9bb519ac50
The documentation for mp(2) claimed we'd return nil on error, when we actually sysfatal. This corrects the documentation to match our actual behavior.
802 lines
14 KiB
Plaintext
802 lines
14 KiB
Plaintext
.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 call sysfatal (see
|
|
.IR perror (2))
|
|
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
|