The sign of the remainder equals the sign of $a$ while the sign of the 
quotient equals the product of the ratios $(a/\vert a \vert) \cdot (b/\vert b \vert)$.  Both $c$ and $d$ can be 
optionally passed as NULL if the value is not desired.  For example, if you want only the quotient of $x/y$ then 
mp\_div(\&x, \&y, \&z, NULL) is acceptable.

This function requires $O(4 \cdot N)$ memory and $O(3 \cdot N^2)$ time.

\subsubsection{mp\_mod(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes $c \equiv a \mbox{ (mod }b\mbox{)}$ but with the added condition that $0 \le c < b$.  That is a normal 
division is performed and if the remainder is negative $b$ is added to it.  Since adding $b$ modulo $b$ is equivalent
to adding zero ($0 \equiv b \mbox{ (mod }b\mbox{)}$) the result is accurate.  The results are undefined 
when $b \le 0$, in theory the routine will still give a properly congruent answer but it will not always be positive. 

This function requires $O(4 \cdot N)$ memory and $O(3 \cdot N^2)$ time.

\subsection{Number Theoretic Functions}

\subsubsection{mp\_addmod, mp\_submod, mp\_mulmod, mp\_sqrmod}
These functions take the time of their host function plus the time it takes to perform a division.  For example, 
mp\_addmod takes $O(N + 3 \cdot N^2)$ time.  Note that if you are performing many modular operations in a row with
the same modulus you should consider Barrett reductions.  

Also note that these functions use mp\_mod which means the result are guaranteed to be positive.

\subsubsection{mp\_invmod(mp\_int *a, mp\_int *b, mp\_int *c)}
This function will find $c = 1/a \mbox{ (mod }b\mbox{)}$ for any value of $a$ such that $(a, b) = 1$ and $b > 0$.  When
$b$ is odd a ``fast'' variant is used which finds the inverse twice as fast.  If no inverse is found (e.g. $(a, b) \ne 1$) then
the function returns \textbf{MP\_VAL} and the result in $c$ is undefined.

\subsubsection{mp\_gcd(mp\_int *a, mp\_int *b, mp\_int *c)}
Finds the greatest common divisor of both $a$ and $b$ and places the result in $c$.  Will work with either positive
or negative inputs.  

Functions requires no additional memory and approximately $O(N \cdot log(N))$ time.

\subsubsection{mp\_lcm(mp\_int *a, mp\_int *b, mp\_int *c)}
Finds the least common multiple of both $a$ and $b$ and places the result in $c$.  Will work with either positive
or negative inputs.  This is calculated by dividing the product of $a$ and $b$ by the greatest common divisor of 
both.  

Functions requires no additional memory and approximately $O(4 \cdot N^2)$ time.

\subsubsection{mp\_n\_root(mp\_int *a, mp\_digit b, mp\_int *c)}
Finds the $b$'th root of $a$ and stores it in $b$.  The roots are found such that $\vert c \vert^b \le \vert a \vert$.  
Uses the Newton approximation approach which means it converges in $O(log \beta^N)$ time to a final result.  Each iteration
requires $b$ multiplications and one division for a total work of $O(6N^2 \cdot log \beta^N) = O(6N^3 \cdot log \beta)$.

If the input $a$ is negative and $b$ is even the function returns \textbf{MP\_VAL}.  Otherwise the function will 
return a root that has a sign that agrees with the sign of $a$.

\subsubsection{mp\_jacobi(mp\_int *a, mp\_int *n, int *c)}
Computes $c = \left ( {a \over n} \right )$ or the Jacobi function of $(a, n)$ and stores the result in an integer addressed
by $c$.  Since the result of the Jacobi function $\left ( {a \over n} \right ) \in \lbrace -1, 0, 1 \rbrace$ it seemed
natural to store the result in a simple C style \textbf{int}.  If $n$ is prime then the Jacobi function produces
the same results as the Legendre function\footnote{Source: Handbook of Applied Cryptography, pp. 73}.  This means if
$n$ is prime then $\left ( {a \over n} \right )$ is equal to $1$ if $a$ is a quadratic residue modulo $n$ or $-1$ if 
it is not.

\subsubsection{mp\_exptmod(mp\_int *a, mp\_int *b, mp\_int *c, mp\_int *d)}
Computes $d = a^b \mbox{ (mod }c\mbox{)}$ using a sliding window $k$-ary exponentiation algorithm.  For an $\alpha$-bit
exponent it performs $\alpha$ squarings and at most $\lfloor \alpha/k \rfloor + 2^{k-1}$ multiplications.  The value of $k$ is
chosen to minimize the number of multiplications required for a given value of $\alpha$.  Barrett or Montgomery 
reductions are used to reduce the squared or multiplied temporary results modulo $c$.

\subsection{Fast Modular Reductions}

\subsubsection{mp\_reduce(mp\_int *a, mp\_int *b, mp\_int *c)}
Computes a Barrett reduction in-place of $a$ modulo $b$ with respect to $c$.  In essence it computes 
$a \equiv a \mbox{ (mod }b\mbox{)}$ provided $0 \le a \le b^2$.  The value of $c$ is precomputed with the 
function mp\_reduce\_setup().  The modulus $b$ must be larger than zero.

The Barrett reduction function has been optimized to use partial multipliers which means compared to MPI it performs
have the number of single precision multipliers (\textit{provided they have the same size digits}).  The partial
multipliers (\textit{one of which is shared with mp\_mul}) both have baseline and comba variants.  Barrett reduction 
can reduce a number modulo a $n-$digit modulus with approximately $2n^2$ single precision multiplications.  

\subsubsection{mp\_montgomery\_reduce(mp\_int *a, mp\_int *m, mp\_digit mp)}
Computes a Montgomery reduction in-place of $a$ modulo $b$ with respect to $mp$.  If $b$ is some $n-$digit modulus then
$R = \beta^{n+1}$.  The result of this function is $aR^{-1} \mbox{ (mod }b\mbox{)}$ provided that $0 \le a \le b^2$.
The value of $mp$ is precomputed with the function mp\_montgomery\_setup().  The modulus $b$ must be odd and larger
than zero.  

The Montgomery reduction comes in two variants.  A standard baseline and a fast comba method.  The baseline routine
is in fact slower than the Barrett reductions, however, the comba routine is much faster.  Montomgery reduction can 
reduce a number modulo a $n-$digit modulus with approximately $n^2 + n$ single precision multiplications.  Compared
to Barrett reductions the montgomery reduction requires half as many multiplications as $n \rightarrow \infty$.  

Note that the final result of a Montgomery reduction is not just the value reduced modulo $b$.  You have to multiply
by $R$ modulo $b$ to get the real result.  At first that may not seem like such a worthwhile routine, however, the
exptmod function can be made to take advantage of this such that only one normalization at the end is required.

This stems from the fact that if $a \rightarrow aR^{-1}$ through Montgomery reduction and if $a = vR$ and $b = uR$ then
$a^2 \rightarrow v^2R^2R^{-1} \equiv v^2R$ and $ab \rightarrow uvRRR^{-1} \equiv uvR$.  The next useful observation is 
that through the reduction $a \rightarrow vRR^{-1} \equiv v$ which means given a final result it can be normalized with
a single reduction.  Now a series of complicated modular operations can be optimized if all the variables are initially
multiplied by $R$ then the final result normalized by performing an extra reduction.

If many variables are to be normalized the simplest method to setup the variables is to first compute $\hat x \equiv R^2 \mbox{ mod }m$.
Now all the variables in the system can be multiplied by $\hat x$ and reduced with Montgomery reduction.  This means that
two long divisions would be required to setup $\hat x$ and a multiplication followed by reduction for each variable.  

A very useful observation is that multiplying by $R = \beta^n$ amounts to performing a left shift by $n$ positions which
requires no single precision multiplications.  

\section{Timing Analysis}

\subsection{Digit Size}
The first major constribution to the time savings is the fact that 28 bits are stored per digit instead of the MPI 
defualt of 16.  This means in many of the algorithms the savings can be considerable.  Consider a baseline multiplier 
with a 1024-bit input.  With MPI the input would be 64 16-bit digits whereas in LibTomMath it would be 37 28-bit digits.
A savings of $64^2 - 37^2 = 2727$ single precision multiplications.  

\subsection{Multiplication Algorithms}
For most inputs a typical baseline $O(n^2)$ multiplier is used which is similar to that of MPI.  There are two variants 
of the baseline multiplier.  The normal and the fast comba variant.  The normal baseline multiplier is the exact same as 
the algorithm from MPI.  The fast comba baseline multiplier is optimized for cases where the number of input digits $N$ 
is less than or equal to $2^{w}/\beta^2$.  Where $w$ is the number of bits in a \textbf{mp\_word} or simply $lg(\beta)$.
By default a mp\_word is 64-bits which means $N \le 256$ is allowed which represents numbers upto $7,168$ bits.  However,
since the Karatsuba multiplier (discussed below) will kick in before that size the slower baseline algorithm (that MPI
uses) should never really be used in a default configuration.  

The fast comba baseline multiplier is optimized by removing the carry operations from the inner loop.  This is often 
referred to as the ``comba'' method since it computes the products a columns first then figures out the carries.  To
accomodate this the result of the inner multiplications must be stored in words large enough not to lose the carry bits.  
This is why there is a limit of $2^{w}/\beta^2$ digits in the input.  This optimization has the effect of making a 
very simple and efficient inner loop.

\subsubsection{Karatsuba Multiplier}
For large inputs, typically 80 digits\footnote{By default that is 2240-bits or more.} or more the Karatsuba multiplication
method is used.  This method has significant overhead but an asymptotic running time of $O(n^{1.584})$ which means for 
fairly large inputs this method is faster than the baseline (or comba) algorithm.  The Karatsuba implementation is 
recursive which means for extremely large inputs they will benefit from the algorithm.

The algorithm is based on the observation that if 

\begin{eqnarray}
x = x_0 + x_1\beta \nonumber \\
y = y_0 + y_1\beta
\end{eqnarray}

Where $x_0, x_1, y_0, y_1$ are half the size of their respective summand than 

\begin{equation}
x \cdot y = x_1y_1\beta^2 + ((x_1 - y_1)(x_0 - y_0) + x_0y_0 + x_1y_1)\beta + x_0y_0
\end{equation}

It is trivial that from this only three products have to be produced: $x_0y_0, x_1y_1, (x_1-y_1)(x_0-y_0)$ which
are all of half size numbers.  A multiplication of two half size numbers requires only $1 \over 4$ of the
original work which means with no recursion the Karatsuba algorithm achieves a running time of ${3n^2}\over 4$.  
The routine provided does recursion which is where the $O(n^{1.584})$ work factor comes from.

The multiplication by $\beta$ and $\beta^2$ amount to digit shift operations.  
The extra overhead in the Karatsuba method comes from extracting the half size numbers $x_0, x_1, y_0, y_1$ and
performing the various smaller calculations.  

The library has been fairly optimized to extract the digits using hard-coded routines instead of the hire
level functions however there is still significant overhead to optimize away.

MPI only implements the slower baseline multiplier where carries are dealt with in the inner loop.  As a result even at
smaller numbers (below the Karatsuba cutoff) the LibTomMath multipliers are faster.

\subsection{Squaring Algorithms}

Similar to the multiplication algorithms there are two baseline squaring algorithms.  Both have an asymptotic 
running time of $O((t^2 + t)/2)$.  The normal baseline squaring is the same from MPI and the fast method is 
a ``comba'' squaring algorithm.  The comba method is used if the number of digits $N$ is less than 
$2^{w-1}/\beta^2$ which by default covers numbers upto $3,584$ bits.  

There is also a Karatsuba squaring method which achieves a running time of $O(n^{1.584})$ after considerably large
inputs.

MPI only implements the slower baseline squaring algorithm.  As a result LibTomMath is considerably faster at squaring
than MPI is.

\subsection{Exponentiation Algorithms}

LibTomMath implements a sliding window $k$-ary left to right exponentiation algorithm.  For a given exponent size $L$ an
appropriate window size $k$ is chosen.  There are always at most $L$ modular squarings and $\lfloor L/k \rfloor$ modular
multiplications.   The $k$-ary method works by precomputing values $g(x) = b^x$ for $2^{k-1} \le x < 2^k$ and a given base 
$b$.  Then the multiplications are grouped in windows of $k$ bits.  The sliding window technique has the benefit 
that it can skip multiplications if there are zero bits following or preceding a window.  Consider the exponent 
$e = 11110001_2$ if $k = 2$ then there will be a two squarings, a multiplication of $g(3)$, two squarings, a multiplication
of $g(3)$, four squarings and and a multiplication by $g(1)$.  In total there are 8 squarings and 3 multiplications.

MPI uses a binary square-multiply method for exponentiation.  For the same exponent $e = 11110001_2$ it would have had to
perform 8 squarings and 5 multiplications.  There is a precomputation phase for the method LibTomMath uses but it 
generally cuts down considerably on the number of multiplications.  Consider a 512-bit exponent.  The worst case for the 
LibTomMath method results in 512 squarings and 124 multiplications.  The MPI method would have 512 squarings 
and 512 multiplications.  Randomly every $2k$ bits another multiplication is saved via the sliding-window 
technique on top of the savings the $k$-ary method provides.

Both LibTomMath and MPI use Barrett reduction instead of division to reduce the numbers modulo the modulus given.
However, LibTomMath can take advantage of the fact that the multiplications required within the Barrett reduction
do not have to give full precision.  As a result the reduction step is much faster and just as accurate.  The LibTomMath 
code will automatically determine at run-time (e.g. when its called) whether the faster multiplier can be used.  The
faster multipliers have also been optimized into the two variants (baseline and comba baseline).

LibTomMath also has a variant of the exptmod function that uses Montgomery reductions instead of Barrett reductions
which is faster.  The code will automatically detect when the Montgomery version can be used (\textit{Requires the
modulus to be odd and below the MONTGOMERY\_EXPT\_CUTOFF size}).  The Montgomery routine is essentially a copy of the 
Barrett exponentiation routine except it uses Montgomery reduction.

As a result of all these changes exponentiation in LibTomMath is much faster than compared to MPI.  On most ALU-strong
processors (AMD Athlon for instance) exponentiation in LibTomMath is often more then ten times faster than MPI.   

\end{document}