# nmod_poly_factor.h – factorisation of univariate polynomials over integers mod n (word-size n)¶

## Types, macros and constants¶

type nmod_poly_factor_struct
type nmod_poly_factor_t

## Factorisation¶

void nmod_poly_factor_init(nmod_poly_factor_t fac)

Initialises fac for use. An nmod_poly_factor_t represents a polynomial in factorised form as a product of polynomials with associated exponents.

void nmod_poly_factor_clear(nmod_poly_factor_t fac)

Frees all memory associated with fac.

void nmod_poly_factor_realloc(nmod_poly_factor_t fac, slong alloc)

Reallocates the factor structure to provide space for precisely alloc factors.

void nmod_poly_factor_fit_length(nmod_poly_factor_t fac, slong len)

Ensures that the factor structure has space for at least len factors. This function takes care of the case of repeated calls by always at least doubling the number of factors the structure can hold.

void nmod_poly_factor_set(nmod_poly_factor_t res, const nmod_poly_factor_t fac)

Sets res to the same factorisation as fac.

void nmod_poly_factor_print(const nmod_poly_factor_t fac)

Prints the entries of fac to standard output.

void nmod_poly_factor_insert(nmod_poly_factor_t fac, const nmod_poly_t poly, slong exp)

Inserts the factor poly with multiplicity exp into the factorisation fac.

If fac already contains poly, then exp simply gets added to the exponent of the existing entry.

void nmod_poly_factor_concat(nmod_poly_factor_t res, const nmod_poly_factor_t fac)

Concatenates two factorisations.

This is equivalent to calling nmod_poly_factor_insert() repeatedly with the individual factors of fac.

Does not support aliasing between res and fac.

void nmod_poly_factor_pow(nmod_poly_factor_t fac, slong exp)

Raises fac to the power exp.

ulong nmod_poly_remove(nmod_poly_t f, const nmod_poly_t p)

Removes the highest possible power of p from f and returns the exponent.

int nmod_poly_is_irreducible(const nmod_poly_t f)

Returns 1 if the polynomial f is irreducible, otherwise returns 0.

int nmod_poly_is_irreducible_ddf(const nmod_poly_t f)

Returns 1 if the polynomial f is irreducible, otherwise returns 0. Uses fast distinct-degree factorisation.

int nmod_poly_is_irreducible_rabin(const nmod_poly_t f)

Returns 1 if the polynomial f is irreducible, otherwise returns 0. Uses Rabin irreducibility test.

int _nmod_poly_is_squarefree(mp_srcptr f, slong len, nmod_t mod)

Returns 1 if (f, len) is squarefree, and 0 otherwise. As a special case, the zero polynomial is not considered squarefree. There are no restrictions on the length.

int nmod_poly_is_squarefree(const nmod_poly_t f)

Returns 1 if f is squarefree, and 0 otherwise. As a special case, the zero polynomial is not considered squarefree.

void nmod_poly_factor_squarefree(nmod_poly_factor_t res, const nmod_poly_t f)

Sets res to a square-free factorization of f.

int nmod_poly_factor_equal_deg_prob(nmod_poly_t factor, flint_rand_t state, const nmod_poly_t pol, slong d)

Probabilistic equal degree factorisation of pol into irreducible factors of degree d. If it passes, a factor is placed in factor and 1 is returned, otherwise 0 is returned and the value of factor is undetermined.

Requires that pol be monic, non-constant and squarefree.

void nmod_poly_factor_equal_deg(nmod_poly_factor_t factors, const nmod_poly_t pol, slong d)

Assuming pol is a product of irreducible factors all of degree d, finds all those factors and places them in factors. Requires that pol be monic, non-constant and squarefree.

void nmod_poly_factor_distinct_deg(nmod_poly_factor_t res, const nmod_poly_t poly, slong *const *degs)

Factorises a monic non-constant squarefree polynomial poly of degree n into factors $$f[d]$$ such that for $$1 \leq d \leq n$$ $$f[d]$$ is the product of the monic irreducible factors of poly of degree $$d$$. Factors $$f[d]$$ are stored in res, and the degree $$d$$ of the irreducible factors is stored in degs in the same order as the factors.

Requires that degs has enough space for (n/2)+1 * sizeof(slong).

void nmod_poly_factor_distinct_deg_threaded(nmod_poly_factor_t res, const nmod_poly_t poly, slong *const *degs)

Multithreaded version of nmod_poly_factor_distinct_deg().

void nmod_poly_factor_cantor_zassenhaus(nmod_poly_factor_t res, const nmod_poly_t f)

Factorises a non-constant polynomial f into monic irreducible factors using the Cantor-Zassenhaus algorithm.

void nmod_poly_factor_berlekamp(nmod_poly_factor_t res, const nmod_poly_t f)

Factorises a non-constant, squarefree polynomial f into monic irreducible factors using the Berlekamp algorithm.

void nmod_poly_factor_kaltofen_shoup(nmod_poly_factor_t res, const nmod_poly_t poly)

Factorises a non-constant polynomial f into monic irreducible factors using the fast version of Cantor-Zassenhaus algorithm proposed by Kaltofen and Shoup (1998). More precisely this algorithm uses a “baby step/giant step” strategy for the distinct-degree factorization step. If flint_get_num_threads() is greater than one nmod_poly_factor_distinct_deg_threaded() is used.

mp_limb_t nmod_poly_factor_with_berlekamp(nmod_poly_factor_t res, const nmod_poly_t f)

Factorises a general polynomial f into monic irreducible factors and returns the leading coefficient of f, or 0 if f is the zero polynomial.

This function first checks for small special cases, deflates f if it is of the form $$p(x^m)$$ for some $$m > 1$$, then performs a square-free factorisation, and finally runs Berlekamp on all the individual square-free factors.

mp_limb_t nmod_poly_factor_with_cantor_zassenhaus(nmod_poly_factor_t res, const nmod_poly_t f)

Factorises a general polynomial f into monic irreducible factors and returns the leading coefficient of f, or 0 if f is the zero polynomial.

This function first checks for small special cases, deflates f if it is of the form $$p(x^m)$$ for some $$m > 1$$, then performs a square-free factorisation, and finally runs Cantor-Zassenhaus on all the individual square-free factors.

mp_limb_t nmod_poly_factor_with_kaltofen_shoup(nmod_poly_factor_t res, const nmod_poly_t f)

Factorises a general polynomial f into monic irreducible factors and returns the leading coefficient of f, or 0 if f is the zero polynomial.

This function first checks for small special cases, deflates f if it is of the form $$p(x^m)$$ for some $$m > 1$$, then performs a square-free factorisation, and finally runs Kaltofen-Shoup on all the individual square-free factors.

mp_limb_t nmod_poly_factor(nmod_poly_factor_t res, const nmod_poly_t f)

Factorises a general polynomial f into monic irreducible factors and returns the leading coefficient of f, or 0 if f is the zero polynomial.

This function first checks for small special cases, deflates f if it is of the form $$p(x^m)$$ for some $$m > 1$$, then performs a square-free factorisation, and finally runs either Cantor-Zassenhaus or Berlekamp on all the individual square-free factors. Currently Cantor-Zassenhaus is used by default unless the modulus is 2, in which case Berlekamp is used.

void _nmod_poly_interval_poly_worker(void *arg_ptr)

Worker function to compute interval polynomials in distinct degree factorisation. Input/output is stored in nmod_poly_interval_poly_arg_t.