MP(2) MP(2)
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
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*)
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MP(2) MP(2)
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 *quo-
tient,
mpint *remainder)
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)
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MP(2) MP(2)
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 arith-
metic. 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 dig-
its as needed.
In general, the result parameters are last in the argument
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MP(2) MP(2)
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, mpbits 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 nor-
malized 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 num-
ber to fill in.
Strtomp and mptoa convert between ASCII and mpint represen-
tations 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 pter-
minates before any digits are found, strtomp return nil.
Mptoa returns a pointer to the filled buffer. If the param-
eter buf is nil, the buffer is allocated. Mpfmt can be used
with fmtinstall(2) and print(2) to print hexadecimal repre-
sentations of mpints. The conventional verb is `B', for
which mp.h provides a `pragma'.
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,
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MP(2) MP(2)
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.
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 number of significant bits in b.
Mplowbits0 returns the number of consecutive zero bits at
the low end of the significant bits. For example, for 0x14,
mpsignif returns 5 and mplowbits0 returns 2. For 0,
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MP(2) MP(2)
mpsignif and mplowbits0 both return 0.
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 sub-
tracts 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 nega-
tive, 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, mul-
tiplication 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 con-
verts it back into the number. It also frees the residue
structure.
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MP(2) MP(2)
Crepre saves a copy of the factors and precomputes the con-
stants 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
/sys/src/libmp
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