# fq_poly.h – univariate polynomials over finite fields¶

Description.

## Types, macros and constants¶

fq_poly_struct
fq_poly_t

Description.

## Memory management¶

void fq_poly_init(fq_poly_t poly, const fq_ctx_t ctx)

Initialises poly for use, with context ctx, and setting its length to zero. A corresponding call to fq_poly_clear() must be made after finishing with the fq_poly_t to free the memory used by the polynomial.

void fq_poly_init2(fq_poly_t poly, slong alloc, const fq_ctx_t ctx)

Initialises poly with space for at least alloc coefficients and sets the length to zero. The allocated coefficients are all set to zero. A corresponding call to fq_poly_clear() must be made after finishing with the fq_poly_t to free the memory used by the polynomial.

void fq_poly_realloc(fq_poly_t poly, slong alloc, const fq_ctx_t ctx)

Reallocates the given polynomial to have space for alloc coefficients. If alloc is zero the polynomial is cleared and then reinitialised. If the current length is greater than alloc the polynomial is first truncated to length alloc.

void fq_poly_fit_length(fq_poly_t poly, slong len, const fq_ctx_t ctx)

If len is greater than the number of coefficients currently allocated, then the polynomial is reallocated to have space for at least len coefficients. No data is lost when calling this function.

The function efficiently deals with the case where fit_length is called many times in small increments by at least doubling the number of allocated coefficients when length is larger than the number of coefficients currently allocated.

void _fq_poly_set_length(fq_poly_t poly, slong newlen, const fq_ctx_t ctx)

Sets the coefficients of poly beyond len to zero and sets the length of poly to len.

void fq_poly_clear(fq_poly_t poly, const fq_ctx_t ctx)

Clears the given polynomial, releasing any memory used. It must be reinitialised in order to be used again.

void _fq_poly_normalise(fq_poly_t poly, const fq_ctx_t ctx)

Sets the length of poly so that the top coefficient is non-zero. If all coefficients are zero, the length is set to zero. This function is mainly used internally, as all functions guarantee normalisation.

void _fq_poly_normalise2(fq_struct *poly, slong *length, const fq_ctx_t ctx)

Sets the length length of (poly,length) so that the top coefficient is non-zero. If all coefficients are zero, the length is set to zero. This function is mainly used internally, as all functions guarantee normalisation.

void fq_poly_truncate(fq_poly_t poly, slong newlen, const fq_ctx_t ctx)

Truncates the polynomial to length at most $$n$$.

void fq_poly_set_trunc(fq_poly_t poly1, fq_poly_t poly2, slong newlen, const fq_ctx_t ctx)

Sets poly1 to poly2 truncated to length $$n$$.

void _fq_poly_reverse(fq_struct* output, const fq_struct* input, slong len, slong m, const fq_ctx_t ctx)

Sets output to the reverse of input, which is of length len, but thinking of it as a polynomial of length m, notionally zero-padded if necessary. The length m must be non-negative, but there are no other restrictions. The polynomial output must have space for m coefficients.

void fq_poly_reverse(fq_poly_t output, const fq_poly_t input, slong m, const fq_ctx_t ctx)

Sets output to the reverse of input, thinking of it as a polynomial of length m, notionally zero-padded if necessary). The length m must be non-negative, but there are no other restrictions. The output polynomial will be set to length m and then normalised.

## Polynomial parameters¶

long fq_poly_degree(fq_poly_t poly, const fq_ctx_t ctx)

Returns the degree of the polynomial poly.

long fq_poly_length(fq_poly_t poly, const fq_ctx_t ctx)

Returns the length of the polynomial poly.

fq_struct * fq_poly_lead(const fq_poly_t poly, const fq_ctx_t ctx)

Returns a pointer to the leading coefficient of poly, or NULL if poly is the zero polynomial.

## Randomisation¶

void fq_poly_randtest(fq_poly_t f, flint_rand_t state, slong len, const fq_ctx_t ctx)

Sets $$f$$ to a random polynomial of length at most len with entries in the field described by ctx.

void fq_poly_randtest_not_zero(fq_poly_t f, flint_rand_t state, slong len, const fq_ctx_t ctx)

Same as fq_poly_randtest but guarantees that the polynomial is not zero.

void fq_poly_randtest_monic(fq_poly_t f, flint_rand_t state, slong len, const fq_ctx_t ctx)

Sets $$f$$ to a random monic polynomial of length len with entries in the field described by ctx.

void fq_poly_randtest_irreducible(fq_poly_t f, flint_rand_t state, slong len, const fq_ctx_t ctx)

Sets $$f$$ to a random monic, irreducible polynomial of length len with entries in the field described by ctx.

## Assignment and basic manipulation¶

void _fq_poly_set(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets (rop, len) to (op, len).

void fq_poly_set(fq_poly_t poly1, const fq_poly_t poly2, const fq_ctx_t ctx)

Sets the polynomial poly1 to the polynomial poly2.

void fq_poly_set_fq(fq_poly_t poly, const fq_t c, const fq_ctx_t ctx)

Sets the polynomial poly to c.

void fq_poly_set_fmpz_mod_poly(fq_poly_t rop, const fmpz_mod_poly_t op, fq_ctx_t ctx)

Sets the polynomial rop to the polynomial op

void fq_poly_set_nmod_poly(fq_poly_t rop, const nmod_poly_t op, fq_ctx_t ctx)

Sets the polynomial rop to the polynomial op

void fq_poly_swap(fq_poly_t op1, fq_poly_t op2, const fq_ctx_t ctx)

Swaps the two polynomials op1 and op2.

void _fq_poly_zero(fq_struct *rop, slong len, const fq_ctx_t ctx)

Sets (rop, len) to the zero polynomial.

void fq_poly_zero(fq_poly_t poly, const fq_ctx_t ctx)

Sets poly to the zero polynomial.

void fq_poly_one(fq_poly_t poly, const fq_ctx_t ctx)

Sets poly to the constant polynomial $$1$$.

void fq_poly_gen(fq_poly_t poly, const fq_ctx_t ctx)

Sets poly to the polynomial $$x$$.

void fq_poly_make_monic(fq_poly_t rop, const fq_poly_t op, const fq_ctx_t ctx)

Sets rop to op, normed to have leading coefficient 1.

void _fq_poly_make_monic(fq_struct *rop, const fq_struct *op, slong length, const fq_ctx_t ctx)

Sets rop to (op,length), normed to have leading coefficient 1. Assumes that rop has enough space for the polynomial, assumes that op is not zero (and thus has an invertible leading coefficient).

## Getting and setting coefficients¶

void fq_poly_get_coeff(fq_t x, const fq_poly_t poly, slong n, const fq_ctx_t ctx)

Sets $$x$$ to the coefficient of $$X^n$$ in poly.

void fq_poly_set_coeff(fq_poly_t poly, slong n, const fq_t x, const fq_ctx_t ctx)

Sets the coefficient of $$X^n$$ in poly to $$x$$.

void fq_poly_set_coeff_fmpz(fq_poly_t poly, slong n, const fmpz_t x, const fq_ctx_t ctx)

Sets the coefficient of $$X^n$$ in the polynomial to $$x$$, assuming $$n \geq 0$$.

## Comparison¶

int fq_poly_equal(const fq_poly_t poly1, const fq_poly_t poly2, const fq_ctx_t ctx)

Returns nonzero if the two polynomials poly1 and poly2 are equal, otherwise returns zero.

int fq_poly_equal_trunc(const fq_poly_t poly1, const fq_poly_t poly2, slong n, const fq_ctx_t ctx)

Notionally truncate poly1 and poly2 to length $$n$$ and return nonzero if they are equal, otherwise return zero.

int fq_poly_is_zero(const fq_poly_t poly, const fq_ctx_t ctx)

Returns whether the polynomial poly is the zero polynomial.

int fq_poly_is_one(const fq_poly_t op)

Returns whether the polynomial poly is equal to the constant polynomial $$1$$.

int fq_poly_is_gen(const fq_poly_t op, const fq_ctx_t ctx)

Returns whether the polynomial poly is equal to the polynomial $$x$$.

int fq_poly_is_unit(const fq_poly_t op, const fq_ctx_t ctx)

Returns whether the polynomial poly is a unit in the polynomial ring $$\mathbf{F}_q[X]$$, i.e. if it has degree $$0$$ and is non-zero.

int fq_poly_equal_fq(const fq_poly_t poly, const fq_t c, const fq_ctx_t ctx)

Returns whether the polynomial poly is equal the (constant) $$\mathbf{F}_q$$ element c

void _fq_poly_add(fq_struct *res, const fq_struct *poly1, slong len1, const fq_struct *poly2, slong len2, const fq_ctx_t ctx)

Sets res to the sum of (poly1,len1) and (poly2,len2).

void fq_poly_add(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, const fq_ctx_t ctx)

Sets res to the sum of poly1 and poly2.

void fq_poly_add_si(fq_poly_t res, const fq_poly_t poly1, slong c, const fq_ctx_t ctx)

Sets res to the sum of poly1 and c.

void fq_poly_add_series(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, slong n, const fq_ctx_t ctx)

Notionally truncate poly1 and poly2 to length n and set res to the sum.

void _fq_poly_sub(fq_struct *res, const fq_struct *poly1, slong len1, const fq_struct *poly2, slong len2, const fq_ctx_t ctx)

Sets res to the difference of (poly1,len1) and (poly2,len2).

void fq_poly_sub(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, const fq_ctx_t ctx)

Sets res to the difference of poly1 and poly2.

void fq_poly_sub_series(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, slong n, const fq_ctx_t ctx)

Notionally truncate poly1 and poly2 to length n and set res to the difference.

void _fq_poly_neg(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets rop to the additive inverse of (poly,len).

void fq_poly_neg(fq_poly_t res, const fq_poly_t poly, const fq_ctx_t ctx)

Sets res to the additive inverse of poly.

## Scalar multiplication and division¶

void _fq_poly_scalar_mul_fq(fq_struct *rop, const fq_struct *op, slong len, const fq_t x, const fq_ctx_t ctx)

Sets (rop,len) to the product of (op,len) by the scalar x, in the context defined by ctx.

void fq_poly_scalar_mul_fq(fq_poly_t rop, const fq_poly_t op, const fq_t x, const fq_ctx_t ctx)

Sets rop to the product of op by the scalar x, in the context defined by ctx.

void _fq_poly_scalar_addmul_fq(fq_struct *rop, const fq_struct *op, slong len, const fq_t x, const fq_ctx_t ctx)

Adds to (rop,len) the product of (op,len) by the scalar x, in the context defined by ctx. In particular, assumes the same length for op and rop.

void fq_poly_scalar_addmul_fq(fq_poly_t rop, const fq_poly_t op, const fq_t x, const fq_ctx_t ctx)

Adds to rop the product of op by the scalar x, in the context defined by ctx.

void _fq_poly_scalar_submul_fq(fq_struct *rop, const fq_struct *op, slong len, const fq_t x, const fq_ctx_t ctx)

Subtracts from (rop,len) the product of (op,len) by the scalar x, in the context defined by ctx. In particular, assumes the same length for op and rop.

void fq_poly_scalar_submul_fq(fq_poly_t rop, const fq_poly_t op, const fq_t x, const fq_ctx_t ctx)

Subtracts from rop the product of op by the scalar x, in the context defined by ctx.

void _fq_poly_scalar_div_fq(fq_struct *rop, const fq_struct *op, slong len, const fq_t x, const fq_ctx_t ctx)

Sets (rop,len) to the quotient of (op,len) by the scalar x, in the context defined by ctx. An exception is raised if x is zero.

void fq_poly_scalar_div_fq(fq_poly_t rop, const fq_poly_t op, const fq_t x, const fq_ctx_t ctx)

Sets rop to the quotient of op by the scalar x, in the context defined by ctx. An exception is raised if x is zero.

## Multiplication¶

void _fq_poly_mul_classical(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets (rop, len1 + len2 - 1) to the product of (op1, len1) and (op2, len2), assuming that len1 is at least len2 and neither is zero.

Permits zero padding. Does not support aliasing of rop with either op1 or op2.

void fq_poly_mul_classical(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2 using classical polynomial multiplication.

void _fq_poly_mul_reorder(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets (rop, len1 + len2 - 1) to the product of (op1, len1) and (op2, len2), assuming that len1 and len2 are non-zero.

void fq_poly_mul_reorder(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2, reordering the two indeterminates $$X$$ and $$Y$$ when viewing the polynomials as elements of $$\mathbf{F}_p[X,Y]$$.

Suppose $$\mathbf{F}_q = \mathbf{F}_p[X]/ (f(X))$$ and recall that elements of $$\mathbf{F}_q$$ are internally represented by elements of type fmpz_poly. For small degree extensions but polynomials in $$\mathbf{F}_q[Y]$$ of large degree $$n$$, we change the representation to

$\begin{split}\begin{split} g(Y) & = \sum_{i=0}^{n} a_i(X) Y^i \\ & = \sum_{j=0}^{d} \sum_{i=0}^{n} \text{Coeff}(a_i(X), j) Y^i. \end{split}\end{split}$

This allows us to use a poor algorithm (such as classical multiplication) in the $$X$$-direction and leverage the existing fast integer multiplication routines in the $$Y$$-direction where the polynomial degree $$n$$ is large.

void _fq_poly_mul_univariate(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets (rop, len1 + len2 - 1) to the product of (op1, len1) and (op2, len2).

Permits zero padding and places no assumptions on the lengths len1 and len2. Supports aliasing.

void fq_poly_mul_univariate(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2 using a bivariate to univariate transformation and reducing this problem to multiplying two univariate polynomials.

void _fq_poly_mul_KS(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets (rop, len1 + len2 - 1) to the product of (op1, len1) and (op2, len2).

Permits zero padding and places no assumptions on the lengths len1 and len2. Supports aliasing.

void fq_poly_mul_KS(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2 using Kronecker substitution, that is, by encoding each coefficient in $$\mathbf{F}_{q}$$ as an integer and reducing this problem to multiplying two polynomials over the integers.

void _fq_poly_mul(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets (rop, len1 + len2 - 1) to the product of (op1, len1) and (op2, len2), choosing an appropriate algorithm.

Permits zero padding. Does not support aliasing.

void fq_poly_mul(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2, choosing an appropriate algorithm.

void _fq_poly_mullow_classical(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, slong n, const fq_ctx_t ctx)

Sets (rop, n) to the first $$n$$ coefficients of (op1, len1) multiplied by (op2, len2).

Assumes 0 < n <= len1 + len2 - 1. Assumes neither len1 nor len2 is zero.

void fq_poly_mullow_classical(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, slong n, const fq_ctx_t ctx)

Sets rop to the product of poly1 and poly2, computed using the classical or schoolbook method.

void _fq_poly_mullow_univariate(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, slong n, const fq_ctx_t ctx)

Sets (rop, n) to the lowest $$n$$ coefficients of the product of (op1, len1) and (op2, len2), computed using a bivariate to univariate transformation.

Assumes that len1 and len2 are positive, but does allow for the polynomials to be zero-padded. The polynomials may be zero, too. Assumes $$n$$ is positive. Supports aliasing between res, poly1 and poly2.

void fq_poly_mullow_univariate(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, slong n, const fq_ctx_t ctx)

Sets rop to the lowest $$n$$ coefficients of the product of op1 and op2, computed using a bivariate to univariate transformation.

void _fq_poly_mullow_KS(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, slong n, const fq_ctx_t ctx)

Sets (rop, n) to the lowest $$n$$ coefficients of the product of (op1, len1) and (op2, len2).

Assumes that len1 and len2 are positive, but does allow for the polynomials to be zero-padded. The polynomials may be zero, too. Assumes $$n$$ is positive. Supports aliasing between rop, op1 and op2.

void fq_poly_mullow_KS(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, slong n, const fq_ctx_t ctx)

Sets rop to the lowest $$n$$ coefficients of the product of op1 and op2.

void _fq_poly_mullow(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, slong n, const fq_ctx_t ctx)

Sets (rop, n) to the lowest $$n$$ coefficients of the product of (op1, len1) and (op2, len2).

Assumes 0 < n <= len1 + len2 - 1. Allows for zero-padding in the inputs. Does not support aliasing between the inputs and the output.

void fq_poly_mullow(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, slong n, const fq_ctx_t ctx)

Sets rop to the lowest $$n$$ coefficients of the product of op1 and op2.

void _fq_poly_mulhigh_classical(fq_struct *res, const fq_struct *poly1, slong len1, const fq_struct *poly2, slong len2, slong start, const fq_ctx_t ctx)

Computes the product of (poly1, len1) and (poly2, len2) and writes the coefficients from start onwards into the high coefficients of res, the remaining coefficients being arbitrary but reduced. Assumes that len1 >= len2 > 0. Aliasing of inputs and output is not permitted. Algorithm is classical multiplication.

void fq_poly_mulhigh_classical(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, slong start, const fq_ctx_t ctx)

Computes the product of poly1 and poly2 and writes the coefficients from start onwards into the high coefficients of res, the remaining coefficients being arbitrary but reduced. Algorithm is classical multiplication.

void _fq_poly_mulhigh(fq_struct *res, const fq_struct *poly1, slong len1, const fq_struct *poly2, slong len2, slong start, const fq_ctx_t ctx)

Computes the product of (poly1, len1) and (poly2, len2) and writes the coefficients from start onwards into the high coefficients of res, the remaining coefficients being arbitrary but reduced. Assumes that len1 >= len2 > 0. Aliasing of inputs and output is not permitted.

void fq_poly_mulhigh(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, slong start, const fq_ctx_t ctx)

Computes the product of poly1 and poly2 and writes the coefficients from start onwards into the high coefficients of res, the remaining coefficients being arbitrary but reduced.

void _fq_poly_mulmod(fq_struct* res, const fq_struct* poly1, slong len1, const fq_struct* poly2, slong len2, const fq_struct* f, slong lenf, const fq_ctx_t ctx)

Sets res to the remainder of the product of poly1 and poly2 upon polynomial division by f.

It is required that len1 + len2 - lenf > 0, which is equivalent to requiring that the result will actually be reduced. Otherwise, simply use _fq_poly_mul instead.

Aliasing of f and res is not permitted.

void fq_poly_mulmod(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, const fq_poly_t f, const fq_ctx_t ctx)

Sets res to the remainder of the product of poly1 and poly2 upon polynomial division by f.

void _fq_poly_mulmod_preinv(fq_struct* res, const fq_struct* poly1, slong len1, const fq_struct* poly2, slong len2, const fq_struct* f, slong lenf, const fq_struct* finv, slong lenfinv, const fq_ctx_t ctx)

Sets res to the remainder of the product of poly1 and poly2 upon polynomial division by f.

It is required that finv is the inverse of the reverse of f mod x^lenf.

Aliasing of res with any of the inputs is not permitted.

void fq_poly_mulmod_preinv(fq_poly_t res, const fq_poly_t poly1, const fq_poly_t poly2, const fq_poly_t f, const fq_poly_t finv, const fq_ctx_t ctx)

Sets res to the remainder of the product of poly1 and poly2 upon polynomial division by f. finv is the inverse of the reverse of f.

## Squaring¶

void _fq_poly_sqr_classical(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets (rop, 2*len - 1) to the square of (op, len), assuming that (op,len) is not zero and using classical polynomial multiplication.

Permits zero padding. Does not support aliasing of rop with either op1 or op2.

void fq_poly_sqr_classical(fq_poly_t rop, const fq_poly_t op, const fq_ctx_t ctx)
Sets rop to the square of op using classical
polynomial multiplication.
void _fq_poly_sqr_reorder(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets (rop, 2*len- 1) to the square of (op, len), assuming that len is not zero reordering the two indeterminates $$X$$ and $$Y$$ when viewing the polynomials as elements of $$\mathbf{F}_p[X,Y]$$.

void fq_poly_sqr_reorder(fq_poly_t rop, const fq_poly_t op, const fq_ctx_t ctx)

Sets rop to the square of op, assuming that len is not zero reordering the two indeterminates $$X$$ and $$Y$$ when viewing the polynomials as elements of $$\mathbf{F}_p[X,Y]$$. See fq_poly_mul_reorder.

void _fq_poly_sqr_KS(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets (rop, 2*len - 1) to the square of (op, len).

Permits zero padding and places no assumptions on the lengths len1 and len2. Supports aliasing.

void fq_poly_sqr_KS(fq_poly_t rop, const fq_poly_t op, const fq_ctx_t ctx)

Sets rop to the square op using Kronecker substitution, that is, by encoding each coefficient in $$\mathbf{F}_{q}$$ as an integer and reducing this problem to multiplying two polynomials over the integers.

void _fq_poly_sqr(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets (rop, 2* len - 1) to the square of (op, len), choosing an appropriate algorithm.

Permits zero padding. Does not support aliasing.

void fq_poly_sqr(fq_poly_t rop, const fq_poly_t op, const fq_ctx_t ctx)

Sets rop to the square of op, choosing an appropriate algorithm.

## Powering¶

void _fq_poly_pow(fq_struct *rop, const fq_struct *op, slong len, ulong e, const fq_ctx_t ctx)

Sets rop = op^e, assuming that e, len > 0 and that rop has space for e*(len - 1) + 1 coefficients. Does not support aliasing.

void fq_poly_pow(fq_poly_t rop, const fq_poly_t op, ulong e, const fq_ctx_t ctx)

Computes rop = op^e. If $$e$$ is zero, returns one, so that in particular 0^0 = 1.

void _fq_poly_powmod_ui_binexp(fq_struct* res, const fq_struct* poly, ulong e, const fq_struct* f, slong lenf, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e > 0.

We require lenf > 1. It is assumed that poly is already reduced modulo f and zero-padded as necessary to have length exactly lenf - 1. The output res must have room for lenf - 1 coefficients.

void fq_poly_powmod_ui_binexp(fq_poly_t res, const fq_poly_t poly, ulong e, const fq_poly_t f, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e >= 0.

void _fq_poly_powmod_ui_binexp_preinv(fq_struct* res, const fq_struct* poly, ulong e, const fq_struct* f, slong lenf, const fq_struct* finv, slong lenfinv, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e > 0. We require finv to be the inverse of the reverse of f.

We require lenf > 1. It is assumed that poly is already reduced modulo f and zero-padded as necessary to have length exactly lenf - 1. The output res must have room for lenf - 1 coefficients.

void fq_poly_powmod_ui_binexp_preinv(fq_poly_t res, const fq_poly_t poly, ulong e, const fq_poly_t f, const fq_poly_t finv, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e >= 0. We require finv to be the inverse of the reverse of f.

void _fq_poly_powmod_fmpz_binexp(fq_struct* res, const fq_struct* poly, fmpz_t e, const fq_struct* f, slong lenf, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e > 0.

We require lenf > 1. It is assumed that poly is already reduced modulo f and zero-padded as necessary to have length exactly lenf - 1. The output res must have room for lenf - 1 coefficients.

void fq_poly_powmod_fmpz_binexp(fq_poly_t res, const fq_poly_t poly, fmpz_t e, const fq_poly_t f, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e >= 0.

void _fq_poly_powmod_fmpz_binexp_preinv(fq_struct* res, const fq_struct* poly, fmpz_t e, const fq_struct* f, slong lenf, const fq_struct* finv, slong lenfinv, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e > 0. We require finv to be the inverse of the reverse of f.

We require lenf > 1. It is assumed that poly is already reduced modulo f and zero-padded as necessary to have length exactly lenf - 1. The output res must have room for lenf - 1 coefficients.

void fq_poly_powmod_fmpz_binexp_preinv(fq_poly_t res, const fq_poly_t poly, fmpz_t e, const fq_poly_t f, const fq_poly_t finv, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e >= 0. We require finv to be the inverse of the reverse of f.

void _fq_poly_powmod_fmpz_sliding_preinv(fq_struct* res, const fq_struct* poly, fmpz_t e, ulong k, const fq_struct* f, slong lenf, const fq_struct* finv, slong lenfinv, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using sliding-window exponentiation with window size k. We require e > 0. We require finv to be the inverse of the reverse of f. If k is set to zero, then an “optimum” size will be selected automatically base on e.

We require lenf > 1. It is assumed that poly is already reduced modulo f and zero-padded as necessary to have length exactly lenf - 1. The output res must have room for lenf - 1 coefficients.

void fq_poly_powmod_fmpz_sliding_preinv(fq_poly_t res, const fq_poly_t poly, fmpz_t e, ulong k, const fq_poly_t f, const fq_poly_t finv, const fq_ctx_t ctx)

Sets res to poly raised to the power e modulo f, using sliding-window exponentiation with window size k. We require e >= 0. We require finv to be the inverse of the reverse of f. If k is set to zero, then an “optimum” size will be selected automatically base on e.

void _fq_poly_powmod_x_fmpz_preinv(fq_struct * res, const fmpz_t e, const fq_struct * f, slong lenf, const fq_struct * finv, slong lenfinv, const fq_ctx_t ctx)

Sets res to x raised to the power e modulo f, using sliding window exponentiation. We require e > 0. We require finv to be the inverse of the reverse of f.

We require lenf > 2. The output res must have room for lenf - 1 coefficients.

void fq_poly_powmod_x_fmpz_preinv(fq_poly_t res, const fmpz_t e, const fq_poly_t f, const fq_poly_t finv, const fq_ctx_t ctx)

Sets res to x raised to the power e modulo f, using sliding window exponentiation. We require e >= 0. We require finv to be the inverse of the reverse of f.

void _fq_poly_pow_trunc_binexp(fq_struct * res, const fq_struct * poly, ulong e, slong trunc, const fq_ctx_t ctx)

Sets res to the low trunc coefficients of poly (assumed to be zero padded if necessary to length trunc) to the power e. This is equivalent to doing a powering followed by a truncation. We require that res has enough space for trunc coefficients, that trunc > 0 and that e > 1. Aliasing is not permitted. Uses the binary exponentiation method.

void fq_poly_pow_trunc_binexp(fq_poly_t res, const fq_poly_t poly, ulong e, slong trunc, const fq_ctx_t ctx)

Sets res to the low trunc coefficients of poly to the power e. This is equivalent to doing a powering followed by a truncation. Uses the binary exponentiation method.

void _fq_poly_pow_trunc(fq_struct * res, const fq_struct * poly, ulong e, slong trunc, const fq_ctx_t mod)

Sets res to the low trunc coefficients of poly (assumed to be zero padded if necessary to length trunc) to the power e. This is equivalent to doing a powering followed by a truncation. We require that res has enough space for trunc coefficients, that trunc > 0 and that e > 1. Aliasing is not permitted.

void fq_poly_pow_trunc(fq_poly_t res, const fq_poly_t poly, ulong e, slong trunc, fq_ctx_t ctx)

Sets res to the low trunc coefficients of poly to the power e. This is equivalent to doing a powering followed by a truncation.

## Shifting¶

void _fq_poly_shift_left(fq_struct *rop, const fq_struct *op, slong len, slong n, const fq_ctx_t ctx)

Sets (rop, len + n) to (op, len) shifted left by $$n$$ coefficients.

Inserts zero coefficients at the lower end. Assumes that len and $$n$$ are positive, and that rop fits len + n elements. Supports aliasing between rop and op.

void fq_poly_shift_left(fq_poly_t rop, const fq_poly_t op, slong n, const fq_ctx_t ctx)

Sets rop to op shifted left by $$n$$ coeffs. Zero coefficients are inserted.

void _fq_poly_shift_right(fq_struct *rop, const fq_struct *op, slong len, slong n, const fq_ctx_t ctx)

Sets (rop, len - n) to (op, len) shifted right by $$n$$ coefficients.

Assumes that len and $$n$$ are positive, that len > n, and that rop fits len - n elements. Supports aliasing between rop and op, although in this case the top coefficients of op are not set to zero.

void fq_poly_shift_right(fq_poly_t rop, const fq_poly_t op, slong n, const fq_ctx_t ctx)

Sets rop to op shifted right by $$n$$ coefficients. If $$n$$ is equal to or greater than the current length of op, rop is set to the zero polynomial.

## Norms¶

long _fq_poly_hamming_weight(const fq_poly *op, slong len, const fq_ctx_t ctx)

Returns the number of non-zero entries in (op, len).

long fq_poly_hamming_weight(const fq_poly_t op, const fq_ctx_t ctx)

Returns the number of non-zero entries in the polynomial op.

## Euclidean division¶

void _fq_poly_divrem_basecase(fq_struct *Q, fq_struct *R, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Computes (Q, lenA - lenB + 1), (R, lenA) such that $$A = B Q + R$$ with $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that the leading coefficient of $$B$$ is invertible and that invB is its inverse.

Assumes that $$\operatorname{len}(A), \operatorname{len}(B) > 0$$. Allows zero-padding in (A, lenA). $$R$$ and $$A$$ may be aliased, but apart from this no aliasing of input and output operands is allowed.

void fq_poly_divrem_basecase(fq_poly_t Q, fq_poly_t R, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Computes $$Q$$, $$R$$ such that $$A = B Q + R$$ with $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that the leading coefficient of $$B$$ is invertible. This can be taken for granted the context is for a finite field, that is, when $$p$$ is prime and $$f(X)$$ is irreducible.

void _fq_poly_divrem(fq_struct *Q, fq_struct *R, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Computes (Q, lenA - lenB + 1), (R, lenA) such that $$A = B Q + R$$ with $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that the leading coefficient of $$B$$ is invertible and that invB is its inverse.

Assumes that $$\operatorname{len}(A), \operatorname{len}(B) > 0$$. Allows zero-padding in (A, lenA). $$R$$ and $$A$$ may be aliased, but apart from this no aliasing of input and output operands is allowed.

void fq_poly_divrem(fq_poly_t Q, fq_poly_t R, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Computes $$Q$$, $$R$$ such that $$A = B Q + R$$ with $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that the leading coefficient of $$B$$ is invertible. This can be taken for granted the context is for a finite field, that is, when $$p$$ is prime and $$f(X)$$ is irreducible.

void fq_poly_divrem_f(fq_t f, fq_poly_t Q, fq_poly_t R, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Either finds a non-trivial factor $$f$$ of the modulus of ctx, or computes $$Q$$, $$R$$ such that $$A = B Q + R$$ and $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

If the leading coefficient of $$B$$ is invertible, the division with remainder operation is carried out, $$Q$$ and $$R$$ are computed correctly, and $$f$$ is set to $$1$$. Otherwise, $$f$$ is set to a non-trivial factor of the modulus and $$Q$$ and $$R$$ are not touched.

Assumes that $$B$$ is non-zero.

void _fq_poly_rem(fq_struct *R, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Sets R to the remainder of the division of (A,lenA) by (B,lenB). Assumes that the leading coefficient of (B,lenB) is invertible and that invB is its inverse.

void fq_poly_rem(fq_poly_t R, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Sets R to the remainder of the division of A by B in the context described by ctx.

void _fq_poly_div_basecase(fq_struct *Q, fq_struct *R, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Notationally, computes $$Q$$, $$R$$ such that $$A = B Q + R$$ with $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$ but only sets (Q, lenA - lenB + 1).

Requires temporary space (R, lenA). If R is NULL, then the temporary space will be allocated. Allows aliasing only between $$A$$ and $$R$$. Allows zero-padding in $$A$$ but not in $$B$$. Assumes that the leading coefficient of $$B$$ is a unit.

void fq_poly_div_basecase(fq_poly_t Q, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Notionally finds polynomials $$Q$$ and $$R$$ such that $$A = B Q + R$$ with $$\operatorname{len}(R) < \operatorname{len}(B)$$, but returns only Q. If $$\operatorname{len}(B) = 0$$ an exception is raised.

void _fq_poly_divrem_divconquer_recursive(fq_struct * Q, fq_struct * BQ, fq_struct * W, const fq_struct * A, const fq_struct * B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Computes (Q, lenB), (BQ, 2 lenB - 1) such that $$BQ = B \times Q$$ and $$A = B Q + R$$ where $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that the leading coefficient of $$B$$ is invertible and that invB is the inverse.

Assumes $$\operatorname{len}(B) > 0$$. Allows zero-padding in (A, lenA). Requires a temporary array (W, 2 lenB - 1). No aliasing of input and output operands is allowed.

This function does not read the bottom $$\operatorname{len}(B) - 1$$ coefficients from $$A$$, which means that they might not even need to exist in allocated memory.

void _fq_poly_divrem_divconquer(fq_struct * Q, fq_struct * R, const fq_struct * A, slong lenA, const fq_struct * B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Computes (Q, lenA - lenB + 1), (R, lenA) such that $$A = B Q + R$$ and $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that the leading coefficient of $$B$$ is invertible and that invB is the inverse.

Assumes $$\operatorname{len}(A) \geq \operatorname{len}(B) > 0$$. Allows zero-padding in (A, lenA). No aliasing of input and output operands is allowed.

void fq_poly_divrem_divconquer(fq_poly_t Q, fq_poly_t R, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Computes $$Q$$, $$R$$ such that $$A = B Q + R$$ and $$0 \leq \operatorname{len}(R) < \operatorname{len}(B)$$.

Assumes that $$B$$ is non-zero and that the leading coefficient of $$B$$ is invertible.

void _fq_poly_div_newton_n_preinv(fq_struct* Q, const fq_struct* A, slong lenA, const fq_struct* B, slong lenB, const fq_struct* Binv, slong lenBinv, const fq_struct ctx_t)

Notionally computes polynomials $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R)$$ less than lenB, where A is of length lenA and B is of length lenB, but return only $$Q$$.

We require that $$Q$$ have space for lenA - lenB + 1 coefficients and assume that the leading coefficient of $$B$$ is a unit. Furthermore, we assume that $$Binv$$ is the inverse of the reverse of $$B$$ mod $$x^{\operatorname{len}(B)}$$.

The algorithm used is to reverse the polynomials and divide the resulting power series, then reverse the result.

void fq_poly_div_newton_n_preinv(fq_poly_t Q, const fq_poly_t A, const fq_poly_t B, const fq_poly_t Binv, const fq_ctx_t ctx)

Notionally computes $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R) < \operatorname{len}(B)$$, but returns only $$Q$$.

We assume that the leading coefficient of $$B$$ is a unit and that $$Binv$$ is the inverse of the reverse of $$B$$ mod $$x^{\operatorname{len}(B)}$$.

It is required that the length of $$A$$ is less than or equal to 2*the length of $$B$$ - 2.

The algorithm used is to reverse the polynomials and divide the resulting power series, then reverse the result.

void _fq_poly_divrem_newton_n_preinv(fq_struct* Q, fq_struct* R, const fq_struct* A, slong lenA, const fq_struct* B, slong lenB, const fq_struct* Binv, slong lenBinv, const fq_ctx_t ctx)

Computes $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R)$$ less than lenB, where $$A$$ is of length lenA and $$B$$ is of length lenB. We require that $$Q$$ have space for lenA - lenB + 1 coefficients. Furthermore, we assume that $$Binv$$ is the inverse of the reverse of $$B$$ mod $$x^{\operatorname{len}(B)}$$. The algorithm used is to call div_newton_n_preinv() and then multiply out and compute the remainder.

void fq_poly_divrem_newton_preinv(fq_poly_t Q, fq_poly_t R, const fq_poly_t A, const fq_poly_t B, const fq_poly_t Binv, const fq_ctx_t ctx)

Computes $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R) < \operatorname{len}(B)$$. We assume $$Binv$$ is the inverse of the reverse of $$B$$ mod $$x^{\operatorname{len}(B)}$$.

It is required that the length of $$A$$ is less than or equal to 2*the length of $$B$$ - 2.

The algorithm used is to call div_newton_n() and then multiply out and compute the remainder.

void _fq_poly_inv_series_newton(fq_struct* Qinv, const fq_struct* Q, slong n, const fq_ctx_t ctx)

Given Q of length n whose constant coefficient is invertible modulo the given modulus, find a polynomial Qinv of length n such that Q * Qinv is 1 modulo $$x^n$$. Requires n > 0. This function can be viewed as inverting a power series via Newton iteration.

void fq_poly_inv_series_newton(fq_poly_t Qinv, const fq_poly_t Q, slong n, const fq_ctx_t ctx)

Given Q find Qinv such that Q * Qinv is 1 modulo $$x^n$$. The constant coefficient of Q must be invertible modulo the modulus of Q. An exception is raised if this is not the case or if n = 0. This function can be viewed as inverting a power series via Newton iteration.

void _fq_poly_inv_series(fq_struct* Qinv, const fq_struct* Q, slong n, const fq_ctx_t ctx)

Given Q of length n whose constant coefficient is invertible modulo the given modulus, find a polynomial Qinv of length n such that Q * Qinv is 1 modulo $$x^n$$. Requires n > 0.

void fq_poly_inv_series(fq_poly_t Qinv, const fq_poly_t Q, slong n, const fq_ctx_t ctx)

Given Q find Qinv such that Q * Qinv is 1 modulo $$x^n$$. The constant coefficient of Q must be invertible modulo the modulus of Q. An exception is raised if this is not the case or if n = 0.

void _fq_poly_div_series(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n, fq_ctx_t ctx)

Set (Q, n) to the quotient of the series (A, Alen) and (B, Blen) assuming Alen, Blen <= n. We assume the bottom coefficient of B is invertible.

void fq_poly_div_series(fmpz_mod_poly_t Q, const fmpz_mod_poly_t A, const fmpz_mod_poly_t B, slong n, fq_ctx_t ctx)

Set $$Q$$ to the quotient of the series $$A$$ by $$B$$, thinking of the series as though they were of length $$n$$. We assume that the bottom coefficient of $$B$$ is invertible.

## Greatest common divisor¶

void fq_poly_gcd(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the greatest common divisor of op1 and op2, using the either the Euclidean or HGCD algorithm. The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial $$P$$ is defined to be $$P$$. Except in the case where the GCD is zero, the GCD $$G$$ is made monic.

long _fq_poly_gcd(fq_struct* G, const fq_struct* A, slong lenA, const fq_struct* B, slong lenB, const fq_ctx_t ctx)

Computes the GCD of $$A$$ of length lenA and $$B$$ of length lenB, where lenA >= lenB > 0 and sets $$G$$ to it. The length of the GCD $$G$$ is returned by the function. No attempt is made to make the GCD monic. It is required that $$G$$ have space for lenB coefficients.

void fq_poly_gcd_euclidean(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the greatest common divisor of op1 and op2, using the Euclidean algorithm. The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial $$P$$ is defined to be $$P$$. Except in the case where the GCD is zero, the GCD $$G$$ is made monic.

long _fq_poly_gcd_euclidean(fq_struct* G, const fq_struct* A, slong lenA, const fq_struct* B, slong lenB, const fq_ctx_t ctx)

Computes the GCD of $$A$$ of length lenA and $$B$$ of length lenB, where lenA >= lenB > 0 and sets $$G$$ to it. The length of the GCD $$G$$ is returned by the function. No attempt is made to make the GCD monic. It is required that $$G$$ have space for lenB coefficients.

slong _fq_poly_hgcd(fq_struct **M, slong *lenM, fq_struct *A, slong *lenA, fq_struct *B, slong *lenB, const fq_struct * a, slong lena, const fq_struct *b, slong lenb, const fq_ctx_t ctx)

Computes the HGCD of $$a$$ and $$b$$, that is, a matrix $$M$$, a sign $$\sigma$$ and two polynomials $$A$$ and $$B$$ such that

$(A,B)^t = \sigma M^{-1} (a,b)^t.$

Assumes that $$\operatorname{len}(a) > \operatorname{len}(b) > 0$$.

Assumes that $$A$$ and $$B$$ have space of size at least $$\operatorname{len}(a)$$ and $$\operatorname{len}(b)$$, respectively. On exit, *lenA and *lenB will contain the correct lengths of $$A$$ and $$B$$.

Assumes that M[0], M[1], M[2], and M[3] each point to a vector of size at least $$\operatorname{len}(a)$$.

void fq_poly_gcd_hgcd(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the greatest common divisor of op1 and op2, using the HGCD algorithm. The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial $$P$$ is defined to be $$P$$. Except in the case where the GCD is zero, the GCD $$G$$ is made monic.

long _fq_poly_gcd_hgcd(fq_struct* G, const fq_struct* A, slong lenA, const fq_struct* B, slong lenB, const fq_ctx_t ctx)

Computes the GCD of $$A$$ of length lenA and $$B$$ of length lenB using the HGCD algorithm, where lenA >= lenB > 0 and sets $$G$$ to it. The length of the GCD $$G$$ is returned by the function. No attempt is made to make the GCD monic. It is required that $$G$$ have space for lenB coefficients.

slong _fq_poly_gcd_euclidean_f(fq_t f, fq_struct *G, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fq_ctx_t ctx)

Either sets $$f = 1$$ and $$G$$ to the greatest common divisor of $$(A,\operatorname{len}(A))$$ and $$(B, \operatorname{len}(B))$$ and returns its length, or sets $$f$$ to a non-trivial factor of the modulus of ctx and leaves the contents of the vector $$(G, lenB)$$ undefined.

Assumes that $$\operatorname{len}(A) \geq \operatorname{len}(B) > 0$$ and that the vector $$G$$ has space for sufficiently many coefficients.

void fq_poly_gcd_euclidean_f(fq_t f, fq_poly_t G, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Either sets $$f = 1$$ and $$G$$ to the greatest common divisor of $$A$$ and $$B$$ or sets $$f$$ to a factor of the modulus of ctx.

slong _fq_poly_xgcd_euclidean(fq_struct *G, fq_struct *S, fq_struct *T, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fmpz_t invB, const fq_ctx_t ctx)

Computes the GCD of $$A$$ and $$B$$ together with cofactors $$S$$ and $$T$$ such that $$S A + T B = G$$. Returns the length of $$G$$.

Assumes that $$\operatorname{len}(A) \geq \operatorname{len}(B) \geq 1$$ and $$(\operatorname{len}(A),\operatorname{len}(B)) \neq (1,1)$$.

No attempt is made to make the GCD monic.

Requires that $$G$$ have space for $$\operatorname{len}(B)$$ coefficients. Writes $$\operatorname{len}(B)-1$$ and $$\operatorname{len}(A)-1$$ coefficients to $$S$$ and $$T$$, respectively. Note that, in fact, $$\operatorname{len}(S) \leq \max(\operatorname{len}(B) - \operatorname{len}(G), 1)$$ and $$\operatorname{len}(T) \leq \max(\operatorname{len}(A) - \operatorname{len}(G), 1)$$.

No aliasing of input and output operands is permitted.

void fq_poly_xgcd_euclidean(fq_poly_t G, fq_poly_t S, fq_poly_t T, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Computes the GCD of $$A$$ and $$B$$. The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial $$P$$ is defined to be $$P$$. Except in the case where the GCD is zero, the GCD $$G$$ is made monic.

Polynomials S and T are computed such that S*A + T*B = G. The length of S will be at most lenB and the length of T will be at most lenA.

slong _fq_poly_xgcd(fq_struct *G, fq_struct *S, fq_struct *T, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fmpz_t invB, const fq_ctx_t ctx)

Computes the GCD of $$A$$ and $$B$$ together with cofactors $$S$$ and $$T$$ such that $$S A + T B = G$$. Returns the length of $$G$$.

Assumes that $$\operatorname{len}(A) \geq \operatorname{len}(B) \geq 1$$ and $$(\operatorname{len}(A),\operatorname{len}(B)) \neq (1,1)$$.

No attempt is made to make the GCD monic.

Requires that $$G$$ have space for $$\operatorname{len}(B)$$ coefficients. Writes $$\operatorname{len}(B)-1$$ and $$\operatorname{len}(A)-1$$ coefficients to $$S$$ and $$T$$, respectively. Note that, in fact, $$\operatorname{len}(S) \leq \max(\operatorname{len}(B) - \operatorname{len}(G), 1)$$ and $$\operatorname{len}(T) \leq \max(\operatorname{len}(A) - \operatorname{len}(G), 1)$$.

No aliasing of input and output operands is permitted.

void fq_poly_xgcd(fq_poly_t G, fq_poly_t S, fq_poly_t T, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Computes the GCD of $$A$$ and $$B$$. The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial $$P$$ is defined to be $$P$$. Except in the case where the GCD is zero, the GCD $$G$$ is made monic.

Polynomials S and T are computed such that S*A + T*B = G. The length of S will be at most lenB and the length of T will be at most lenA.

slong _fq_poly_xgcd_euclidean_f(fq_t f, fq_struct *G, fq_struct *S, fq_struct *T, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fmpz_t invB, const fq_ctx_t ctx)

Either sets $$f = 1$$ and computes the GCD of $$A$$ and $$B$$ together with cofactors $$S$$ and $$T$$ such that $$S A + T B = G$$; otherwise, sets $$f$$ to a non-trivial factor of the modulus of ctx and leaves $$G$$, $$S$$, and $$T$$ undefined. Returns the length of $$G$$.

Assumes that $$\operatorname{len}(A) \geq \operatorname{len}(B) \geq 1$$ and $$(\operatorname{len}(A),\operatorname{len}(B)) \neq (1,1)$$.

No attempt is made to make the GCD monic.

Requires that $$G$$ have space for $$\operatorname{len}(B)$$ coefficients. Writes $$\operatorname{len}(B)-1$$ and $$\operatorname{len}(A)-1$$ coefficients to $$S$$ and $$T$$, respectively. Note that, in fact, $$\operatorname{len}(S) \leq \max(\operatorname{len}(B) - \operatorname{len}(G), 1)$$ and $$\operatorname{len}(T) \leq \max(\operatorname{len}(A) - \operatorname{len}(G), 1)$$.

No aliasing of input and output operands is permitted.

void fq_poly_xgcd_euclidean_f(fq_t f, fq_poly_t G, fq_poly_t S, fq_poly_t T, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Either sets $$f = 1$$ and computes the GCD of $$A$$ and $$B$$ or sets $$f$$ to a non-trivial factor of the modulus of ctx.

If the GCD is computed, polynomials S and T are computed such that S*A + T*B = G; otherwise, they are undefined. The length of S will be at most lenB and the length of T will be at most lenA.

The GCD of zero polynomials is defined to be zero, whereas the GCD of the zero polynomial and some other polynomial $$P$$ is defined to be $$P$$. Except in the case where the GCD is zero, the GCD $$G$$ is made monic.

## Divisibility testing¶

int _fq_poly_divides(fq_struct *Q, const fq_struct *A, slong lenA, const fq_struct *B, slong lenB, const fq_t invB, const fq_ctx_t ctx)

Returns $$1$$ if (B, lenB) divides (A, lenA) exactly and sets $$Q$$ to the quotient, otherwise returns $$0$$.

It is assumed that $$\operatorname{len}(A) \geq \operatorname{len}(B) > 0$$ and that $$Q$$ has space for $$\operatorname{len}(A) - \operatorname{len}(B) + 1$$ coefficients.

Aliasing of $$Q$$ with either of the inputs is not permitted.

This function is currently unoptimised and provided for convenience only.

int fq_poly_divides(fq_poly_t Q, const fq_poly_t A, const fq_poly_t B, const fq_ctx_t ctx)

Returns $$1$$ if $$B$$ divides $$A$$ exactly and sets $$Q$$ to the quotient, otherwise returns $$0$$.

This function is currently unoptimised and provided for convenience only.

## Derivative¶

void _fq_poly_derivative(fq_struct *rop, const fq_struct *op, slong len, const fq_ctx_t ctx)

Sets (rop, len - 1) to the derivative of (op, len). Also handles the cases where len is $$0$$ or $$1$$ correctly. Supports aliasing of rop and op.

void fq_poly_derivative(fq_poly_t rop, const fq_poly_t op, const fq_ctx_t ctx)

Sets rop to the derivative of op.

## Evaluation¶

void _fq_poly_evaluate_fq(fq_t rop, const fq_struct *op, slong len, const fq_t a, const fq_ctx_t ctx)

Sets rop to (op, len) evaluated at $$a$$.

Supports zero padding. There are no restrictions on len, that is, len is allowed to be zero, too.

void fq_poly_evaluate_fq(fq_t rop, const fq_poly_t f, const fq_t a, const fq_ctx_t ctx)

Sets rop to the value of $$f(a)$$.

As the coefficient ring $$\mathbf{F}_q$$ is finite, Horner’s method is sufficient.

## Composition¶

void _fq_poly_compose_divconquer(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Computes the composition of (op1, len1) and (op2, len2) using a divide and conquer approach and places the result into rop, assuming rop can hold the output of length (len1 - 1) * (len2 - 1) + 1.

Assumes len1, len2 > 0. Does not support aliasing between rop and any of (op1, len1) and (op2, len2).

void fq_poly_compose_divconquer(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the composition of op1 and op2. To be precise about the order of composition, denoting rop, op1, and op2 by $$f$$, $$g$$, and $$h$$, respectively, sets $$f(t) = g(h(t))$$.

void _fq_poly_compose_horner(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets rop to the composition of (op1, len1) and (op2, len2).

Assumes that rop has space for (len1-1)*(len2-1) + 1 coefficients. Assumes that op1 and op2 are non-zero polynomials. Does not support aliasing between any of the inputs and the output.

void fq_poly_compose_horner(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the composition of op1 and op2. To be more precise, denoting rop, op1, and op2 by $$f$$, $$g$$, and $$h$$, sets $$f(t) = g(h(t))$$.

This implementation uses Horner’s method.

void _fq_poly_compose(fq_struct *rop, const fq_struct *op1, slong len1, const fq_struct *op2, slong len2, const fq_ctx_t ctx)

Sets rop to the composition of (op1, len1) and (op2, len2).

Assumes that rop has space for (len1-1)*(len2-1) + 1 coefficients. Assumes that op1 and op2 are non-zero polynomials. Does not support aliasing between any of the inputs and the output.

void fq_poly_compose(fq_poly_t rop, const fq_poly_t op1, const fq_poly_t op2, const fq_ctx_t ctx)

Sets rop to the composition of op1 and op2. To be precise about the order of composition, denoting rop, op1, and op2 by $$f$$, $$g$$, and $$h$$, respectively, sets $$f(t) = g(h(t))$$.

void _fq_poly_compose_mod_horner(fq_struct * res, const fq_struct * f, slong lenf, const fq_struct * g, const fq_struct * h, slong lenh, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that the length of $$g$$ is one less than the length of $$h$$ (possibly with zero padding). The output is not allowed to be aliased with any of the inputs.

The algorithm used is Horner’s rule.

void fq_poly_compose_mod_horner(fq_poly_t res, const fq_poly_t f, const fq_poly_t g, const fq_poly_t h, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero. The algorithm used is Horner’s rule.

void _fq_poly_compose_mod_horner_preinv(fq_struct * res, const fq_struct * f, slong lenf, const fq_struct * g, const fq_struct * h, slong lenh, const fq_struct * hinv, slong lenhiv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that the length of $$g$$ is one less than the length of $$h$$ (possibly with zero padding). We also require that the length of $$f$$ is less than the length of $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

The algorithm used is Horner’s rule.

void fq_poly_compose_mod_horner_preinv(fq_poly_t res, const fq_poly_t f, const fq_poly_t g, const fq_poly_t h, const fq_poly_t hinv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that $$f$$ has smaller degree than $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. The algorithm used is Horner’s rule.

void _fq_poly_compose_mod_brent_kung(fq_struct * res, const fq_struct * f, slong lenf, const fq_struct * g, const fq_struct * h, slong lenh, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that the length of $$g$$ is one less than the length of $$h$$ (possibly with zero padding). We also require that the length of $$f$$ is less than the length of $$h$$. The output is not allowed to be aliased with any of the inputs.

The algorithm used is the Brent-Kung matrix algorithm.

void fq_poly_compose_mod_brent_kung(fq_poly_t res, const fq_poly_t f, const fq_poly_t g, const fq_poly_t h, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that $$f$$ has smaller degree than $$h$$. The algorithm used is the Brent-Kung matrix algorithm.

void _fq_poly_compose_mod_brent_kung_preinv(fq_struct * res, const fq_struct * f, slong lenf, const fq_struct * g, const fq_struct * h, slong lenh, const fq_struct * hinv, slong lenhiv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that the length of $$g$$ is one less than the length of $$h$$ (possibly with zero padding). We also require that the length of $$f$$ is less than the length of $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

The algorithm used is the Brent-Kung matrix algorithm.

void fq_poly_compose_mod_brent_kung_preinv(fq_poly_t res, const fq_poly_t f, const fq_poly_t g, const fq_poly_t h, const fq_poly_t hinv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that $$f$$ has smaller degree than $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. The algorithm used is the Brent-Kung matrix algorithm.

void _fq_poly_compose_mod(fq_struct * res, const fq_struct * f, slong lenf, const fq_struct * g, const fq_struct * h, slong lenh, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that the length of $$g$$ is one less than the length of $$h$$ (possibly with zero padding). The output is not allowed to be aliased with any of the inputs.

void fq_poly_compose_mod(fq_poly_t res, const fq_poly_t f, const fq_poly_t g, const fq_poly_t h, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero.

void _fq_poly_compose_mod_preinv(fq_struct * res, const fq_struct * f, slong lenf, const fq_struct * g, const fq_struct * h, slong lenh, const fq_struct * hinv, slong lenhiv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that the length of $$g$$ is one less than the length of $$h$$ (possibly with zero padding). We also require that the length of $$f$$ is less than the length of $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

void fq_poly_compose_mod_preinv(fq_poly_t res, const fq_poly_t f, const fq_poly_t g, const fq_poly_t h, const fq_poly_t hinv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero and that $$f$$ has smaller degree than $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h.

void _fq_poly_reduce_matrix_mod_poly(fq_mat_t A, const fq_mat_t B, const fq_poly_t f, const fq_ctx_t ctx)

Sets the ith row of A to the reduction of the ith row of $$B$$ modulo $$f$$ for $$i=1,\ldots,\sqrt{\deg(f)}$$. We require $$B$$ to be at least a $$\sqrt{\deg(f)}\times \deg(f)$$ matrix and $$f$$ to be nonzero.

void _fq_poly_precompute_matrix(fq_mat_t A, const fq_struct* f, const fq_struct* g, slong leng, const fq_struct* ginv, slong lenginv, const fq_ctx_t ctx)

Sets the ith row of A to $$f^i$$ modulo $$g$$ for $$i=1,\ldots,\sqrt{\deg(g)}$$. We require $$A$$ to be a $$\sqrt{\deg(g)}\times \deg(g)$$ matrix. We require ginv to be the inverse of the reverse of g and $$g$$ to be nonzero.

void fq_poly_precompute_matrix(fq_mat_t A, const fq_poly_t f, const fq_poly_t g, const fq_poly_t ginv, const fq_ctx_t ctx)

Sets the ith row of A to $$f^i$$ modulo $$g$$ for $$i=1,\ldots,\sqrt{\deg(g)}$$. We require $$A$$ to be a $$\sqrt{\deg(g)}\times \deg(g)$$ matrix. We require ginv to be the inverse of the reverse of g.

void _fq_poly_compose_mod_brent_kung_precomp_preinv(fq_struct* res, const fq_struct* f, slong lenf, const fq_mat_t A, const fq_struct* h, slong lenh, const fq_struct* hinv, slong lenhinv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that $$h$$ is nonzero. We require that the ith row of $$A$$ contains $$g^i$$ for $$i=1,\ldots,\sqrt{\deg(h)}$$, i.e. $$A$$ is a $$\sqrt{\deg(h)}\times \deg(h)$$ matrix. We also require that the length of $$f$$ is less than the length of $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

The algorithm used is the Brent-Kung matrix algorithm.

void fq_poly_compose_mod_brent_kung_precomp_preinv(fq_poly_t res, const fq_poly_t f, const fq_mat_t A, const fq_poly_t h, const fq_poly_t hinv, const fq_ctx_t ctx)

Sets res to the composition $$f(g)$$ modulo $$h$$. We require that the ith row of $$A$$ contains $$g^i$$ for $$i=1,\ldots,\sqrt{\deg(h)}$$, i.e. $$A$$ is a $$\sqrt{\deg(h)}\times \deg(h)$$ matrix. We require that $$h$$ is nonzero and that $$f$$ has smaller degree than $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. This version of Brent-Kung modular composition is particularly useful if one has to perform several modular composition of the form $$f(g)$$ modulo $$h$$ for fixed $$g$$ and $$h$$.

## Output¶

int _fq_poly_fprint_pretty(FILE *file, const fq_struct *poly, slong len, const char *x, const fq_ctx_t ctx)

Prints the pretty representation of (poly, len) to the stream file, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_poly_fprint_pretty(FILE * file, const fq_poly_t poly, const char *x, const fq_ctx_t ctx)

Prints the pretty representation of poly to the stream file, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int _fq_poly_print_pretty(const fq_struct *poly, slong len, const char *x, const fq_ctx_t ctx)

Prints the pretty representation of (poly, len) to stdout, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_poly_print_pretty(const fq_poly_t poly, const char *x, const fq_ctx_t ctx)

Prints the pretty representation of poly to stdout, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int _fq_poly_fprint(FILE *file, const fq_struct *poly, slong len, const fq_ctx_t ctx)

Prints the pretty representation of (poly, len) to the stream file.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_poly_fprint(FILE * file, const fq_poly_t poly, const fq_ctx_t ctx)

Prints the pretty representation of poly to the stream file.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int _fq_poly_print(const fq_struct *poly, slong len, const fq_ctx_t ctx)

Prints the pretty representation of (poly, len) to stdout.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fq_poly_print(const fq_poly_t poly, const fq_ctx_t ctx)

Prints the representation of poly to stdout.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

char * _fq_poly_get_str(const fq_struct * poly, slong len, const fq_ctx_t ctx)

Returns the plain FLINT string representation of the polynomial (poly, len).

char * fq_poly_get_str(const fq_poly_t poly, const fq_ctx_t ctx)

Returns the plain FLINT string representation of the polynomial poly.

char * _fq_poly_get_str_pretty(const fq_struct * poly, slong len, const char * x, const fq_ctx_t ctx)

Returns a pretty representation of the polynomial (poly, len) using the null-terminated string x as the variable name.

char * fq_poly_get_str_pretty(const fq_poly_t poly, const char * x, const fq_ctx_t ctx)

Returns a pretty representation of the polynomial poly using the null-terminated string x as the variable name

## Inflation and deflation¶

void fq_poly_inflate(fq_poly_t result, const fq_poly_t input, ulong inflation, const fq_ctx_t ctx)

Sets result to the inflated polynomial $$p(x^n)$$ where $$p$$ is given by input and $$n$$ is given by inflation.

void fq_poly_deflate(fq_poly_t result, const fq_poly_t input, ulong deflation, const fq_ctx_t ctx)

Sets result to the deflated polynomial $$p(x^{1/n})$$ where $$p$$ is given by input and $$n$$ is given by deflation. Requires $$n > 0$$.

ulong fq_poly_deflation(const fq_poly_t input, const fq_ctx_t ctx)

Returns the largest integer by which input can be deflated. As special cases, returns 0 if input is the zero polynomial and 1 of input is a constant polynomial.