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, 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

          #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)

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     MP(2)                                                       MP(2)

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

          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*)

          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)

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     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-
                  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)

          mpint*  mpfactorial(ulong n)

          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

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     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

          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, 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

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     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

          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

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     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.

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     MP(2)                                                       MP(2)

          mpfactorial  returns n!.

          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:

          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

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     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
          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

          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

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     MP(2)                                                       MP(2)


     Page 9                       Plan 9             (printed 7/23/24)