.TH MP 2
.SH NAME
mpsetminbits, mpnew, mpfree, mpbits, mpnorm, mpcopy, mpassign, mprand, strtomp, mpfmt,mptoa, betomp, mptobe, letomp, mptole, mptoui, uitomp, mptoi, itomp, uvtomp, mptouv, vtomp, mptov, mpdigdiv, mpadd, mpsub, mpleft, mpright, mpmul, mpexp, mpmod, mpdiv, mpfactorial, mpcmp, mpextendedgcd, mpinvert, mpsignif, mplowbits0, mpvecdigmuladd, mpvecdigmulsub, mpvecadd, mpvecsub, mpveccmp, mpvecmul, mpmagcmp, mpmagadd, mpmagsub, crtpre, crtin, crtout, crtprefree, crtresfree \- extended precision arithmetic
.SH SYNOPSIS
.B #include <u.h>
.br
.B #include <libc.h>
.br
.B #include <mp.h>
.PP
.ta +\w'\fLCRTpre* \fP'u
.B
mpint*	mpnew(int n)
.PP
.B
void	mpfree(mpint *b)
.PP
.B
void	mpsetminbits(int n)
.PP
.B
void	mpbits(mpint *b, int n)
.PP
.B
void	mpnorm(mpint *b)
.PP
.B
mpint*	mpcopy(mpint *b)
.PP
.B
void	mpassign(mpint *old, mpint *new)
.PP
.B
mpint*	mprand(int bits, void (*gen)(uchar*, int), mpint *b)
.PP
.B
mpint*	strtomp(char *buf, char **rptr, int base, mpint *b)
.PP
.B
char*	mptoa(mpint *b, int base, char *buf, int blen)
.PP
.B
int	mpfmt(Fmt*)
.PP
.B
mpint*	betomp(uchar *buf, uint blen, mpint *b)
.PP
.B
int	mptobe(mpint *b, uchar *buf, uint blen, uchar **bufp)
.PP
.B
mpint*	letomp(uchar *buf, uint blen, mpint *b)
.PP
.B
int	mptole(mpint *b, uchar *buf, uint blen, uchar **bufp)
.PP
.B
uint	mptoui(mpint*)
.PP
.B
mpint*	uitomp(uint, mpint*)
.PP
.B
int	mptoi(mpint*)
.PP
.B
mpint*	itomp(int, mpint*)
.PP
.B
mpint*	vtomp(vlong, mpint*)
.PP
.B
vlong	mptov(mpint*)
.PP
.B
mpint*	uvtomp(uvlong, mpint*)
.PP
.B
uvlong	mptouv(mpint*)
.PP
.B
void	mpadd(mpint *b1, mpint *b2, mpint *sum)
.PP
.B
void	mpmagadd(mpint *b1, mpint *b2, mpint *sum)
.PP
.B
void	mpsub(mpint *b1, mpint *b2, mpint *diff)
.PP
.B
void	mpmagsub(mpint *b1, mpint *b2, mpint *diff)
.PP
.B
void	mpleft(mpint *b, int shift, mpint *res)
.PP
.B
void	mpright(mpint *b, int shift, mpint *res)
.PP
.B
void	mpmul(mpint *b1, mpint *b2, mpint *prod)
.PP
.B
void	mpexp(mpint *b, mpint *e, mpint *m, mpint *res)
.PP
.B
void	mpmod(mpint *b, mpint *m, mpint *remainder)
.PP
.B
void	mpdiv(mpint *dividend, mpint *divisor,  mpint *quotient,
.br
.B
	mpint *remainder)
.PP
.B
mpint*	mpfactorial(ulong n)
.PP
.B
int	mpcmp(mpint *b1, mpint *b2)
.PP
.B
int	mpmagcmp(mpint *b1, mpint *b2)
.PP
.B
void	mpextendedgcd(mpint *a, mpint *b, mpint *d, mpint *x,
.br
.B
	mpint *y)
.PP
.B
void	mpinvert(mpint *b, mpint *m, mpint *res)
.PP
.B
int	mpsignif(mpint *b)
.PP
.B
int	mplowbits0(mpint *b)
.PP
.B
void	mpdigdiv(mpdigit *dividend, mpdigit divisor,
.br
.B
	mpdigit *quotient)
.PP
.B
void	mpvecadd(mpdigit *a, int alen, mpdigit *b, int blen,
.br
.B
	mpdigit *sum)
.PP
.B
void	mpvecsub(mpdigit *a, int alen, mpdigit *b, int blen,
.br
.B
	mpdigit *diff)
.PP
.B
void	mpvecdigmuladd(mpdigit *b, int n, mpdigit m, mpdigit *p)
.PP
.B
int	mpvecdigmulsub(mpdigit *b, int n, mpdigit m, mpdigit *p)
.PP
.B
void	mpvecmul(mpdigit *a, int alen, mpdigit *b, int blen,
.br
.B
	mpdigit *p)
.PP
.B
int	mpveccmp(mpdigit *a, int alen, mpdigit *b, int blen)
.PP
.B
CRTpre*	crtpre(int nfactors, mpint **factors)
.PP
.B
CRTres*	crtin(CRTpre *crt, mpint *x)
.PP
.B
void	crtout(CRTpre *crt, CRTres *r, mpint *x)
.PP
.B
void	crtprefree(CRTpre *cre)
.PP
.B
void	crtresfree(CRTres *res)
.PP
.B
mpint	*mpzero, *mpone, *mptwo
.DT
.SH DESCRIPTION
These routines perform extended precision integer arithmetic.
The basic type is
.BR mpint ,
which points to an array of
.BR mpdigit s,
stored in little-endian order:
.IP
.EX
typedef struct mpint mpint;
struct mpint
{
	int	sign;   /* +1 or -1 */
	int	size;   /* allocated digits */
	int	top;    /* significant digits */
	mpdigit	*p;
	char	flags;
};
.EE
.PP
The sign of 0 is +1.
.PP
The size of
.B mpdigit
is architecture-dependent and defined in
.BR /$cputype/include/u.h .
.BR Mpint s
are dynamically allocated and must be explicitly freed.  Operations
grow the array of digits as needed.
.PP
In general, the result parameters are last in the
argument list.
.PP
Routines that return an
.B mpint
will allocate the
.B mpint
if the result parameter is
.BR nil .
This includes
.IR strtomp ,
.IR itomp ,
.IR uitomp ,
and
.IR btomp .
These functions, in addition to
.I mpnew
and
.IR mpcopy ,
will return
.B nil
if the allocation fails.
.PP
Input and result parameters may point to the same
.BR mpint .
The routines check and copy where necessary.
.PP
.I Mpnew
creates an
.B mpint
with an initial allocation of
.I n
bits.
If
.I n
is zero, the allocation will be whatever was specified in the
last call to
.I mpsetminbits
or to the initial value, 1056.
.I Mpfree
frees an
.BR mpint .
.I Mpbits
grows the allocation of
.I b
to fit at least
.I n
bits.  If
.B b->top
doesn't cover
.I n
bits,
.I mpbits
increases it to do so.
Unless you are writing new basic operations, you
can restrict yourself to
.B mpnew(0)
and
.BR mpfree(b) .
.PP
.I Mpnorm
normalizes the representation by trimming any high order zero
digits.  All routines except
.B mpbits
return normalized results.
.PP
.I Mpcopy
creates a new
.B mpint
with the same value as
.I b
while
.I mpassign
sets the value of
.I new
to be that of
.IR old .
.PP
.I Mprand
creates an
.I n
bit random number using the generator
.IR gen .
.I Gen
takes a pointer to a string of uchar's and the number
to fill in.
.PP
.I Strtomp
and
.I mptoa
convert between
.SM ASCII
and
.B mpint
representations using the base indicated.
Only the bases 10, 16, 32, and 64 are
supported.  Anything else defaults to 16.
.IR Strtomp
skips any leading spaces or tabs.
.IR Strtomp 's
scan stops when encountering a digit not valid in the
base.  If
.I rptr
is not zero,
.I *rptr
is set to point to the character immediately after the
string converted.
If the parse pterminates before any digits are found,
.I strtomp
return
.BR nil .
.I Mptoa
returns a pointer to the filled buffer.
If the parameter
.I buf
is
.BR nil ,
the buffer is allocated.
.I Mpfmt
can be used with
.IR fmtinstall (2)
and
.IR print (2)
to print hexadecimal representations of
.BR mpint s.
The conventional verb is
.LR B ,
for which
.I mp.h
provides a
.LR pragma .
.PP
.I Mptobe
and
.I mptole
convert an
.I mpint
to a byte array.  The former creates a big endian representation,
the latter a little endian one.
If the destination
.I buf
is not
.BR nil ,
it specifies the buffer of length
.I blen
for the result.  If the representation
is less than
.I blen
bytes, the rest of the buffer is zero filled.
If
.I buf
is
.BR nil ,
then a buffer is allocated and a pointer to it is
deposited in the location pointed to by
.IR bufp .
Sign is ignored in these conversions, i.e., the byte
array version is always positive.
.PP
.IR Betomp ,
and
.I letomp
convert from a big or little endian byte array at
.I buf
of length
.I blen
to an
.IR mpint .
If
.I b
is not
.IR nil ,
it refers to a preallocated
.I mpint
for the result.
If
.I b
is
.BR nil ,
a new integer is allocated and returned as the result.
.PP
The integer conversions are:
.TF Mptouv
.TP
.I mptoui
.BR mpint -> "unsigned int"
.TP
.I uitomp
.BR "unsigned int" -> mpint
.TP
.I mptoi
.BR mpint -> "int"
.TP
.I itomp
.BR "int" -> mpint
.TP
.I mptouv
.BR mpint -> "unsigned vlong"
.TP
.I uvtomp
.BR "unsigned vlong" -> mpint
.TP
.I mptov
.BR mpint -> "vlong"
.TP
.I vtomp
.BR "vlong" -> mpint
.PD
.PP
When converting to the base integer types, if the integer is too large,
the largest integer of the appropriate sign
and size is returned.
.PP
The mathematical functions are:
.TF mpmagadd
.TP
.I mpadd
.BR "sum = b1 + b2" .
.TP
.I mpmagadd
.BR "sum = abs(b1) + abs(b2)" . 
.TP
.I mpsub
.BR "diff = b1 - b2" .
.TP
.I mpmagsub
.BR "diff = abs(b1) - abs(b2)" .
.TP
.I mpleft
.BR "res = b<<shift" .
.TP
.I mpright
.BR "res = b>>shift" .
.TP
.I mpmul
.BR "prod = b1*b2" .
.TP
.I mpexp
if
.I m
is nil,
.BR "res = b**e" .
Otherwise,
.BR "res = b**e mod m" .
.TP
.I mpmod
.BR "remainder = b % m" .
.TP
.I mpdiv
.BR "quotient = dividend/divisor" .
.BR "remainder = dividend % divisor" .
.TP
.I mpfactorial
Return a newly allocated
.BR "fact = n!" .
.I mpcmp
returns -1, 0, or +1 as
.I b1
is less than, equal to, or greater than
.IR b2 .
.TP
.I mpmagcmp
the same as
.I mpcmp
but ignores the sign and just compares magnitudes.
.PD
.PP
.I Mpextendedgcd
computes the greatest common denominator,
.IR d ,
of
.I a
and
.IR b .
It also computes
.I x
and
.I y
such that
.BR "a*x + b*y = d" .
Both
.I a
and
.I b
are required to be positive.
If called with negative arguments, it will
return a gcd of 0.
.PP
.I Mpinverse
computes the multiplicative inverse of
.I b
.B mod
.IR m .
.PP
.I Mpsignif
returns the number of significant bits in
.IR b .
.I Mplowbits0
returns the number of consecutive zero bits
at the low end of the significant bits.
For example, for 0x14,
.I mpsignif
returns 5 and
.I mplowbits0
returns 2.
For 0, 
.I mpsignif
and
.I mplowbits0
both return 0.
.PP
The remaining routines all work on arrays of
.B mpdigit
rather than
.BR mpint 's.
They are the basis of all the other routines.  They are separated out
to allow them to be rewritten in assembler for each architecture.  There
is also a portable C version for each one.
.TF mpvecdigmuladd
.TP
.I mpdigdiv
.BR "quotient = dividend[0:1] / divisor" .
.TP
.I mpvecadd
.BR "sum[0:alen] = a[0:alen-1] + b[0:blen-1]" .
We assume alen >= blen and that sum has room for alen+1 digits.
.TP
.I mpvecsub
.BR "diff[0:alen-1] = a[0:alen-1] - b[0:blen-1]" .
We assume that alen >= blen and that diff has room for alen digits.
.TP
.I mpvecdigmuladd
.BR "p[0:n] += m * b[0:n-1]" .
This multiplies a an array of digits times a scalar and adds it to another array.
We assume p has room for n+1 digits.
.TP
.I mpvecdigmulsub
.BR "p[0:n] -= m * b[0:n-1]" .
This multiplies a an array of digits times a scalar and subtracts it fromo another array.
We assume p has room for n+1 digits.  It returns +1 is the result is positive and
-1 if negative.
.TP
.I mpvecmul
.BR "p[0:alen*blen] = a[0:alen-1] * b[0:blen-1]" .
We assume that p has room for alen*blen+1 digits.
.TP
.I mpveccmp
This returns -1, 0, or +1 as a - b is negative, 0, or positive.
.PD
.PP
.IR mptwo ,
.I mpone
and
.I mpzero
are the constants 2, 1 and 0.  These cannot be freed.
.SS "Chinese remainder theorem
.PP
When computing in a non-prime modulus, 
.IR n,
it is possible to perform the computations on the residues modulo the prime
factors of
.I n
instead.  Since these numbers are smaller, multiplication and exponentiation
can be much faster.
.PP
.I Crtin
computes the residues of
.I x
and returns them in a newly allocated structure:
.IP
.EX
typedef struct CRTres	CRTres;	
{
	int	n;	/* number of residues */
	mpint	*r[n];	/* residues */
};
.EE
.PP
.I Crtout
takes a residue representation of a number and converts it back into
the number.  It also frees the residue structure.
.PP
.I Crepre
saves a copy of the factors and precomputes the constants necessary
for converting the residue form back into a number modulo
the product of the factors.  It returns a newly allocated structure
containing values.
.PP
.I Crtprefree
and
.I crtresfree
free
.I CRTpre
and
.I CRTres
structures respectively.
.SH SOURCE
.B /sys/src/libmp