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## 8 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: mp_limb_t mpn_neg (mp_limb_t *rp, const mp_limb_t *sp, mp_size_t n)

Perform the negation of {sp, n}, and write the result to {rp, n}. Return carry-out.

— 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.

If the two input operands are the same, use `mpn_sqr`.

— 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 (s1n+s2n)-limb result to rp. Return the most significant limb of the result.

The destination has to have space for s1n + s2n limbs, even if the product's most significant limb is zero. No overlap is permitted between the destination and either source.

This function requires that s1n is greater than or equal to s2n.

— Function: void mpn_sqr (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n)

Compute the square of {s1p, n} and write the 2*n-limb result to rp.

The destination has to have space for 2*n limbs, even if the result's most significant limb is zero. No overlap is permitted between the destination and the 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 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 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, plus 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: 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, nndn+1} and the remainder at {rp, dn}. The quotient is rounded towards 0.

No overlap is permitted between arguments, except that np might equal rp. The dividend size nn must be greater than or equal to divisor size dn. The most significant limb of the divisor 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 rs2ns3n + 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 (const 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_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 *xp, mp_size_t xn, mp_limb_t *yp, mp_size_t yn)

Set {rp, retval} to the greatest common divisor of {xp, xn} and {yp, yn}. The result can be up to yn limbs, the return value is the actual number produced. Both source operands are destroyed.

It is required that xn >= yn > 0, and the most significant limb of {yp, yn} must be non-zero. No overlap is permitted between {xp, xn} and {yp, yn}.

— Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *xp, mp_size_t xn, mp_limb_t ylimb)

Return the greatest common divisor of {xp, xn} and ylimb. Both operands must be non-zero.

— Function: mp_size_t mpn_gcdext (mp_limb_t *gp, mp_limb_t *sp, mp_size_t *sn, mp_limb_t *up, mp_size_t un, mp_limb_t *vp, mp_size_t vn)

Let U be defined by {up, un} and let V be defined by {vp, vn}.

Compute the greatest common divisor G of U and V. Compute a cofactor S such that G = US + VT. The second cofactor T is not computed but can easily be obtained from (G - U*S) / V (the division will be exact). It is required that un >= vn > 0, and the most significant limb of {vp, vn} must be non-zero.

S satisfies S = 1 or abs(S) < V / (2 G). S = 0 if and only if V divides U (i.e., G = V).

Store G at gp and let the return value define its limb count. Store S at sp and let |*sn| define its limb count. S can be negative; when this happens *sn will be negative. The area at gp should have room for vn limbs and the area at sp should have room for vn+1 limbs.

Both source operands are destroyed.

Compatibility notes: GMP 4.3.0 and 4.3.1 defined S less strictly. Earlier as well as later GMP releases define S as described here. GMP releases before GMP 4.3.0 required additional space for both input and output areas. More precisely, the areas {up, un+1} and {vp, vn+1} were destroyed (i.e. the operands plus an extra limb past the end of each), and the areas pointed to by gp and sp should each have room for un+1 limbs.

— 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 `mpn_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: mp_bitcnt_t mpn_scan0 (const mp_limb_t *s1p, mp_bitcnt_t 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: mp_bitcnt_t mpn_scan1 (const mp_limb_t *s1p, mp_bitcnt_t 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: mp_bitcnt_t mpn_popcount (const mp_limb_t *s1p, mp_size_t n)

Count the number of set bits in {s1p, n}.

— Function: mp_bitcnt_t 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. The most significant limb of the input {s1p, n} must be non-zero.

— Function: void mpn_and_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical and of {s1p, n} and {s2p, n}, and write the result to {rp, n}.

— Function: void mpn_ior_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical inclusive or of {s1p, n} and {s2p, n}, and write the result to {rp, n}.

— Function: void mpn_xor_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical exclusive or of {s1p, n} and {s2p, n}, and write the result to {rp, n}.

— Function: void mpn_andn_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical and of {s1p, n} and the bitwise complement of {s2p, n}, and write the result to {rp, n}.

— Function: void mpn_iorn_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical inclusive or of {s1p, n} and the bitwise complement of {s2p, n}, and write the result to {rp, n}.

— Function: void mpn_nand_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical and of {s1p, n} and {s2p, n}, and write the bitwise complement of the result to {rp, n}.

— Function: void mpn_nior_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical inclusive or of {s1p, n} and {s2p, n}, and write the bitwise complement of the result to {rp, n}.

— Function: void mpn_xnor_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)

Perform the bitwise logical exclusive or of {s1p, n} and {s2p, n}, and write the bitwise complement of the result to {rp, n}.

— Function: void mpn_com (mp_limb_t *rp, const mp_limb_t *sp, mp_size_t n)

Perform the bitwise complement of {sp, n}, and write the result to {rp, n}.

— Function: void mpn_copyi (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n)

Copy from {s1p, n} to {rp, n}, increasingly.

— Function: void mpn_copyd (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n)

Copy from {s1p, n} to {rp, n}, decreasingly.

— Function: void mpn_zero (mp_limb_t *rp, mp_size_t n)

Zero {rp, n}.

```
```

### 8.1 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.
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.