# mpn_extras.h – support functions for limb arrays¶

## Macros¶

MPN_NORM(a, an)

Normalise (a, an) so that either an is zero or a[an - 1] is nonzero.

MPN_SWAP(a, an, b, bn)

Swap (a, an) and (b, bn), i.e. swap pointers and sizes.

## Utility functions¶

void flint_mpn_debug(mp_srcptr x, mp_size_t xsize)

Prints debug information about (x, xsize) to stdout. In particular, this will print binary representations of all the limbs.

int flint_mpn_zero_p(mp_srcptr x, mp_size_t xsize)

Returns $$1$$ if all limbs of (x, xsize) are zero, otherwise $$0$$.

## Multiplication¶

mp_size_t flint_mpn_fmms1(mp_ptr y, mp_limb_t a1, mp_srcptr x1, mp_limb_t a2, mp_srcptr x2, mp_size_t n)

Given not-necessarily-normalized $$x_1$$ and $$x_2$$ of length $$n > 0$$ and output $$y$$ of length $$n$$, try to compute $$y = a_1*x_1 - a_2*x_2$$. Return the normalized length of $$y$$ if $$y \ge 0$$ and $$y$$ fits into $$n$$ limbs. Otherwise, return $$-1$$. $$y$$ may alias $$x1$$ but is not allowed to alias $$x_2$$.

## Divisibility¶

int flint_mpn_divisible_1_p(x, xsize, d)

Expression determining whether (x, xsize) is divisible by the mp_limb_t d which is assumed to be odd-valued and at least~3.

This function is implemented as a macro.

mp_size_t flint_mpn_divexact_1(mp_ptr x, mp_size_t xsize, mp_limb_t d)

Divides $$x$$ once by a known single-limb divisor, returns the new size.

mp_size_t flint_mpn_remove_2exp(mp_ptr x, mp_size_t xsize, flint_bitcnt_t *bits)

Divides (x, xsize) by $$2^n$$ where $$n$$ is the number of trailing zero bits in $$x$$. The new size of $$x$$ is returned, and $$n$$ is stored in the bits argument. $$x$$ may not be zero.

mp_size_t flint_mpn_remove_power_ascending(mp_ptr x, mp_size_t xsize, mp_ptr p, mp_size_t psize, ulong *exp)

Divides (x, xsize) by the largest power $$n$$ of (p, psize) that is an exact divisor of $$x$$. The new size of $$x$$ is returned, and $$n$$ is stored in the exp argument. $$x$$ may not be zero, and $$p$$ must be greater than $$2$$.

This function works by testing divisibility by ascending squares $$p, p^2, p^4, p^8, \dotsc$$, making it efficient for removing potentially large powers. Because of its high overhead, it should not be used as the first stage of trial division.

int flint_mpn_factor_trial(mp_srcptr x, mp_size_t xsize, slong start, slong stop)

Searches for a factor of (x, xsize) among the primes in positions start, ..., stop-1 of flint_primes. Returns $$i$$ if flint_primes[i] is a factor, otherwise returns $$0$$ if no factor is found. It is assumed that start >= 1.

int flint_mpn_factor_trial_tree(slong * factors, mp_srcptr x, mp_size_t xsize, slong num_primes)

Searches for a factor of (x, xsize) among the primes in positions approximately in the range 0, ..., num_primes - 1 of flint_primes.

Returns the number of prime factors found and fills factors with their indices in flint_primes. It is assumed that num_primes is in the range 0, ..., 3512.

If the input fits in a small fmpz the number is fully factored instead.

The algorithm used is a tree based gcd with a product of primes, the tree for which is cached globally (it is threadsafe).

## Division¶

int flint_mpn_divides(mp_ptr q, mp_srcptr array1, mp_size_t limbs1, mp_srcptr arrayg, mp_size_t limbsg, mp_ptr temp)

If (arrayg, limbsg) divides (array1, limbs1) then (q, limbs1 - limbsg + 1) is set to the quotient and 1 is returned, otherwise 0 is returned. The temporary space temp must have space for limbsg limbs.

Assumes limbs1 limbs1 >= limbsg > 0.

mp_limb_t flint_mpn_preinv1(mp_limb_t d, mp_limb_t d2)

Computes a precomputed inverse from the leading two limbs of the divisor b, n to be used with the preinv1 functions. We require the most significant bit of b, n to be 1.

mp_limb_t flint_mpn_divrem_preinv1(mp_ptr q, mp_ptr a, mp_size_t m, mp_srcptr b, mp_size_t n, mp_limb_t dinv)

Divide a, m by b, n, returning the high limb of the quotient (which will either be 0 or 1), storing the remainder in-place in a, n and the rest of the quotient in q, m - n. We require the most significant bit of b, n to be 1. dinv must be computed from b[n - 1], b[n - 2] by flint_mpn_preinv1. We also require m >= n >= 2.

void flint_mpn_mulmod_preinv1(mp_ptr r, mp_srcptr a, mp_srcptr b, mp_size_t n, mp_srcptr d, mp_limb_t dinv, ulong norm)

Given a normalised integer $$d$$ with precomputed inverse dinv provided by flint_mpn_preinv1, computes $$ab \pmod{d}$$ and stores the result in $$r$$. Each of $$a$$, $$b$$ and $$r$$ is expected to have $$n$$ limbs of space, with zero padding if necessary.

The value norm is provided for convenience. If $$a$$, $$b$$ and $$d$$ have been shifted left by norm bits so that $$d$$ is normalised, then $$r$$ will be shifted right by norm bits so that it has the same shift as all the inputs.

We require $$a$$ and $$b$$ to be reduced modulo $$n$$ before calling the function.

void flint_mpn_preinvn(mp_ptr dinv, mp_srcptr d, mp_size_t n)

Compute an $$n$$ limb precomputed inverse dinv of the $$n$$ limb integer $$d$$.

We require that $$d$$ is normalised, i.e. with the most significant bit of the most significant limb set.

void flint_mpn_mod_preinvn(mp_ptr r, mp_srcptr a, mp_size_t m, mp_srcptr d, mp_size_t n, mp_srcptr dinv)

Given a normalised integer $$d$$ of $$n$$ limbs, with precomputed inverse dinv provided by flint_mpn_preinvn and integer $$a$$ of $$m$$ limbs, computes $$a \pmod{d}$$ and stores the result in-place in the lower $$n$$ limbs of $$a$$. The remaining limbs of $$a$$ are destroyed.

We require $$m \geq n$$. No aliasing of $$a$$ with any of the other operands is permitted.

Note that this function is not always as fast as ordinary division.

mp_limb_t flint_mpn_divrem_preinvn(mp_ptr q, mp_ptr r, mp_srcptr a, mp_size_t m, mp_srcptr d, mp_size_t n, mp_srcptr dinv)

Given a normalised integer $$d$$ with precomputed inverse dinv provided by flint_mpn_preinvn, computes the quotient of $$a$$ by $$d$$ and stores the result in $$q$$ and the remainder in the lower $$n$$ limbs of $$a$$. The remaining limbs of $$a$$ are destroyed.

The value $$q$$ is expected to have space for $$m - n$$ limbs and we require $$m >= n$$. No aliasing is permitted between $$q$$ and $$a$$ or between these and any of the other operands.

Note that this function is not always as fast as ordinary division.

void flint_mpn_mulmod_preinvn(mp_ptr r, mp_srcptr a, mp_srcptr b, mp_size_t n, mp_srcptr d, mp_srcptr dinv, ulong norm)

Given a normalised integer $$d$$ with precomputed inverse dinv provided by flint_mpn_preinvn, computes $$ab \pmod{d}$$ and stores the result in $$r$$. Each of $$a$$, $$b$$ and $$r$$ is expected to have $$n$$ limbs of space, with zero padding if necessary.

The value norm is provided for convenience. If $$a$$, $$b$$ and $$d$$ have been shifted left by norm bits so that $$d$$ is normalised, then $$r$$ will be shifted right by norm bits so that it has the same shift as all the inputs.

We require $$a$$ and $$b$$ to be reduced modulo $$n$$ before calling the function.

Note that this function is not always as fast as ordinary division.

## GCD¶

mp_size_t flint_mpn_gcd_full2(mp_ptr arrayg, mp_ptr array1, mp_size_t limbs1, mp_ptr array2, mp_size_t limbs2, mp_ptr temp)
Sets (arrayg, retvalue) to the gcd of (array1, limbs1) and
(array2, limbs2).

The only assumption is that neither limbs1 or limbs2 is zero.

The function must be supplied with limbs1 + limbs2 limbs of temporary space, or NULL must be passed to temp if the function should allocate its own space.

mp_size_t flint_mpn_gcd_full(mp_ptr arrayg, mp_ptr array1, mp_size_t limbs1, mp_ptr array2, mp_size_t limbs2)

Sets (arrayg, retvalue) to the gcd of (array1, limbs1) and (array2, limbs2).

The only assumption is that neither limbs1 or limbs2 is zero.

## Random Number Generation¶

void flint_mpn_rrandom(mp_limb_t *rp, gmp_randstate_t state, mp_size_t n)

Generates a random number with n limbs and stores it on rp. The number it generates will tend to have long strings of zeros and ones in the binary representation.

Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs.

void flint_mpn_urandomb(mp_limb_t *rp, gmp_randstate_t state, flint_bitcnt_t n)

Generates a uniform random number n bits and stores it on rp.