Subversion Repositories planix.SVN

.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, 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

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 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