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 precision 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

You are viewing proxied material from sdf.org. The copyright of proxied material belongs to its original authors. Any comments or complaints in relation to proxied material should be directed to the original authors of the content concerned. Please see the disclaimer for more details.