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added most of the missing doumentation to doc/bn.tex

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czurnieden 2018-12-09 23:11:38 +01:00
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doc/bn.tex
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@ -1,3 +1,13 @@
\def\fixedpdfdate{D:20181209230255+01'00'}
\pdfinfo{
/CreationDate (\fixedpdfdate)
/ModDate (\fixedpdfdate)
}
\def\fixedpdfdate{D:20181209230255+01'00'}
\pdfinfo{
/CreationDate (\fixedpdfdate)
/ModDate (\fixedpdfdate)
}
\documentclass[synpaper]{book}
\usepackage{hyperref}
\usepackage{makeidx}
@ -546,6 +556,25 @@ int main(void)
\end{alltt} \end{small}
\section{Maintenance Functions}
\subsection{Clear Leading Zeros}
This is used to ensure that leading zero digits are trimed and the leading "used" digit will be non-zero.
It also fixes the sign if there are no more leading digits.
\index{mp\_clamp}
\begin{alltt}
void mp_clamp(mp_int *a);
\end{alltt}
\subsection{Zero Out}
This function will set the ``bigint'' to zeros without changing the amount of allocated memory.
\index{mp\_zero}
\begin{alltt}
void mp_zero(mp_int *a);
\end{alltt}
\subsection{Reducing Memory Usage}
When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
@ -640,6 +669,39 @@ int main(void)
\end{alltt} \end{small}
\chapter{Basic Operations}
\section{Copying}
A so called ``deep copy'', where new memory is allocated and all contents of $a$ are copied verbatim into $b$ such that $b = a$ at the end.
\index{mp\_copy}
\begin{alltt}
int mp_copy (mp_int * a, mp_int *b);
\end{alltt}
You can also just swap $a$ and $b$. It does the normal pointer changing with a temporary pointer variable, just that you do not have to.
\index{mp\_exch}
\begin{alltt}
void mp_exch (mp_int * a, mp_int *b);
\end{alltt}
\section{Bit Counting}
To get the position of the lowest bit set (LSB, the Lowest Significant Bit; the number of bits which are zero before the first zero bit )
\index{mp\_cnt\_lsb}
\begin{alltt}
int mp_cnt_lsb(const mp_int *a);
\end{alltt}
To get the position of the highest bit set (MSB, the Most Significant Bit; the number of bits in teh ``bignum'')
\index{mp\_count\_bits}
\begin{alltt}
int mp_count_bits(const mp_int *a);
\end{alltt}
\section{Small Constants}
Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two
small constant assignment functions. The first function is used to set a single digit constant while the second sets
@ -1103,6 +1165,21 @@ function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may b
value to signal that the remainder is not desired. The division itself is implemented as a left-shift
operation of $a$ by $b$ bits.
\index{mp\_tc\_div\_2d}\label{arithrightshift}
\begin{alltt}
int mp_tc_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
\end{alltt}
The two-co,mplement version of the function above. This can be used to implement arbitrary-precision two-complement integers together with the two-complement bit-wise operations at page \ref{tcbitwiseops}.
It is also not very uncommon to need just the power of two $2^b$; for example the startvalue for the Newton method.
\index{mp\_2expt}
\begin{alltt}
int mp_2expt(mp_int *a, int b);
\end{alltt}
It is faster than doing it by shifting $1$ with \texttt{mp_mul_2d}.
\subsection{Polynomial Basis Operations}
Strictly speaking the organization of the integers within the mp\_int structures is what is known as a
@ -1128,19 +1205,32 @@ void mp_rshd (mp_int * a, int b)
This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations
in place and no new digits are required to complete it.
\subsection{AND, OR and XOR Operations}
\subsection{AND, OR, XOR and COMPLEMENT Operations}
While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The
three functions are prototyped as follows.
four functions are prototyped as follows.
\index{mp\_or} \index{mp\_and} \index{mp\_xor}
\index{mp\_or} \index{mp\_and} \index{mp\_xor} \index {mp\_complement}
\begin{alltt}
int mp_or (mp_int * a, mp_int * b, mp_int * c);
int mp_and (mp_int * a, mp_int * b, mp_int * c);
int mp_xor (mp_int * a, mp_int * b, mp_int * c);
int mp_complement(const mp_int *a, mp_int *b);
\end{alltt}
Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.
Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR and $ b = \sim a $.
There are also three functions that act as if the ``bignum'' would be a two-complement number.
\index{mp\_tc\_or} \index{mp\_tc\_and} \index{mp\_tc\_xor}\label{tcbitwiseops}
\begin{alltt}
int mp_tc_or (mp_int * a, mp_int * b, mp_int * c);
int mp_tc_and (mp_int * a, mp_int * b, mp_int * c);
int mp_tc_xor (mp_int * a, mp_int * b, mp_int * c);
\end{alltt}
The compute $c = a \odot b$ as above if both $a$ and $b$ are positive, negative values are converted into their two-complement representation first. This can be used to implement arbitrary-precision two-complement integers together with the arithmetic right-shift at page \ref{arithrightshift}.
\section{Addition and Subtraction}
@ -1170,7 +1260,7 @@ Which assigns $-a$ to $b$.
\subsection{Absolute}
Simple integer absolutes can be performed with the following.
\index{mp\_neg}
\index{mp\_abs}
\begin{alltt}
int mp_abs (mp_int * a, mp_int * b);
\end{alltt}
@ -1587,6 +1677,33 @@ int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is
slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.
\section{Combined Modular Reduction}
Some of the combinations of an arithmetic operations followed by a modular reduction can be done in a faster way. The ones implemented are:
Addition $d = (a + b) \mod c$
\index{mp\_addmod}
\begin{alltt}
int mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d);
\end{alltt}
Subtraction $d = (a - b) \mod c$
\begin{alltt}
int mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d);
\end{alltt}
Multiplication $d = (ab) \mod c$
\begin{alltt}
int mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d);
\end{alltt}
Squaring $d = (a^2) \mod c$
\begin{alltt}
int mp_sqrmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d);
\end{alltt}
\chapter{Exponentiation}
\section{Single Digit Exponentiation}
\index{mp\_expt\_d\_ex}
@ -1628,6 +1745,13 @@ detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix b
moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
and the other two algorithms.
\section{Modulus a Power of Two}
\index{mp\_mod_2d}
\begin{alltt}
int mp_mod_2d(const mp_int *a, int b, mp_int *c)
\end{alltt}
It calculates $c = a \mod 2^b$.
\section{Root Finding}
\index{mp\_n\_root}
\begin{alltt}
@ -1645,6 +1769,15 @@ values of $b$. If particularly large roots are required then a factor method co
$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply
$\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$
The square root $c = a^{1/2}$ (with the same conditions $c^2 \le a$ and $(c+1)^2 > a$) is implemented with a faster algorithm.
\index{mp\_sqrt}
\begin{alltt}
int mp_sqrt (mp_int * a, mp_digit b, mp_int * c)
\end{alltt}
\chapter{Prime Numbers}
\section{Trial Division}
\index{mp\_prime\_is\_divisible}
@ -1693,6 +1826,13 @@ require ten tests whereas a 1024-bit number would only require four tests.
You should always still perform a trial division before a Miller-Rabin test though.
\section{Primality Testing}
Testing if a number is a square can be done a bit faster than just by calculating the square root. It is used by the primality testing function described below.
\index{mp\_is\_square}
\begin{alltt}
int mp_is_square(const mp_int *arg, int *ret);
\end{alltt}
\index{mp\_prime\_is\_prime}
\begin{alltt}
int mp_prime_is_prime (mp_int * a, int t, int *result)
@ -1762,6 +1902,17 @@ mp\_prime\_random().
\label{fig:primeopts}
\end{figure}
\chapter{Random Number Generation}
\section{PRNG}
\index{mp\_rand}
\begin{alltt}
int mp_rand(mp_int *a, int digits)
\end{alltt}
The function generates a random number of \texttt{digits} bits.
This random number is cryptographically secure if the source of random numbers the operating systems offers is cryptographically secure. It will use \texttt{arc4random()} if the OS is a BSD flavor, Wincrypt on Windows, and \texttt{\dev\urandom} on all operating systems that have it.
\chapter{Input and Output}
\section{ASCII Conversions}
\subsection{To ASCII}
@ -1773,6 +1924,13 @@ This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This
to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required
by the conversion before storing any data use the following function.
\index{mp\_toradix\_n}
\begin{alltt}
int mp_toradix_n (mp_int * a, char *str, int radix, int maxlen);
\end{alltt}
Like \texttt{mp\_toradix} but stores upto maxlen-1 chars and always a NULL byte.
\index{mp\_radix\_size}
\begin{alltt}
int mp_radix_size (mp_int * a, int radix, int *size)
@ -1780,6 +1938,13 @@ int mp_radix_size (mp_int * a, int radix, int *size)
This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this
function returns an error code and ``size'' will be zero.
If \texttt{LTM\_NO\_FILE} is not defined a function to write to a file is also available.
\index{mp\_fwrite}
\begin{alltt}
int mp_fwrite(const mp_int *a, int radix, FILE *stream);
\end{alltt}
\subsection{From ASCII}
\index{mp\_read\_radix}
\begin{alltt}
@ -1789,6 +1954,13 @@ This will read the base-``radix'' NUL terminated string from ``str'' into $a$.
character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign
can be used to denote a negative number.
If \texttt{LTM\_NO\_FILE} is not defined a function to read from a file is also available.
\index{mp\_fread}
\begin{alltt}
int mp_fread(mp_int *a, int radix, FILE *stream);
\end{alltt}
\section{Binary Conversions}
Converting an mp\_int to and from binary is another keen idea.
@ -1807,6 +1979,13 @@ int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?)
requires. It does not store the sign of the integer.
\index{mp\_to\_unsigned\_bin\_n}
\begin{alltt}
int mp_to_unsigned_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen)
\end{alltt}
Like \texttt{mp\_to\_unsigned\_bin} but checks if the value at \texttt{*outlen} is larger than or equal to the output of \texttt{mp\_unsigned\_bin\_size(a)} and sets \texttt{*outlen} to the output of \texttt{mp\_unsigned\_bin\_size(a)} or returns \texttt{MP\_VAL} if the test failed.
\index{mp\_read\_unsigned\_bin}
\begin{alltt}
int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
@ -1816,7 +1995,7 @@ integer $a$ will always be positive.
For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the
previous functions.
\index{mp\_signed\_bin\_size} \index{mp\_to\_signed\_bin} \index{mp\_read\_signed\_bin}
\begin{alltt}
int mp_signed_bin_size(mp_int *a);
int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
@ -1826,6 +2005,13 @@ They operate essentially the same as the unsigned copies except they prefix the
byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
is non--zero.
The two functions \texttt{mp\_import} and \texttt{mp\_export} implement the corresponding GMP functions as described at \url{http://gmplib.org/manual/Integer-Import-and-Export.html}.
\index{mp\_import} \index{mp\_export}
\begin{alltt}
int mp_import(mp_int *rop, size_t count, int order, size_t size, int endian, size_t nails, const void *op);
int mp_export(void *rop, size_t *countp, int order, size_t size, int endian, size_t nails, const mp_int *op);
\end{alltt}
\chapter{Algebraic Functions}
\section{Extended Euclidean Algorithm}
\index{mp\_exteuclid}
@ -1911,6 +2097,111 @@ These work like the full mp\_int capable variants except the second parameter $b
functions fairly handy if you have to work with relatively small numbers since you will not have to allocate
an entire mp\_int to store a number like $1$ or $2$.
The division by three can be made faster by replacing the division with a multiplication by the multiplicative inverse of three.
\index{mp\_div\_3}
\begin{alltt}
int mp_div_3(const mp_int *a, mp_int *c, mp_digit *d);
\end{alltt}
\chapter{Little Helpers}
It is never wrong to have some useful little shortcuts at hand.
\section{Function Macros}
To make this overview simpler the macros are given as function prototypes. The return of logic macros is \texttt{MP\_NO} or \texttt{MP\_YES} respectively.
\index{mp\_iseven}
\begin{alltt}
int mp_iseven(mp_int *a)
\end{alltt}
Checks if $a = 0 mod 2$
\index{mp\_isodd}
\begin{alltt}
int mp_isodd(mp_int *a)
\end{alltt}
Checks if $a = 1 mod 2$
\index{mp\_isneg}
\begin{alltt}
int mp_isneg(mp_int *a)
\end{alltt}
Checks if $a < 0$
\index{mp\_iszero}
\begin{alltt}
int mp_iszero(mp_int *a)
\end{alltt}
Checks if $a = 0$. It does not check if the amount of memory allocated for $a$ is also minimal.
Other macros which are either shortcuts to normal functions or just other names for them do have their place in a programmer's life, too!
\subsection{Renamings}
\index{mp\_mag\_size}
\begin{alltt}
#define mp_mag_size(mp) mp_unsigned_bin_size(mp)
\end{alltt}
\index{mp\_raw\_size}
\begin{alltt}
#define mp_raw_size(mp) mp_signed_bin_size(mp)
\end{alltt}
\index{mp\_read\_mag}
\begin{alltt}
#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len))
\end{alltt}
\index{mp\_read\_raw}
\begin{alltt}
#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len))
\end{alltt}
\index{mp\_tomag}
\begin{alltt}
#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str))
\end{alltt}
\index{mp\_toraw}
\begin{alltt}
#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str))
\end{alltt}
\subsection{Shortcuts}
\index{mp\_tobinary}
\begin{alltt}
#define mp_tobinary(M, S) mp_toradix((M), (S), 2)
\end{alltt}
\index{mp\_tooctal}
\begin{alltt}
#define mp_tooctal(M, S) mp_toradix((M), (S), 8)
\end{alltt}
\index{mp\_todecimal}
\begin{alltt}
#define mp_todecimal(M, S) mp_toradix((M), (S), 10)
\end{alltt}
\index{mp\_tohex}
\begin{alltt}
#define mp_tohex(M, S) mp_toradix((M), (S), 16)
\end{alltt}
\input{bn.ind}
\end{document}