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# (gmp.info.gz) Low-level Functions

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Low-level Functions
*******************

This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.

These functions start with the prefix `mpn_'.

The `mpn' functions are designed to be as fast as possible, *not* to
provide a coherent calling interface.  The different functions have
somewhat similar interfaces, but there are variations that make them
hard to use.  These functions do as little as possible apart from the
real multiple precision computation, so that no time is spent on things
that not all callers need.

A source operand is specified by a pointer to the least significant
limb and a limb count.  A destination operand is specified by just a
pointer.  It is the responsibility of the caller to ensure that the
destination has enough space for storing the result.

With this way of specifying operands, it is possible to perform
computations on subranges of an argument, and store the result into a
subrange of a destination.

A common requirement for all functions is that each source area
needs at least one limb.  No size argument may be zero.  Unless
otherwise stated, in-place operations are allowed where source and
destination are the same, but not where they only partly overlap.

The `mpn' functions are the base for the implementation of the
`mpz_', `mpf_', and `mpq_' functions.

This example adds the number beginning at S1P and the number
beginning at S2P and writes the sum at DESTP.  All areas have N limbs.

cy = mpn_add_n (destp, s1p, s2p, n)

It should be noted that the `mpn' functions make no attempt to
identify high or low zero limbs on their operands, or other special
forms.  On random data such cases will be unlikely and it'd be wasteful
for every function to check every time.  An application knowing
something about its data can take steps to trim or perhaps split its
calculations.

In the notation used below, a source operand is identified by the
pointer to the least significant limb, and the limb count in braces.
For example, {S1P, S1N}.

- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
of the result to RP.  Return carry, either 0 or 1.

This is the lowest-level function for addition.  It is the
preferred function for addition, since it is written in assembly
for most CPUs.  For addition of a variable to itself (i.e., S1P
equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.

- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Add {S1P, N} and S2LIMB, and write the N least significant limbs
of the result to RP.  Return carry, either 0 or 1.

- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
limbs of the result to RP.  Return carry, either 0 or 1.

This function requires that S1N is greater than or equal to S2N.

- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Subtract {S2P, N} from {S1P, N}, and write the N least significant
limbs of the result to RP.  Return borrow, either 0 or 1.

This is the lowest-level function for subtraction.  It is the
preferred function for subtraction, since it is written in
assembly for most CPUs.

- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Subtract S2LIMB from {S1P, N}, and write the N least significant
limbs of the result to RP.  Return borrow, either 0 or 1.

- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
significant limbs of the result to RP.  Return borrow, either 0 or
1.

This function requires that S1N is greater than or equal to S2N.

- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const
mp_limb_t *S2P, mp_size_t N)
Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
RP.

The destination has to have space for 2*N limbs, even if the
product's most significant limb is zero.  No overlap is permitted
between the destination and either source.

- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} by S2LIMB, and write the N least significant
limbs of the product to RP.  Return the most significant limb of
the product.  {S1P, N} and {RP, N} are allowed to overlap provided
RP <= S1P.

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP.  It is written
in assembly for most CPUs.

Don't call this function if S2LIMB is a power of 2; use
`mpn_lshift' with a count equal to the logarithm of S2LIMB
instead, for optimal speed.

- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and add the N least significant
limbs of the product to {RP, N} and write the result to RP.
Return the most significant limb of the product, plus carry-out

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP.  It is written
in assembly for most CPUs.

- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and subtract the N least significant
limbs of the product from {RP, N} and write the result to RP.
Return the most significant limb of the product, minus borrow-out
from the subtraction.

This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP.
It is written in assembly for most CPUs.

- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Multiply {S1P, S1N} and {S2P, S2N}, and write the result to RP.
Return the most significant limb of the result.

The destination has to have space for S1N + S2N limbs, even if the
result might be one limb smaller.

This function requires that S1N is greater than or equal to S2N.
The destination must be distinct from both input operands.

- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
mp_size_t DN)
Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
and the remainder at {RP, DN}.  The quotient is rounded towards 0.

No overlap is permitted between arguments.  NN must be greater
than or equal to DN.  The most significant limb of DP must be
non-zero.  The QXN operand must be zero.

- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
mp_size_t S3N)
[This function is obsolete.  Please call `mpn_tdiv_qr' instead for
best performance.]

Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
with the exception of the most significant limb, which is
returned.  The remainder replaces the dividend at RS2P; it will be
S3N limbs long (i.e., as many limbs as the divisor).

In addition to an integer quotient, QXN fraction limbs are
developed, and stored after the integral limbs.  For most usages,
QXN will be zero.

It is required that RS2N is greater than or equal to S3N.  It is
required that the most significant bit of the divisor is set.

If the quotient is not needed, pass RS2P + S3N as R1P.  Aside from
that special case, no overlap between arguments is permitted.

Return the most significant limb of the quotient, either 0 or 1.

The area at R1P needs to be RS2N - S3N + QXN limbs large.

- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
mp_size_t S2N, mp_limb_t S3LIMB)
Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
Return the remainder.

The integer quotient is written to {R1P+QXN, S2N} and in addition
QXN fraction limbs are developed and written to {R1P, QXN}.
Either or both S2N and QXN can be zero.  For most usages, QXN will
be zero.

`mpn_divmod_1' exists for upward source compatibility and is
simply a macro calling `mpn_divrem_1' with a QXN of 0.

The areas at R1P and S2P have to be identical or completely
separate, not partially overlapping.

- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
[This function is obsolete.  Please call `mpn_tdiv_qr' instead for
best performance.]

- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t N)
- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t N, mp_limb_t CARRY)
Divide {SP, N} by 3, expecting it to divide exactly, and writing
the result to {RP, N}.  If 3 divides exactly, the return value is
zero and the result is the quotient.  If not, the return value is
non-zero and the result won't be anything useful.

`mpn_divexact_by3c' takes an initial carry parameter, which can be
the return value from a previous call, so a large calculation can
be done piece by piece from low to high.  `mpn_divexact_by3' is
simply a macro calling `mpn_divexact_by3c' with a 0 carry
parameter.

These routines use a multiply-by-inverse and will be faster than
`mpn_divrem_1' on CPUs with fast multiplication but slow division.

The source a, result q, size n, initial carry i, and return value
c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS.  The return
c is always 0, 1 or 2, and the initial carry i must also be 0, 1
or 2 (these are both borrows really).  When c=0 clearly q=(a-i)/3.
When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b
== 1 mod 3 (when `mp_bits_per_limb' is even, which is always so
currently).

- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Divide {S1P, S1N} by S2LIMB, and return the remainder.  S1N can be
zero.

- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, unsigned
long int D)
This function puts the low floor(D/mp_bits_per_limb) limbs of Q =
{S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod
`mp_bits_per_limb' bits of Q.

{S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed
at S1P.  Since the low floor(D/mp_bits_per_limb) limbs of this
difference are zero, it is possible to overwrite the low limbs at
S1P with this difference, provided RP <= S1P.

This function requires that S1N * mp_bits_per_limb >= D, and that
{S2P, S2N} is odd.

*This interface is preliminary.  It might change incompatibly in
future revisions.*

- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsigned int COUNT)
Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
The bits shifted out at the left are returned in the least
significant COUNT bits of the return value (the rest of the return
value is zero).

COUNT must be in the range 1 to mp_bits_per_limb-1.  The regions
{SP, N} and {RP, N} may overlap, provided RP >= SP.

This function is written in assembly for most CPUs.

- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsigned int COUNT)
Shift {SP, N} right by COUNT bits, and write the result to {RP,
N}.  The bits shifted out at the right are returned in the most
significant COUNT bits of the return value (the rest of the return
value is zero).

COUNT must be in the range 1 to mp_bits_per_limb-1.  The regions
{SP, N} and {RP, N} may overlap, provided RP <= SP.

This function is written in assembly for most CPUs.

- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
mp_size_t N)
Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
S2, 0 if they are equal, or a negative value if S1 < S2.

- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N)
Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and
{S2P, S2N}.  The result can be up to S2N limbs, the return value
is the actual number produced.  Both source operands are destroyed.

{S1P, S1N} must have at least as many bits as {S2P, S2N}.  {S2P,
S2N} must be odd.  Both operands must have non-zero most
significant limbs.  No overlap is permitted between {S1P, S1N} and
{S2P, S2N}.

- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Return the greatest common divisor of {S1P, S1N} and S2LIMB.  Both
operands must be non-zero.

- Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,
mp_size_t *R2N, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t
*S2P, mp_size_t S2N)
Calculate the greatest common divisor of {S1P, S1N} and {S2P,
S2N}.  Store the gcd at {R1P, RETVAL} and the first cofactor at
{R2P, *R2N}, with *R2N negative if the cofactor is negative.  R1P
and R2P should each have room for S1N+1 limbs, but the return
value and value stored through R2N indicate the actual number
produced.

{S1P, S1N} >= {S2P, S2N} is required, and both must be non-zero.
The regions {S1P, S1N+1} and {S2P, S2N+1} are destroyed (i.e. the
operands plus an extra limb past the end of each).

The cofactor R1 will satisfy R2*S1 + K*S2 = R1.  The second
cofactor K is not calculated but can easily be obtained from (R1 -
R2*S1) / S2.

- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
const mp_limb_t *SP, mp_size_t N)
Compute the square root of {SP, N} and put the result at {R1P,
ceil(N/2)} and the remainder at {R2P, RETVAL}.  R2P needs space
for N limbs, but the return value indicates how many are produced.

The most significant limb of {SP, N} must be non-zero.  The areas
{R1P, ceil(N/2)} and {SP, N} must be completely separate.  The
areas {R2P, N} and {SP, N} must be either identical or completely
separate.

If the remainder is not wanted then R2P can be `NULL', and in this
case the return value is zero or non-zero according to whether the
remainder would have been zero or non-zero.

A return value of zero indicates a perfect square.  See also
`mpz_perfect_square_p'.

- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
mp_limb_t *S1P, mp_size_t S1N)
Convert {S1P, S1N} to a raw unsigned char array at STR in base
BASE, and return the number of characters produced.  There may be
leading zeros in the string.  The string is not in ASCII; to
convert it to printable format, add the ASCII codes for `0' or
`A', depending on the base and range.  BASE can vary from 2 to 256.

The most significant limb of the input {S1P, S1N} must be
non-zero.  The input {S1P, S1N} is clobbered, except when BASE is
a power of 2, in which case it's unchanged.

The area at STR has to have space for the largest possible number
represented by a S1N long limb array, plus one extra character.

- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
*STR, size_t STRSIZE, int BASE)
Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.

STR[0] is the most significant byte and STR[STRSIZE-1] is the
least significant.  Each byte should be a value in the range 0 to
BASE-1, not an ASCII character.  BASE can vary from 2 to 256.

The return value is the number of limbs written to RP.  If the most
significant input byte is non-zero then the high limb at RP will be
non-zero, and only that exact number of limbs will be required
there.

If the most significant input byte is zero then there may be high
zero limbs written to RP and included in the return value.

STRSIZE must be at least 1, and no overlap is permitted between
{STR,STRSIZE} and the result at RP.

- Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,
unsigned long int BIT)
Scan S1P from bit position BIT for the next clear bit.

It is required that there be a clear bit within the area at S1P at
or beyond bit position BIT, so that the function has something to
return.

- Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P,
unsigned long int BIT)
Scan S1P from bit position BIT for the next set bit.

It is required that there be a set bit within the area at S1P at or
beyond bit position BIT, so that the function has something to
return.

- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
Generate a random number of length R1N and store it at R1P.  The
most significant limb is always non-zero.  `mpn_random' generates
uniformly distributed limb data, `mpn_random2' generates long
strings of zeros and ones in the binary representation.

`mpn_random2' is intended for testing the correctness of the `mpn'
routines.

- Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,
mp_size_t N)
Count the number of set bits in {S1P, N}.

- Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const
mp_limb_t *S2P, mp_size_t N)
Compute the hamming distance between {S1P, N} and {S2P, N}, which
is the number of bit positions where the two operands have
different bit values.

- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
N)
Return non-zero iff {S1P, N} is a perfect square.

Nails
=====

*Everything in this section is highly experimental and may disappear or
be subject to incompatible changes in a future version of GMP.*

Nails are an experimental feature whereby a few bits are left unused
at the top of each `mp_limb_t'.  This can significantly improve carry
handling on some processors.

All the `mpn' functions accepting limb data will expect the nail
bits to be zero on entry, and will return data with the nails similarly
all zero.  This applies both to limb vectors and to single limb
arguments.

Nails can be enabled by configuring with `--enable-nails'.  By
default the number of bits will be chosen according to what suits the
host processor, but a particular number can be selected with
`--enable-nails=N'.

At the mpn level, a nail build is neither source nor binary
compatible with a non-nail build, strictly speaking.  But programs
acting on limbs only through the mpn functions are likely to work
equally well with either build, and judicious use of the definitions
below should make any program compatible with either build, at the
source level.

For the higher level routines, meaning `mpz' etc, a nail build
should be fully source and binary compatible with a non-nail build.

- Macro: GMP_NAIL_BITS
- Macro: GMP_NUMB_BITS
- Macro: GMP_LIMB_BITS
`GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
not in use.  `GMP_NUMB_BITS' is the number of data bits in a limb.
`GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'.  In
all cases

GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS

Bit masks for the nail and number parts of a limb.
`GMP_NAIL_MASK' is 0 when nails are not in use.

`GMP_NAIL_MASK' is not often needed, since the nail part can be
obtained with `x >> GMP_NUMB_BITS', and that means one less large
constant, which can help various RISC chips.

- Macro: GMP_NUMB_MAX
The maximum value that can be stored in the number part of a limb.
This is the same as `GMP_NUMB_MASK', but can be used for clarity
when doing comparisons rather than bit-wise operations.

The term "nails" comes from finger or toe nails, which are at the
ends of a limb (arm or leg).  "numb" is short for number, but is also
how the developers felt after trying for a long time to come up with
sensible names for these things.

In the future (the distant future most likely) a non-zero nail might
be permitted, giving non-unique representations for numbers in a limb
vector.  This would help vector processors since carries would only
ever need to propagate one or two limbs.

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