MP(2)                                                       MP(2)

          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, mpdigdiv, mpadd,
          mpsub, mpleft, mpright, mpmul, mpexp, mpmod, mpmodadd,
          mpmodsub, mpmodmul, mpdiv, mpcmp, mpsel, mpextendedgcd,
          mpinvert, mpsignif, mplowbits0, mpvecdigmuladd,
          mpvecdigmulsub, mpvecadd, mpvecsub, mpveccmp, mpvecmul,
          mpmagcmp, mpmagadd, mpmagsub, crtpre, crtin, crtout,
          crtprefree, crtresfree - extended precision arithmetic

          #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

          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

          void    mptober(mpint *b, uchar *buf, int blen)

          mpint*  letomp(uchar *buf, uint blen, mpint *b)

     MP(2)                                                       MP(2)

          int     mptole(mpint *b, uchar *buf, uint blen, uchar

          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*)

          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)

          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)

     MP(2)                                                       MP(2)

          void    mpmod(mpint *b, mpint *m, mpint *remainder)

          void    mpdiv(mpint *dividend, mpint *divisor,  mpint *quo-
                  mpint *remainder)

          void    mpmodadd(mpint *b1, mpint *b2, mpint *m, mpint *sum)

          void    mpmodsub(mpint *b1, mpint *b2, mpint *m, mpint

          void    mpmodmul(mpint *b1, mpint *b2, mpint *m, mpint

          int     mpcmp(mpint *b1, mpint *b2)

          int     mpmagcmp(mpint *b1, mpint *b2)

          void    mpsel(int s, mpint *b1, mpint *b2, mpint *res)

          void    mpextendedgcd(mpint *a, mpint *b, mpint *d, mpint
                  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

          int     mpvecdigmulsub(mpdigit *b, int n, mpdigit m, mpdigit

          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)

     MP(2)                                                       MP(2)

          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 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

          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 mpfree(b).

     MP(2)                                                       MP(2)

          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.  Base 0 defaults to 16.
          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 hexadecimal, 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
          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 `B', 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.

          Mptober and mptolel fill blen lower bytes of an mpint into a
          fixed length byte array.  Mptober fills the bytes right
          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.

     MP(2)                                                       MP(2)

          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.
          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.

          Logical operations (treating negative numbers using two's

          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
          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:

     MP(2)                                                       MP(2)

          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
                          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
          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

     MP(2)                                                       MP(2)

          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

          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