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MP(2)									 MP(2)

       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,	mpcmp,	mpextendedgcd,
       mpinvert,   mpsignif,   mplowbits0,   mpvecdigmuladd,   mpvecdigmulsub,
       mpvecadd, mpvecsub, mpveccmp, mpvecmul, mpmagcmp,  mpmagadd,  mpmagsub,
       crtpre,	crtin,	crtout,	 crtprefree,  crtresfree  - extended precision

       #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)

       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

       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
		   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 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, mpbits increases it to do	so.   Unless  you  are
       writing new basic operations, you can restrict yourself to mpnew(0) and

       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(2) and print(2) to
       print hexadecimal representations of mpints.  The conventional verb  is
       for which mp.h provides a

       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 preal‐
       located 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.

	      sum = abs(b1) + abs(b2).

       mpsub  diff = b1 - b2.

	      diff = abs(b1) - abs(b2).

       mpleft res = b<<shift.

	      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.

	      the  same as mpcmp but ignores the sign and just compares magni‐

       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  number  of  significant	bits in b.  Mplowbits0
       returns the number of consecutive zero bits at the low end of the  sig‐
       nificant	 bits.	 For  example, for 0x14, mpsignif returns 5 and mplow‐
       bits0 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 sepa‐
       rated out to allow them to be rewritten in assembler for each architec‐
       ture.  There is also a portable C version for each one.

	      quotient = dividend[0:1] / divisor.

	      sum[0:alen] = a[0:alen-1] + b[0:blen-1].	We assume alen >= blen
	      and that sum has room for alen+1 digits.

	      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.

	      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.

	      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 posi‐
	      tive and -1 if negative.

	      p[0:alen*blen] = a[0:alen-1] * b[0:blen-1].  We  assume  that  p
	      has room for alen*blen+1 digits.

	      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

   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

	      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 neces‐
       sary for converting the residue form back  into	a  number  modulo  the
       product	of  the	 factors.  It returns a newly allocated structure con‐
       taining values.

       Crtprefree and crtresfree free CRTpre  and  CRTres  structures  respec‐


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