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 `size` limbs.

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

In the notation used here, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {s1p, s1size}.

__Function:__mp_limb_t**mpn_add_n***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, const mp_limb_t *`s2p`, mp_size_t`size`)-
Add {
`s1p`,`size`} and {`s2p`,`size`}, and write the`size`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 targets. 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`size`, mp_limb_t`s2limb`)-
Add {
`s1p`,`size`} and`s2limb`, and write the`size`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`s1size`, const mp_limb_t *`s2p`, mp_size_t`s2size`)-
Add {
`s1p`,`s1size`} and {`s2p`,`s2size`}, and write the`s1size`least significant limbs of the result to`rp`. Return carry, either 0 or 1.This function requires that

`s1size`is greater than or equal to`s2size`.

__Function:__mp_limb_t**mpn_sub_n***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, const mp_limb_t *`s2p`, mp_size_t`size`)-
Subtract {
`s2p`,`s2size`} from {`s1p`,`size`}, and write the`size`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 targets.

__Function:__mp_limb_t**mpn_sub_1***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, mp_size_t`size`, mp_limb_t`s2limb`)-
Subtract
`s2limb`from {`s1p`,`size`}, and write the`size`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`s1size`, const mp_limb_t *`s2p`, mp_size_t`s2size`)-
Subtract {
`s2p`,`s2size`} from {`s1p`,`s1size`}, and write the`s1size`least significant limbs of the result to`rp`. Return borrow, either 0 or 1.This function requires that

`s1size`is greater than or equal to`s2size`.

__Function:__void**mpn_mul_n***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, const mp_limb_t *`s2p`, mp_size_t`size`)-
Multiply {
`s1p`,`size`} and {`s2p`,`size`}, and write the**entire**result to`rp`.The destination has to have space for 2*

`size`limbs, even if the significant result might be one limb smaller.

__Function:__mp_limb_t**mpn_mul_1***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, mp_size_t`size`, mp_limb_t`s2limb`)-
Multiply {
`s1p`,`size`} and`s2limb`, and write the`size`least significant limbs of the product to`rp`. Return the most significant limb of the product.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 targets.

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`size`, mp_limb_t`s2limb`)-
Multiply {
`s1p`,`size`} and`s2limb`, and add the`size`least significant limbs of the product to {`rp`,`size`} and write the result to`rp`. Return the most significant limb of the product, plus carry-out from the addition.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 targets.

__Function:__mp_limb_t**mpn_submul_1***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, mp_size_t`size`, mp_limb_t`s2limb`)-
Multiply {
`s1p`,`size`} and`s2limb`, and subtract the`size`least significant limbs of the product from {`rp`,`size`} 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 targets.

__Function:__mp_limb_t**mpn_mul***(mp_limb_t **`rp`, const mp_limb_t *`s1p`, mp_size_t`s1size`, const mp_limb_t *`s2p`, mp_size_t`s2size`)-
Multiply {
`s1p`,`s1size`} and {`s2p`,`s2size`}, and write the result to`rp`. Return the most significant limb of the result.The destination has to have space for

`s1size`+`s2size`limbs, even if the result might be one limb smaller.This function requires that

`s1size`is greater than or equal to`s2size`. The destination must be distinct from either 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`}. Write the quotient at`qp`and the remainder at`rp`.The quotient written at

`qp`will be`nn`-`dn`+ 1 limbs. The remainder written at`rp`will be`dn`limbs.It is required that

`nn`is greater than or equal to`dn`. The`qxn`operand must be zero.The quotient is rounded towards 0.

No overlap between arguments is permitted.

__Function:__mp_limb_t**mpn_divrem***(mp_limb_t **`r1p`, mp_size_t`xsize`, mp_limb_t *`rs2p`, mp_size_t`rs2size`, const mp_limb_t *`s3p`, mp_size_t`s3size`)-
[This function is obsolete. Please call
`mpn_tdiv_qr`

instead for best performance.]Divide {

`rs2p`,`rs2size`} by {`s3p`,`s3size`}, 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`s3size`limbs long (i.e., as many limbs as the divisor).In addition to an integer quotient,

`xsize`fraction limbs are developed, and stored after the integral limbs. For most usages,`xsize`will be zero.It is required that

`rs2size`is greater than or equal to`s3size`. It is required that the most significant bit of the divisor is set.If the quotient is not needed, pass

`rs2p`+`s3size`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`rs2size`-`s3size`+`xsize`limbs large.

__Function:__mp_limb_t**mpn_divrem_1***(mp_limb_t **`r1p`, mp_size_t`xsize`, mp_limb_t *`s2p`, mp_size_t`s2size`, mp_limb_t`s3limb`)__Macro:__mp_limb_t**mpn_divmod_1***(mp_limb_t **`r1p`, mp_limb_t *`s2p`, mp_size_t`s2size`, mp_limb_t`s3limb`)-
Divide {
`s2p`,`s2size`} by`s3limb`, and write the quotient at`r1p`. Return the remainder.The integer quotient is written to {

`r1p`+`xsize`,`s2size`} and in addition`xsize`fraction limbs are developed and written to {`r1p`,`xsize`}. Either or both`s2size`and`xsize`can be zero. For most usages,`xsize`will be zero.`mpn_divmod_1`

exists for upward source compatibility and is simply a macro calling`mpn_divrem_1`

with an`xsize`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`rs2size`, const mp_limb_t *`s3p`, mp_size_t`s3size`)-
**This interface is obsolete. It will disappear from future releases. Use**`mpn_divrem`

in its stead.

__Macro:__mp_limb_t**mpn_divexact_by3***(mp_limb_t **`rp`, mp_limb_t *`sp`, mp_size_t`size`)__Function:__mp_limb_t**mpn_divexact_by3c***(mp_limb_t **`rp`, mp_limb_t *`sp`, mp_size_t`size`, mp_limb_t`carry`)-
Divide {
`sp`,`size`} by 3, expecting it to divide exactly, and writing the result to {`rp`,`size`}. 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.`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 @ifnottex*c*b^n + a-i = 3*q*, where*b*is the size of a limb @ifnottex (*2^32*or*2^64*).*c*is always 0, 1 or 2, and the initial carry must also be 0, 1 or 2 (these are both borrows really). When*c=0*, clearly*q=(a-i)/3*. When @ifnottex*c!=0*, the remainder*(a-i) mod 3*is given by*3-c*, because @ifnottex*b == 1 mod 3*.

__Function:__mp_limb_t**mpn_mod_1***(mp_limb_t **`s1p`, mp_size_t`s1size`, mp_limb_t`s2limb`)-
Divide {
`s1p`,`s1size`} by`s2limb`, and return the remainder.`s1size`can be zero.

__Function:__mp_limb_t**mpn_preinv_mod_1***(mp_limb_t **`s1p`, mp_size_t`s1size`, mp_limb_t`s2limb`, mp_limb_t`s3limb`)-
**This interface is obsolete. It will disappear from future releases. Use**`mpn_mod_1`

in its stead.

__Function:__mp_limb_t**mpn_bdivmod***(mp_limb_t **`rp`, mp_limb_t *`s1p`, mp_size_t`s1size`, const mp_limb_t *`s2p`, mp_size_t`s2size`, unsigned long int`d`)-
The function puts the low [
`d`/`BITS_PER_MP_LIMB`] limbs of`q`= {`s1p`,`s1size`}/{`s2p`,`s2size`} mod 2^`d`at`rp`, and returns the high`d`mod`BITS_PER_MP_LIMB`bits of`q`.{

`s1p`,`s1size`} -`q`* {`s2p`,`s2size`} mod 2^(`s1size`*`BITS_PER_MP_LIMB`) is placed at`s1p`. Since the low [`d`/`BITS_PER_MP_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

`s1size`*`BITS_PER_MP_LIMB`>=`D`, and that {`s2p`,`s2size`} 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 *`src_ptr`, mp_size_t`src_size`, unsigned long int`count`)-
Shift {
`src_ptr`,`src_size`}`count`bits to the left, and write the`src_size`least significant limbs of the result to`rp`.`count`might be in the range 1 to n - 1, on an n-bit machine. The bits shifted out to the left are returned.Overlapping of the destination space and the source space is allowed in this function, provided

`rp`>=`src_ptr`.This function is written in assembly for most targets.

__Function:__mp_limp_t**mpn_rshift***(mp_limb_t **`rp`, const mp_limb_t *`src_ptr`, mp_size_t`src_size`, unsigned long int`count`)-
Shift {
`src_ptr`,`src_size`}`count`bits to the right, and write the`src_size`most significant limbs of the result to`rp`.`count`might be in the range 1 to n - 1, on an n-bit machine. The bits shifted out to the right are returned.Overlapping of the destination space and the source space is allowed in this function, provided

`rp`<=`src_ptr`.This function is written in assembly for most targets.

__Function:__int**mpn_cmp***(const mp_limb_t **`s1p`, const mp_limb_t *`s2p`, mp_size_t`size`)-
Compare {
`s1p`,`size`} and {`s2p`,`size`} and return a positive value if s1 > src2, 0 of they are equal, and a negative value if s1 < src2.

__Function:__mp_size_t**mpn_gcd***(mp_limb_t **`rp`, mp_limb_t *`s1p`, mp_size_t`s1size`, mp_limb_t *`s2p`, mp_size_t`s2size`)-
Puts at
`rp`the greatest common divisor of {`s1p`,`s1size`} and {`s2p`,`s2size`}; both source operands are destroyed by the operation. The size in limbs of the greatest common divisor is returned.{

`s1p`,`s1size`} must have at least as many bits as {`s2p`,`s2size`}, and {`s2p`,`s2size`} must be odd.

__Function:__mp_limb_t**mpn_gcd_1***(const mp_limb_t **`s1p`, mp_size_t`s1size`, mp_limb_t`s2limb`)-
Return the greatest common divisor of {
`s1p`,`s1size`} and`s2limb`, where`s2limb`(as well as`s1size`) must be different from 0.

__Function:__mp_size_t**mpn_gcdext***(mp_limb_t **`r1p`, mp_limb_t *`r2p`, mp_size_t *`r2size`, mp_limb_t *`s1p`, mp_size_t`s1size`, mp_limb_t *`s2p`, mp_size_t`s2size`)-
Compute the greatest common divisor of {
`s1p`,`s1size`} and {`s2p`,`s2size`}. Store the gcd at`r1p`and return its size in limbs. Write the first cofactor at`r2p`and store its size in *`r2size`. If the cofactor is negative, *`r2size`is negative and`r2p`is the absolute value of the cofactor.{

`s1p`,`s1size`} must be greater than or equal to {`s2p`,`s2size`}. Neither operand may equal 0. Both source operands are destroyed, plus one limb past the end of each, ie. {`s1p`,`s1size`+1} and {`s2p`,`s2size`+1}.

__Function:__mp_size_t**mpn_sqrtrem***(mp_limb_t **`r1p`, mp_limb_t *`r2p`, const mp_limb_t *`sp`, mp_size_t`size`)-
Compute the square root of {
`sp`,`size`} and put the result at`r1p`. Write the remainder at`r2p`, unless`r2p`is`NULL`

.Return the size of the remainder, whether

`r2p`was`NULL`

or non-`NULL`

. Iff the operand was a perfect square, the return value will be 0.The areas at

`r1p`and`sp`have to be distinct. The areas at`r2p`and`sp`have to be identical or completely separate, not partially overlapping.@ifnottex The area at

`r1p`needs to have space for ceil(`size`/2) limbs. The area at`r2p`needs to be`size`limbs large.

__Function:__mp_size_t**mpn_get_str***(unsigned char **`str`, int`base`, mp_limb_t *`s1p`, mp_size_t`s1size`)-
Convert {
`s1p`,`s1size`} to a raw unsigned char array in base`base`. 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. There may be leading zeros in the string.The area at

`s1p`is clobbered.Return the number of characters in

`str`.The area at

`str`has to have space for the largest possible number represented by a`s1size`long limb array, plus one extra character.

__Function:__mp_size_t**mpn_set_str***(mp_limb_t **`r1p`, const char *`str`, size_t`strsize`, int`base`)-
Convert the raw unsigned char array at
`str`of length`strsize`to a limb array {`s1p`,`s1size`}. The base of`str`is`base`.Return the number of limbs stored in

`r1p`.

__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`r1size`)__Function:__void**mpn_random2***(mp_limb_t **`r1p`, mp_size_t`r1size`)-
Generate a random number of length
`r1size`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`, unsigned long int`size`)-
Count the number of set bits in {
`s1p`,`size`}.

__Function:__unsigned long int**mpn_hamdist***(const mp_limb_t **`s1p`, const mp_limb_t *`s2p`, unsigned long int`size`)-
Compute the hamming distance between {
`s1p`,`size`} and {`s2p`,`size`}.

__Function:__int**mpn_perfect_square_p***(const mp_limb_t **`s1p`, mp_size_t`size`)-
Return non-zero iff {
`s1p`,`size`} is a perfect square.

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