nmod_poly_factor.h – factorisation of univariate polynomials over integers mod n (word-size n)¶
Types, macros and constants¶
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type nmod_poly_factor_struct¶
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type nmod_poly_factor_t¶
Factorisation¶
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void nmod_poly_factor_init(nmod_poly_factor_t fac)¶
Initialises
fac
for use. Annmod_poly_factor_t
represents a polynomial in factorised form as a product of polynomials with associated exponents.
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void nmod_poly_factor_clear(nmod_poly_factor_t fac)¶
Frees all memory associated with
fac
.
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void nmod_poly_factor_realloc(nmod_poly_factor_t fac, slong alloc)¶
Reallocates the factor structure to provide space for precisely
alloc
factors.
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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.
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void nmod_poly_factor_set(nmod_poly_factor_t res, const nmod_poly_factor_t fac)¶
Sets
res
to the same factorisation asfac
.
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void nmod_poly_factor_print(const nmod_poly_factor_t fac)¶
Prints the entries of
fac
to standard output.
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void nmod_poly_factor_insert(nmod_poly_factor_t fac, const nmod_poly_t poly, slong exp)¶
Inserts the factor
poly
with multiplicityexp
into the factorisationfac
.If
fac
already containspoly
, thenexp
simply gets added to the exponent of the existing entry.
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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 offac
.Does not support aliasing between
res
andfac
.
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void nmod_poly_factor_pow(nmod_poly_factor_t fac, slong exp)¶
Raises
fac
to the powerexp
.
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int nmod_poly_is_irreducible(const nmod_poly_t f)¶
Returns 1 if the polynomial
f
is irreducible, otherwise returns 0.
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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.
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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.
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int _nmod_poly_is_squarefree(nn_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.
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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.
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void nmod_poly_factor_squarefree(nmod_poly_factor_t res, const nmod_poly_t f)¶
Sets
res
to a square-free factorization off
.
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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 degreed
. 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.
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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 degreed
, finds all those factors and places them in factors. Requires thatpol
be monic, non-constant and squarefree.
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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 ofpoly
of degree \(d\). Factors \(f[d]\) are stored inres
, and the degree \(d\) of the irreducible factors is stored indegs
in the same order as the factors.Requires that
degs
has enough space for(n/2)+1 * sizeof(slong)
.
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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()
.
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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.
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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.
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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. Ifflint_get_num_threads()
is greater than onenmod_poly_factor_distinct_deg_threaded()
is used.
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ulong 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 off
, or 0 iff
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.
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ulong 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 off
, or 0 iff
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.
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ulong 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 off
, or 0 iff
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.
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ulong 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 off
, or 0 iff
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.
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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
.