| MP(3) | MP(3)

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, mpfactorial, 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*)

` `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
`

DESCRIPTION

These routines perform extended precision integer arithmetic. The basic type is `mpint`, which points to an array of `mpdigit`s, 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`. `Mpint`s 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 `mpint`s.

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`.
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`.
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:

`typedef struct CRTres CRTres;
{

| int n; // number of residues mpint *r[n]; // residues

};

`

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.

SOURCE

| `/usr/local/plan9/src/libmp`