MP(2) MP(2)
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
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)
mpint* mpnorm(mpint *b)
mpint* mpcopy(mpint *b)
void mpassign(mpint *old, mpint *new)
mpint* mprand(int bits, void (*gen)(uchar*, int), mpint *b)
mpint* mpnrand(mpint *n, 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)
void mptober(mpint *b, uchar *buf, int blen)
Page 1 Plan 9 (printed 12/23/24)
MP(2) MP(2)
mpint* letomp(uchar *buf, uint blen, mpint *b)
int mptole(mpint *b, uchar *buf, uint blen, uchar
**bufp)
void mptolel(mpint *b, uchar *buf, int blen)
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*)
mpint* dtomp(double, mpint*)
double mptod(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 mpand(mpint *b1, mpint *b2, mpint *res)
void mpbic(mpint *b1, mpint *b2, mpint *res)
void mpor(mpint *b1, mpint *b2, mpint *res)
void mpnot(mpint *b, mpint *res)
void mpxor(mpint *b1, mpint *b2, mpint *res)
void mptrunc(mpint *b, int n, mpint *res)
void mpxtend(mpint *b, int n, mpint *res)
Page 2 Plan 9 (printed 12/23/24)
MP(2) MP(2)
void mpasr(mpint *b, int n, 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)
void mpmodadd(mpint *b1, mpint *b2, mpint *m, mpint *sum)
void mpmodsub(mpint *b1, mpint *b2, mpint *m, mpint
*diff)
void mpmodmul(mpint *b1, mpint *b2, mpint *m, mpint
*prod)
int mpcmp(mpint *b1, mpint *b2)
int mpmagcmp(mpint *b1, mpint *b2)
void mpsel(int s, mpint *b1, mpint *b2, mpint *res)
mpint* mpfactorial(ulong n)
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)
Page 3 Plan 9 (printed 12/23/24)
MP(2) MP(2)
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
list.
Routines that return an mpint will allocate the mpint if the
result parameter is nil. This includes strtomp, itomp,
uitomp, btomp, and dtomp. 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
Page 4 Plan 9 (printed 12/23/24)
MP(2) MP(2)
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.
Mpnrand uses gen to generate a uniform random number x, 0 ≤
x < n.
Strtomp and mptoa convert between ASCII and mpint represen-
tations using the base indicated. Only the bases 2, 4, 8,
10, 16, 32, and 64 are supported. Strtomp skips any leading
spaces or tabs. Strtomp's scan stops when encountering a
digit not valid in the base. If base is zero then C-style
prefixes are interpreted to find the base: 0x for hexadeci-
mal, 0b for binary and 0 for octal. Otherwise decimal is
assumed. rptr is not zero, *rptr is set to point to the
character immediately after the string converted. If the
parse terminates before any digits are found, strtomp return
nil. Mptoa returns a pointer to the ASCII filled buffer.
If the parameter buf is nil, the buffer is allocated. Set-
ting base to zero uses hexadecimal default. Mpfmt can be
used with fmtinstall(2) and print(2) to print ASCII repre-
sentations of mpints. The conventional verb is `B', for
which mp.h provides a `pragma'. The precision in the format
string changes the base, defaulting to hexadecimal when
omited.
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.
Mptober and mptolel fill blen lower bytes of an mpint into a
fixed length byte array. Mptober fills the bytes right
Page 5 Plan 9 (printed 12/23/24)
MP(2) MP(2)
adjusted in big endian order so that the least significant
byte is at buf[blen-1] while mptolel fills in little endian
order; left adjusted; so that the least significat byte is
filled into buf[0].
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 (and floating point) 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
mptod mpint->double
dtomp double->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.
When converting to and from floating point, results are
rounded using IEEE 754 "round to nearest". If the integer
is too large in magnitude, mptod returns infinity of the
appropriate sign.
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.
mpsel assigns b1 to res when s is not zero, otherwise
b2 is assigned to res.
Page 6 Plan 9 (printed 12/23/24)
MP(2) MP(2)
mpfactorial returns n!.
Logical operations (treating negative numbers using two's
complement):
mpand res = b1 & b2.
mpbic res = b1 & ~b2.
mpor res = b1 | b2.
mpxor res = b1 ^ b2.
mpnot res = ~b1.
mpasr res = b>>shift (mpasr, unlike mpright, uses two's
complement).
mptrunc truncates b to n bits and stores the result in
res. The result is never negative.
mpxtend truncates b to n bits, sign extends the MSB and
stores the result in res.
Modular arithmetic:
mpmodadd sum = b1+b2 mod m.
mpmodsub diff = b1-b2 mod m.
mpmodmul prod = b1*b2 mod m.
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.
Mpinvert 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,
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
Page 7 Plan 9 (printed 12/23/24)
MP(2) MP(2)
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 from 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.
Time invariant computation
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
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 MPtimesafe flag set
and is not normalized.
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.
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.
Page 8 Plan 9 (printed 12/23/24)
MP(2) MP(2)
SOURCE
/sys/src/libmp
Page 9 Plan 9 (printed 12/23/24)