.TH MP 2
.SH NAME
mpsetminbits, mpnew, mpfree, mpbits, mpnorm, mpcopy, mpassign, mprand, mpnrand, strtomp, mpfmt, mptoa, betomp, mptobe, mptober, letomp, mptole, mptolel, mptoui, uitomp, mptoi, itomp, uvtomp, mptouv, vtomp, mptov, mptod, dtomp, mpdigdiv, mpadd, mpsub, mpleft, mpright, mpmul, mpexp, mpmod, mpmodadd, mpmodsub, mpmodmul, mpdiv, mpcmp, mpsel, mpfactorial, 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
mpint*	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*	mpnrand(mpint *n, 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
void	mptober(mpint *b, uchar *buf, int blen)
.PP
.B
mpint*	letomp(uchar *buf, uint blen, mpint *b)
.PP
.B
int	mptole(mpint *b, uchar *buf, uint blen, uchar **bufp)
.PP
.B
void	mptolel(mpint *b, uchar *buf, int blen)
.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
mpint*	dtomp(double, mpint*)
.PP
.B
double	mptod(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	mpand(mpint *b1, mpint *b2, mpint *res)
.PP
.B
void	mpbic(mpint *b1, mpint *b2, mpint *res)
.PP
.B
void	mpor(mpint *b1, mpint *b2, mpint *res)
.PP
.B
void	mpnot(mpint *b, mpint *res)
.PP
.B
void	mpxor(mpint *b1, mpint *b2, mpint *res)
.PP
.B
void	mptrunc(mpint *b, int n, mpint *res)
.PP
.B
void	mpxtend(mpint *b, int n, mpint *res)
.PP
.B
void	mpasr(mpint *b, int n, 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
void	mpmodadd(mpint *b1, mpint *b2, mpint *m, mpint *sum)
.PP
.B
void	mpmodsub(mpint *b1, mpint *b2, mpint *m, mpint *diff)
.PP
.B
void	mpmodmul(mpint *b1, mpint *b2, mpint *m, mpint *prod)
.PP
.B
int	mpcmp(mpint *b1, mpint *b2)
.PP
.B
int	mpmagcmp(mpint *b1, mpint *b2)
.PP
.B
void	mpsel(int s, mpint *b1, mpint *b2, mpint *res)
.PP
.B
mpint*	mpfactorial(ulong n)
.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 ,
.IR btomp ,
and
.IR dtomp .
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 Mpnrand
uses
.I gen
to generate a uniform random number
.IR x ,
.if t 0 ≤ \fIx\fR < \fIn\fR.
.if n 0 ≤ x < n.
.PP
.I Strtomp
and
.I mptoa
convert between
.SM ASCII
and
.B mpint
representations using the base indicated.
Only the bases 2, 4, 8, 10, 16, 32, and 64 are
supported.
.IR Strtomp
skips any leading spaces or tabs.
.IR Strtomp 's
scan stops when encountering a digit not valid in the
base.  If
.I base
is zero then C-style prefixes are interpreted to
find the base:
.B 0x
for hexadecimal,
.B 0b
for binary and
.B 0
for octal. Otherwise decimal is assumed.
.I rptr
is not zero,
.I *rptr
is set to point to the character immediately after the
string converted.
If the parse terminates before any digits are found,
.I strtomp
return
.BR nil .
.I Mptoa
returns a pointer to the
.SM ASCII
filled buffer.
If the parameter
.I buf
is
.BR nil ,
the buffer is allocated.
Setting
.I base
to zero uses hexadecimal default.
.I Mpfmt
can be used with
.IR fmtinstall (2)
and
.IR print (2)
to print
.SM ASCII
representations of
.BR mpint s.
The conventional verb is
.LR B ,
for which
.I mp.h
provides a
.LR pragma .
The precisition in the format string changes the base,
defaulting to hexadecimal when omited.
.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
.I Mptober
and
.I mptolel
fill
.I blen
lower bytes of an
.I mpint
into a fixed length byte array.
.I Mptober
fills the bytes right adjusted in big endian order so that the least
significant byte is at
.I buf[blen-1]
while
.I mptolel
fills in little endian order; left adjusted; so that the least
significat byte is filled into
.IR buf[0] .
.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 (and floating point) 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
.TP
.I mptod
.BR mpint -> "double"
.TP
.I dtomp
.BR "double" -> 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
When converting to and from floating point, results are rounded using IEEE 754 "round to nearest".
If the integer is too large in magnitude,
.I mptod 
returns infinity of the appropriate sign.
.PP
The mathematical functions are:
.TF mpfactorial
.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 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.
.TP
.I mpsel
assigns
.I b1
to
.I res
when
.I s
is not zero, otherwise
.I b2
is assigned to
.IR res .
.TP
.I mpfactorial
returns \fIn\fR!.
.PD
.PP
Logical operations (treating negative numbers using two's complement):
.TF mpxtend_
.TP
.I mpand
.BR "res = b1 & b2" .
.TP
.I mpbic
.BR "res = b1 & ~b2" .
.TP
.I mpor
.BR "res = b1 | b2" .
.TP
.I mpxor
.BR "res = b1 ^ b2" .
.TP
.I mpnot
.BR "res = ~b1" .
.TP
.I mpasr
.BR "res = b>>shift"
(\fImpasr\fR, unlike
.IR mpright ,
uses two's complement).
.TP
.I mptrunc
truncates
.I b
to 
.I n
bits and stores the result in
.IR res .
The result is never negative.
.TP
.I mpxtend
truncates
.I b
to
.I n
bits, sign extends the MSB and stores the result in
.IR res .
.PD
.PP
Modular arithmetic:
.TF mpmodmul_
.TP
.I mpmodadd	
.BR "sum = b1+b2 mod m" .
.TP
.I mpmodsub	
.BR "diff = b1-b2 mod m" .
.TP
.I mpmodmul	
.BR "prod = b1*b2 mod m" .
.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 Mpinvert
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 from 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 "Time invariant computation"
.PP
In the field of cryptography, it is sometimes neccesary to implement
algorithms such that the runtime of the algorithm is not depdenent on
the input data. This library provides partial support for time
invariant computation with the
.I MPtimesafe
flag that can be set on input or destination operands to request timing
safe operation. The result of a timing safe operation will also have the
.I MPtimesafe
flag set and is not normalized.
.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