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NAME
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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
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SYNOPSIS
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#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
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DESCRIPTION
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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
{
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int sign; /* +1 or −1 */
int size; /* allocated digits */
int top; /* significant digits */
mpdigit *p;
char flags;
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};
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.
mpmagaddsum = 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.
mpfactorialreturns factorial of n.
mpcmp returns -1, 0, or +1 as b1 is less than, equal to, or greater
than b2.
mpmagcmpthe 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:
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typedef struct CRTres CRTres;
{
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int n; // number of residues
mpint *r[n]; // residues
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};
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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.
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SOURCE
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