MP(3)

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

SYNOPSIS
#include <u.h>
#include <libc.h>
#include <mp.h>

mpint*    mpnew(int n)

void    mpfree(mpint *b)

void    mpsetminbits(int n)

void    mpbits(mpint *b, int n)

void    mpnorm(mpint *b)

mpint*    mpcopy(mpint *b)

void    mpassign(mpint *old, mpint *new)

mpint*    mprand(int bits, void (*gen)(uchar*, int), mpint *b)

mpint*    strtomp(char *buf, char **rptr, int base, mpint *b)

char*    mptoa(mpint *b, int base, char *buf, int blen)

int    mpfmt(Fmt*)

mpint*    betomp(uchar *buf, uint blen, mpint *b)

int    mptobe(mpint *b, uchar *buf, uint blen, uchar **bufp)

mpint*    letomp(uchar *buf, uint blen, mpint *b)

int    mptole(mpint *b, uchar *buf, uint blen, uchar **bufp)

uint    mptoui(mpint*)

mpint*    uitomp(uint, mpint*)

int    mptoi(mpint*)

mpint*    itomp(int, mpint*)

mpint*    vtomp(vlong, mpint*)

vlong    mptov(mpint*)

mpint*    uvtomp(uvlong, mpint*)

uvlong    mptouv(mpint*)

void    mpadd(mpint *b1, mpint *b2, mpint *sum)

void    mpmagadd(mpint *b1, mpint *b2, mpint *sum)

void    mpsub(mpint *b1, mpint *b2, mpint *diff)

void    mpmagsub(mpint *b1, mpint *b2, mpint *diff)

void    mpleft(mpint *b, int shift, mpint *res)

void    mpright(mpint *b, int shift, mpint *res)

void    mpmul(mpint *b1, mpint *b2, mpint *prod)

void    mpexp(mpint *b, mpint *e, mpint *m, mpint *res)

void    mpmod(mpint *b, mpint *m, mpint *remainder)

void    mpdiv(mpint *dividend, mpint *divisor,  mpint *quotient, mpint *remainder)

mpint*    mpfactorial(ulong n)

int    mpcmp(mpint *b1, mpint *b2)

int    mpmagcmp(mpint *b1, mpint *b2)

void    mpextendedgcd(mpint *a, mpint *b, mpint *d, mpint *x, mpint *y)

void    mpinvert(mpint *b, mpint *m, mpint *res)

int    mpsignif(mpint *b)

int    mplowbits0(mpint *b)

void    mpdigdiv(mpdigit *dividend, mpdigit divisor, mpdigit *quotient)

void    mpvecadd(mpdigit *a, int alen, mpdigit *b, int blen, mpdigit *sum)

void    mpvecsub(mpdigit *a, int alen, mpdigit *b, int blen, mpdigit *diff)

void    mpvecdigmuladd(mpdigit *b, int n, mpdigit m, mpdigit *p)

int    mpvecdigmulsub(mpdigit *b, int n, mpdigit m, mpdigit *p)

void    mpvecmul(mpdigit *a, int alen, mpdigit *b, int blen, mpdigit *p)

int    mpveccmp(mpdigit *a, int alen, mpdigit *b, int blen)

CRTpre*    crtpre(int nfactors, mpint **factors)

CRTres*    crtin(CRTpre *crt, mpint *x)

void    crtout(CRTpre *crt, CRTres *r, mpint *x)

void    crtprefree(CRTpre *cre)

void    crtresfree(CRTres *res)

mpint    *mpzero, *mpone, *mptwo

DESCRIPTION

These routines perform extended precision integer arithmetic. The basic type is mpint, which points to an array of mpdigits, stored in little-endian order:

    typedef struct mpint mpint;
    struct mpint
    {
    	int	sign;   /* +1 or -1 */
    	int	size;   /* allocated digits */
    	int	top;    /* significant digits */
    	mpdigit	*p;
    	char	flags;
    };

The sign of 0 is +1.

The size of mpdigit is architecture-dependent and defined in /$cputype/include/u.h. Mpints are dynamically allocated and must be explicitly freed. Operations grow the array of digits as needed.

In general, the result parameters are last in the argument list.

Routines that return an mpint will allocate the mpint if the result parameter is nil. This includes strtomp, itomp, uitomp, and btomp. These functions, in addition to mpnew and mpcopy, will return nil if the allocation fails.

Input and result parameters may point to the same mpint. The routines check and copy where necessary.

Mpnew creates an mpint with an initial allocation of n bits. If n is zero, the allocation will be whatever was specified in the last call to mpsetminbits or to the initial value, 1056. Mpfree frees an mpint. Mpbits grows the allocation of b to fit at least n bits. If b->top doesn't cover n bits it increases it to do so. Unless you are writing new basic operations, you can restrict yourself to mpnew(0) and mpfree(b).

Mpnorm normalizes the representation by trimming any high order zero digits. All routines except mpbits return normalized results.

Mpcopy creates a new mpint with the same value as b while mpassign sets the value of new to be that of old.

Mprand creates an n bit random number using the generator gen. Gen takes a pointer to a string of uchar's and the number to fill in.

Strtomp and mptoa convert between ASCII and mpint representations using the base indicated. Only the bases 10, 16, 32, and 64 are supported. Anything else defaults to 16. Strtomp skips any leading spaces or tabs. Strtomp's scan stops when encountering a digit not valid in the base. If rptr is not zero, *rptr is set to point to the character immediately after the string converted. If the parse pterminates before any digits are found, strtomp return nil. Mptoa returns a pointer to the filled buffer. If the parameter buf is nil, the buffer is allocated. Mpfmt can be used with fmtinstall((3)) and print((3)) to print hexadecimal representations of mpints.

Mptobe and mptole convert an mpint to a byte array. The former creates a big endian representation, the latter a little endian one. If the destination buf is not nil, it specifies the buffer of length blen for the result. If the representation is less than blen bytes, the rest of the buffer is zero filled. If buf is nil, then a buffer is allocated and a pointer to it is deposited in the location pointed to by bufp. Sign is ignored in these conversions, i.e., the byte array version is always positive.

Betomp, and letomp convert from a big or little endian byte array at buf of length blen to an mpint. If b is not nil, it refers to a preallocated mpint for the result. If b is nil, a new integer is allocated and returned as the result.

The integer conversions are:

mptoui
    mpint->unsigned int

uitomp
    unsigned int->mpint

mptoi
    mpint->int

itomp
    int->mpint

mptouv
    mpint->unsigned vlong

uvtomp
    unsigned vlong->mpint

mptov
    mpint->vlong

vtomp
    vlong->mpint

When converting to the base integer types, if the integer is too large, the largest integer of the appropriate sign and size is returned.

The mathematical functions are:

mpadd
    sum = b1 + b2.

mpmagadd
    sum = abs(b1) + abs(b2).

mpsub
    diff = b1 - b2.

mpmagsub
    diff = abs(b1) - abs(b2).

mpleft
    res = b<<shift.

mpright
    res = b>>shift.

mpmul
    prod = b1*b2.

mpexp
    if m is nil, res = b**e. Otherwise, res = b**e mod m.

mpmod
    remainder = b % m.

mpdiv
    quotient = dividend/divisor. remainder = dividend % divisor.

mpfactorial
    returns factorial of n.

mpcmp
    returns -1, 0, or +1 as b1 is less than, equal to, or greater than b2.

mpmagcmp
    the same as mpcmp but ignores the sign and just compares magnitudes.

Mpextendedgcd computes the greatest common denominator, d, of a and b. It also computes x and y such that a*x + b*y = d. Both a and b are required to be positive. If called with negative arguments, it will return a gcd of 0.

Mpinverse computes the multiplicative inverse of b mod m.

Mpsignif returns the bit offset of the left most 1 bit in b. Mplowbits0 returns the bit offset of the right most 1 bit. For example, for 0x14, mpsignif would return 4 and mplowbits0 would return 2.

The remaining routines all work on arrays of mpdigit rather than 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.

mpdigdiv
    quotient = dividend[0:1] / divisor.

mpvecadd
    sum[0:alen] = a[0:alen-1] + b[0:blen-1]. We assume alen >= blen and that sum has room for alen+1 digits.

mpvecsub
    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.

mpvecdigmuladd
    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.

mpvecdigmulsub
    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.

mpvecmul
    p[0:alen*blen] = a[0:alen-1] * b[0:blen-1]. We assume that p has room for alen*blen+1 digits.

mpveccmp
    This returns -1, 0, or +1 as a - b is negative, 0, or positive.

mptwo, mpone and mpzero are the constants 2, 1 and 0. These cannot be freed.

  Chinese remainder theorem

When computing in a non-prime modulus, n, it is possible to perform the computations on the residues modulo the prime factors of n instead. Since these numbers are smaller, multiplication and exponentiation can be much faster.

Crtin computes the residues of x and returns them in a newly allocated structure:
    	typedef struct CRTres	CRTres;
    	{
    		int	n;	// number of residues
    		mpint	*r[n];	// residues
    	};

Crtout takes a residue representation of a number and converts it back into the number. It also frees the residue structure.

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.

Crtprefree and crtresfree free CRTpre and CRTres structures respectively.

SOURCE
/src/libmp