# fq_poly_factor.h – factorisation of univariate polynomials over finite fields¶

Description.

## Types, macros and constants¶

fq_poly_factor_struct
fq_poly_factor_t

Description.

## Memory Management¶

void fq_poly_factor_init(fq_poly_factor_t fac, const fq_ctx_t ctx)

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

void fq_poly_factor_clear(fq_poly_factor_t fac, const fq_ctx_t ctx)

Frees all memory associated with fac.

void fq_poly_factor_realloc(fq_poly_factor_t fac, slong alloc, const fq_ctx_t ctx)

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

void fq_poly_factor_fit_length(fq_poly_factor_t fac, slong len, const fq_ctx_t ctx)

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

## Basic Operations¶

void fq_poly_factor_set(fq_poly_factor_t res, const fq_poly_factor_t fac, const fq_ctx_t ctx)

Sets res to the same factorisation as fac.

void fq_poly_factor_print_pretty(const fq_poly_factor_t fac, const fq_ctx_t ctx)

Pretty-prints the entries of fac to standard output.

void fq_poly_factor_print(const fq_poly_factor_t fac, const fq_ctx_t ctx)

Prints the entries of fac to standard output.

void fq_poly_factor_insert(fq_poly_factor_t fac, const fq_poly_t poly, slong exp, const fq_ctx_t ctx)

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 fq_poly_factor_concat(fq_poly_factor_t res, const fq_poly_factor_t fac, const fq_ctx_t ctx)

Concatenates two factorisations.

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

Does not support aliasing between res and fac.

void fq_poly_factor_pow(fq_poly_factor_t fac, slong exp, const fq_ctx_t ctx)

Raises fac to the power exp.

ulong fq_poly_remove(fq_poly_t f, const fq_poly_t p, const fq_ctx_t ctx)

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

## Irreducibility Testing¶

int fq_poly_is_irreducible(const fq_poly_t f, const fq_ctx_t ctx)

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

int fq_poly_is_irreducible_ddf(const fq_poly_t f, const fq_ctx_t ctx)

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

int fq_poly_is_irreducible_ben_or(const fq_poly_t f, const fq_ctx_t ctx)

Returns 1 if the polynomial f is irreducible, otherwise returns 0. Uses Ben-Or’s irreducibility test.

int _fq_poly_is_squarefree(const fq_struct * f, slong len, const fq_ctx_t ctx)

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 fq_poly_is_squarefree(const fq_poly_t f, const fq_ctx_t ctx)

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

## Factorisation¶

int fq_poly_factor_equal_deg_prob(fq_poly_t factor, flint_rand_t state, const fq_poly_t pol, slong d, const fq_ctx_t ctx)

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 fq_poly_factor_equal_deg(fq_poly_factor_t factors, const fq_poly_t pol, slong d, const fq_ctx_t ctx)

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 fq_poly_factor_split_single(fq_poly_t linfactor, const fq_poly_t input, const fq_ctx_t ctx)

Assuming input is a product of factors all of degree 1, finds a single linear factor of input and places it in linfactor. Requires that input be monic and non-constant.

void fq_poly_factor_distinct_deg(fq_poly_factor_t res, const fq_poly_t poly, slong * const *degs, const fq_ctx_t ctx)

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 are stored in res, associated powers of irreducible polynomials are stored in degs in the same order as factors.

Requires that degs have enough space for irreducible polynomials’ powers (maximum space required is n * sizeof(slong)).

void fq_poly_factor_squarefree(fq_poly_factor_t res, const fq_poly_t f, const fq_ctx_t ctx)

Sets res to a squarefree factorization of f.

void fq_poly_factor(fq_poly_factor_t res, fq_t lead, const fq_poly_t f, const fq_ctx_t ctx)

Factorises a non-constant polynomial f into monic irreducible factors choosing the best algorithm for given modulo and degree. The output lead is set to the leading coefficient of $$f$$ upon return. Choice of algorithm is based on heuristic measurements.

void fq_poly_factor_cantor_zassenhaus(fq_poly_factor_t res, const fq_poly_t f, const fq_ctx_t ctx)

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

void fq_poly_factor_kaltofen_shoup(fq_poly_factor_t res, const fq_poly_t poly, const fq_ctx_t ctx)

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.

void fq_poly_factor_berlekamp(fq_poly_factor_t factors, const fq_poly_t f, const fq_ctx_t ctx)

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

void fq_poly_factor_with_berlekamp(fq_poly_factor_t res, fq_t leading_coeff, const fq_poly_t f, const fq_ctx_t)

Factorises a general polynomial f into monic irreducible factors and sets leading_coeff to 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 factorisation on all the individual square-free factors.

void fq_poly_factor_with_cantor_zassenhaus(fq_poly_factor_t res, fq_t leading_coeff, const fq_poly_t f, const fq_ctx_t ctx)

Factorises a general polynomial f into monic irreducible factors and sets leading_coeff to 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.

void fq_poly_factor_with_kaltofen_shoup(fq_poly_factor_t res, fq_t leading_coeff, const fq_poly_t f, const fq_ctx_t ctx)

Factorises a general polynomial f into monic irreducible factors and sets leading_coeff to 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.

void fq_poly_iterated_frobenius_preinv(fq_poly_t *rop, slong n, const fq_poly_t v, const fq_poly_t vinv, const fq_ctx_t ctx)

Sets rop[i] to be $$x^{q^i}\mod v$$ for $$0 <= i < n$$.

It is required that vinv is the inverse of the reverse of v mod x^lenv.

## Root Finding¶

void fq_poly_roots(fq_poly_factor_t r, const fq_poly_t f, int with_multiplicity, const fq_ctx_t ctx)

Fill $$r$$ with factors of the form $$x - r_i$$ where the $$r_i$$ are the distinct roots of a nonzero $$f$$ in $$F_q$$. If $$with_multiplicity$$ is zero, the exponent $$e_i$$ of the factor $$x - r_i$$ is $$1$$. Otherwise, it is the largest $$e_i$$ such that $$(x-r_i)^e_i$$ divides $$f$$. This function throws if $$f$$ is zero, but is otherwise always successful.