fmpz_mod_poly_init poly ctx

Initialises poly for use with context ctx and set it to zero. A corresponding call to fmpz_mod_poly_clear must be made to free the memory used by the polynomial.

fmpz_mod_poly_init2 poly alloc ctx

Initialises poly with space for at least alloc coefficients and sets the length to zero. The allocated coefficients are all set to zero.

fmpz_mod_poly_clear poly ctx

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

fmpz_mod_poly_realloc poly alloc 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.

fmpz_mod_poly_fit_length poly len 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 it 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.

_fmpz_mod_poly_normalise poly

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.

_fmpz_mod_poly_set_length poly len

Demotes the coefficients of poly beyond len and sets the length of poly to len.

fmpz_mod_poly_truncate poly len ctx

If the current length of poly is greater than len, it is truncated to have the given length. Discarded coefficients are not necessarily set to zero.

fmpz_mod_poly_set_trunc res poly n ctx

Notionally truncate poly to length \(n\) and set res to the result. The result is normalised.

fmpz_mod_poly_randtest f state len ctx

Sets the polynomial~`f` to a random polynomial of length up~len.

fmpz_mod_poly_randtest_irreducible f state len ctx

Sets the polynomial~`f` to a random irreducible polynomial of length up~len, assuming len is positive.

fmpz_mod_poly_randtest_not_zero f state len ctx

Sets the polynomial~`f` to a random polynomial of length up~len, assuming len is positive.

fmpz_mod_poly_randtest_monic poly state len ctx

Generates a random monic polynomial with length len.

fmpz_mod_poly_randtest_monic_irreducible poly state len ctx

Generates a random monic irreducible polynomial with length len.

fmpz_mod_poly_randtest_monic_primitive poly state len ctx

Generates a random monic irreducible primitive polynomial with length len.

fmpz_mod_poly_randtest_trinomial poly state len ctx

Generates a random monic trinomial of length len.

fmpz_mod_poly_randtest_trinomial_irreducible poly state len max_attempts ctx

Attempts to set poly to a monic irreducible trinomial of length len. It will generate up to max_attempts trinomials in attempt to find an irreducible one. If max_attempts is 0, then it will keep generating trinomials until an irreducible one is found. Returns \(1\) if one is found and \(0\) otherwise.

fmpz_mod_poly_randtest_pentomial poly state len ctx

Generates a random monic pentomial of length len.

fmpz_mod_poly_randtest_pentomial_irreducible poly state len max_attempts ctx

Attempts to set poly to a monic irreducible pentomial of length len. It will generate up to max_attempts pentomials in attempt to find an irreducible one. If max_attempts is 0, then it will keep generating pentomials until an irreducible one is found. Returns \(1\) if one is found and \(0\) otherwise.

fmpz_mod_poly_randtest_sparse_irreducible poly state len ctx

Attempts to set poly to a sparse, monic irreducible polynomial with length len. It attempts to find an irreducible trinomial. If that does not succeed, it attempts to find a irreducible pentomial. If that fails, then poly is just set to a random monic irreducible polynomial.

fmpz_mod_poly_degree poly ctx

Returns the degree of the polynomial. The degree of the zero polynomial is defined to be \(-1\).

fmpz_mod_poly_length poly ctx

Returns the length of the polynomial, which is one more than its degree.

fmpz_mod_poly_lead poly ctx

Returns a pointer to the first leading coefficient of poly if this is non-zero, otherwise returns NULL.

fmpz_mod_poly_set poly1 poly2 ctx

Sets the polynomial poly1 to the value of poly2.

fmpz_mod_poly_swap poly1 poly2 ctx

Swaps the two polynomials. This is done efficiently by swapping pointers rather than individual coefficients.

fmpz_mod_poly_zero poly ctx

Sets poly to the zero polynomial.

fmpz_mod_poly_one poly ctx

Sets poly to the constant polynomial \(1\).

fmpz_mod_poly_zero_coeffs poly i j ctx

Sets the coefficients of \(X^k\) for \(k \in [i, j)\) in the polynomial to zero.

fmpz_mod_poly_reverse res poly n ctx

This function considers the polynomial poly to be of length \(n\), notionally truncating and zero padding if required, and reverses the result. Since the function normalises its result res may be of length less than \(n\).

fmpz_mod_poly_set_ui f c ctx

Sets the polynomial \(f\) to the constant \(c\) reduced modulo \(p\).

fmpz_mod_poly_set_fmpz f c ctx

Sets the polynomial \(f\) to the constant \(c\) reduced modulo \(p\).

fmpz_mod_poly_set_fmpz_poly f g ctx

Sets \(f\) to \(g\) reduced modulo \(p\), where \(p\) is the modulus that is part of the data structure of \(f\).

fmpz_mod_poly_get_fmpz_poly f g ctx

Sets \(f\) to \(g\). This is done simply by lifting the coefficients of \(g\) taking representatives \([0, p) \subset \mathbf{Z}\).

fmpz_mod_poly_get_nmod_poly f g

Sets \(f\) to \(g\) assuming the modulus of both polynomials is the same (no checking is performed).

fmpz_mod_poly_set_nmod_poly f g

Sets \(f\) to \(g\) assuming the modulus of both polynomials is the same (no checking is performed).

fmpz_mod_poly_equal poly1 poly2 ctx

Returns non-zero if the two polynomials are equal, otherwise returns zero.

fmpz_mod_poly_equal_trunc poly1 poly2 n ctx

Notionally truncates the two polynomials to length \(n\) and returns non-zero if the two polynomials are equal, otherwise returns zero.

fmpz_mod_poly_is_zero poly ctx

Returns non-zero if the polynomial is zero.

fmpz_mod_poly_is_one poly ctx

Returns non-zero if the polynomial is the constant \(1\).

fmpz_mod_poly_is_gen poly ctx

Returns non-zero if the polynomial is the degree \(1\) polynomial \(x\).

fmpz_mod_poly_set_coeff_fmpz poly n x ctx

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

fmpz_mod_poly_set_coeff_ui poly n x ctx

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

fmpz_mod_poly_get_coeff_fmpz x poly n ctx

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

_fmpz_mod_poly_shift_left res poly len n ctx

Sets (res, len + n) to (poly, len) shifted left by \(n\) coefficients.

Inserts zero coefficients at the lower end. Assumes that len and \(n\) are positive, and that res fits len + n elements. Supports aliasing between res and poly.

fmpz_mod_poly_shift_left f g n ctx

Sets res to poly shifted left by \(n\) coeffs. Zero coefficients are inserted.

_fmpz_mod_poly_shift_right res poly len n ctx

Sets (res, len - n) to (poly, len) shifted right by \(n\) coefficients.

Assumes that len and \(n\) are positive, that len > n, and that res fits len - n elements. Supports aliasing between res and poly, although in this case the top coefficients of poly are not set to zero.

fmpz_mod_poly_shift_right f g n ctx

Sets res to poly shifted right by \(n\) coefficients. If \(n\) is equal to or greater than the current length of poly, res is set to the zero polynomial.

_fmpz_mod_poly_add res poly1 len1 poly2 len2 p

Sets res to the sum of (poly1, len1) and (poly2, len2). It is assumed that res has sufficient space for the longer of the two polynomials.

fmpz_mod_poly_add res poly1 poly2 ctx

Sets res to the sum of poly1 and poly2.

fmpz_mod_poly_add_series res poly1 poly2 n ctx

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

_fmpz_mod_poly_sub res poly1 len1 poly2 len2 p

Sets res to (poly1, len1) minus (poly2, len2). It is assumed that res has sufficient space for the longer of the two polynomials.

fmpz_mod_poly_sub res poly1 poly2 ctx

Sets res to poly1 minus poly2.

fmpz_mod_poly_sub_series res poly1 poly2 n ctx

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

_fmpz_mod_poly_neg res poly len p

Sets (res, len) to the negative of (poly, len) modulo \(p\).

fmpz_mod_poly_neg res poly ctx

Sets res to the negative of poly modulo \(p\).

_fmpz_mod_poly_scalar_mul_fmpz res poly len x p

Sets (res, len) to (poly, len) multiplied by \(x\), reduced modulo \(p\).

fmpz_mod_poly_scalar_mul_fmpz res poly x ctx

Sets res to poly multiplied by \(x\).

fmpz_mod_poly_scalar_addmul_fmpz rop op x ctx

Adds to rop the product of op by the scalar x.

_fmpz_mod_poly_scalar_div_fmpz res poly len x p

Sets (res, len) to (poly, len) divided by \(x\) (i.e. multiplied by the inverse of \(x \pmod{p}\)). The result is reduced modulo \(p\).

fmpz_mod_poly_scalar_div_fmpz res poly x ctx

Sets res to poly divided by \(x\), (i.e. multiplied by the inverse of \(x \pmod{p}\)). The result is reduced modulo \(p\).

_fmpz_mod_poly_mul res poly1 len1 poly2 len2 p

Sets (res, len1 + len2 - 1) to the product of (poly1, len1) and (poly2, len2). Assumes len1 >= len2 > 0. Allows zero-padding of the two input polynomials.

fmpz_mod_poly_mul res poly1 poly2 ctx

Sets res to the product of poly1 and poly2.

_fmpz_mod_poly_mullow res poly1 len1 poly2 len2 p n

Sets (res, n) to the lowest \(n\) coefficients of the product of (poly1, len1) and (poly2, len2).

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

fmpz_mod_poly_mullow res poly1 poly2 n ctx

Sets res to the lowest \(n\) coefficients of the product of poly1 and poly2.

_fmpz_mod_poly_sqr res poly len p

Sets res to the square of poly.

fmpz_mod_poly_sqr res poly ctx

Computes res as the square of poly.

fmpz_mod_poly_mulhigh res poly1 poly2 start 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.

_fmpz_mod_poly_mulmod res poly1 len1 poly2 len2 f lenf p

Sets res, len1 + len2 - 1 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 _fmpz_mod_poly_mul instead.

Aliasing of f and res is not permitted.

fmpz_mod_poly_mulmod res poly1 poly2 f ctx

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

_fmpz_mod_poly_mulmod_preinv res poly1 len1 poly2 len2 f lenf finv lenfinv p

Sets res, len1 + len2 - 1 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. It is required that len1 + len2 - lenf > 0, which is equivalent to requiring that the result will actually be reduced. It is required that len1 < lenf and len2 < lenf. Otherwise, simply use _fmpz_mod_poly_mul instead.

Aliasing of f or finv and res is not permitted.

fmpz_mod_poly_mulmod_preinv res poly1 poly2 f finv 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. It is required that poly1 and poly2 are reduced modulo f.

_fmpz_mod_poly_product_roots_fmpz_vec poly xs n f

Sets (poly, n + 1) to the monic polynomial which is the product of \((x - x_0)(x - x_1) \cdots (x - x_{n-1})\), the roots \(x_i\) being given by xs. It is required that the roots are canonical.

Aliasing of the input and output is not allowed.

fmpz_mod_poly_product_roots_fmpz_vec poly xs n f ctx

Sets poly to the monic polynomial which is the product of \((x - x_0)(x - x_1) \cdots (x - x_{n-1})\), the roots \(x_i\) being given by xs. It is required that the roots are canonical.

fmpz_mod_poly_find_distinct_nonzero_roots roots A ctx

If A has \(\deg(A)\) distinct nonzero roots in \(\mathbb{F}_p\), write these roots out to roots[0] to roots[deg(A) - 1] and return 1. Otherwise, return 0. It is assumed that A is nonzero and that the modulus of A is prime. This function uses Rabin's probabilistic method via gcd's with \((x + \delta)^{\frac{p-1}{2}} - 1\).

_fmpz_mod_poly_pow rop op len e p

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

fmpz_mod_poly_pow rop op e ctx

Computes rop = poly^e. If \(e\) is zero, returns one, so that in particular 0^0 = 1.

_fmpz_mod_poly_pow_trunc res poly e trunc p

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.

fmpz_mod_poly_pow_trunc res poly e trunc 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.

_fmpz_mod_poly_pow_trunc_binexp res poly e trunc p

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.

fmpz_mod_poly_pow_trunc_binexp res poly e trunc 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.

_fmpz_mod_poly_powmod_ui_binexp res poly e f lenf p

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.

fmpz_mod_poly_powmod_ui_binexp res poly e f ctx

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

_fmpz_mod_poly_powmod_ui_binexp_preinv res poly e f lenf finv lenfinv p

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.

fmpz_mod_poly_powmod_ui_binexp_preinv res poly e f finv 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.

_fmpz_mod_poly_powmod_fmpz_binexp res poly e f lenf p

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.

fmpz_mod_poly_powmod_fmpz_binexp res poly e f ctx

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

_fmpz_mod_poly_powmod_fmpz_binexp_preinv res poly e f lenf finv lenfinv p

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.

fmpz_mod_poly_powmod_fmpz_binexp_preinv res poly e f finv 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.

_fmpz_mod_poly_powmod_x_fmpz_preinv res e f lenf finv lenfinv p

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.

fmpz_mod_poly_powmod_x_fmpz_preinv res e f finv 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 ``

_fmpz_mod_poly_powers_mod_preinv_naive res f flen n g glen ginv ginvlen p

Compute f^0, f^1, ..., f^(n-1) mod g, where g has length glen and f is reduced mod g and has length flen (possibly zero spaced). Assumes res is an array of n arrays each with space for at least glen - 1 coefficients and that flen > 0. We require that ginv of length ginvlen is set to the power series inverse of the reverse of g.

fmpz_mod_poly_powers_mod_naive res f n g ctx

Set the entries of the array res to f^0, f^1, ..., f^(n-1) mod g. No aliasing is permitted between the entries of res and either of the inputs.

_fmpz_mod_poly_powers_mod_preinv_threaded_pool res f flen n g glen ginv ginvlen p threads num_threads

Compute f^0, f^1, ..., f^(n-1) mod g, where g has length glen and f is reduced mod g and has length flen (possibly zero spaced). Assumes res is an array of n arrays each with space for at least glen - 1 coefficients and that flen > 0. We require that ginv of length ginvlen is set to the power series inverse of the reverse of g.

fmpz_mod_poly_powers_mod_bsgs res f n g ctx

Set the entries of the array res to f^0, f^1, ..., f^(n-1) mod g. No aliasing is permitted between the entries of res and either of the inputs.

fmpz_mod_poly_frobenius_powers_2exp_precomp pow f finv m ctx

If p = f->p, compute \(x^{(p^1)}\), \(x^{(p^2)}\), \(x^{(p^4)}\), ..., \(x^{(p^{(2^l)})} \pmod{f}\) where \(2^l\) is the greatest power of \(2\) less than or equal to \(m\).

Allows construction of \(x^{(p^k)}\) for \(k = 0\), \(1\), ..., \(x^{(p^m)} \pmod{f}\) using fmpz_mod_poly_frobenius_power.

Requires precomputed inverse of \(f\), i.e. newton inverse.

fmpz_mod_poly_frobenius_powers_2exp_clear pow ctx

Clear resources used by the fmpz_mod_poly_frobenius_powers_2exp_t struct.

fmpz_mod_poly_frobenius_power res pow f m ctx

If p = f->p, compute \(x^{(p^m)} \pmod{f}\).

Requires precomputed frobenius powers supplied by fmpz_mod_poly_frobenius_powers_2exp_precomp.

If \(m == 0\) and \(f\) has degree \(0\) or \(1\), this performs a division. However an impossible inverse by the leading coefficient of \(f\) will have been caught by fmpz_mod_poly_frobenius_powers_2exp_precomp.

fmpz_mod_poly_frobenius_powers_precomp pow f finv m ctx

If p = f->p, compute \(x^{(p^0)}\), \(x^{(p^1)}\), \(x^{(p^2)}\), \(x^{(p^3)}\), ..., \(x^{(p^m)} \pmod{f}\).

Requires precomputed inverse of \(f\), i.e. newton inverse.

fmpz_mod_poly_frobenius_powers_clear pow ctx

Clear resources used by the fmpz_mod_poly_frobenius_powers_t struct.

_fmpz_mod_poly_divrem_basecase Q R A lenA B lenB invB p

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 modulo \(p\), and that invB is the 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.

fmpz_mod_poly_divrem_basecase Q R A B 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 modulo \(p\).

_fmpz_mod_poly_divrem_newton_n_preinv Q R A lenA B lenB Binv lenBinv mod

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.

fmpz_mod_poly_divrem_newton_n_preinv Q R A B Binv 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.

_fmpz_mod_poly_div_newton_n_preinv Q A lenA B lenB Binv lenBinv mod

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.

fmpz_mod_poly_div_newton_n_preinv Q A B Binv 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.

fmpz_mod_poly_remove f g ctx

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

_fmpz_mod_poly_rem_basecase R A lenA B lenB invB p

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

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 modulo \(p\).

fmpz_mod_poly_rem_basecase R A B ctx

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) with \(0 \leq \operatorname{len}(R) < \operatorname{len}(B)\) assuming that the leading term of \(B\) is a unit.

_fmpz_mod_poly_div Q A lenA B lenB p

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).

Assumes that the leading coefficient of \(B\) is a unit modulo \(p\).

fmpz_mod_poly_div Q A B ctx

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) with \(0 \leq \operatorname{len}(R) < \operatorname{len}(B)\) assuming that the leading term of \(B\) is a unit.

_fmpz_mod_poly_divrem Q R A lenA B lenB invB p

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

Assumes that \(B\) is non-zero, that the leading coefficient of \(B\) is invertible modulo \(p\) 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.

fmpz_mod_poly_divrem Q R A B 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 modulo \(p\).

fmpz_mod_poly_divrem_f f Q R A B ctx

Either finds a non-trivial factor~`f` of the modulus~`p`, 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 in \(\mathbf{Z}/(p)\), 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 \(p\) and \(Q\) and \(R\) are not touched.

Assumes that \(B\) is non-zero.

_fmpz_mod_poly_rem R A lenA B lenB invB p

Notationally, computes (Q, lenA - lenB + 1), (R, lenB - 1) such that \(A = B Q + R\) and \(0 \leq \operatorname{len}(R) < \operatorname{len}(B)\), returning only the remainder part.

Assumes that \(B\) is non-zero, that the leading coefficient of \(B\) is invertible modulo \(p\) 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.

_fmpz_mod_poly_divides_classical Q A lenA B lenB mod

Returns \(1\) if \((B, lenB)\) divides \((A, lenA)\) and sets \((Q, lenA - lenB + 1)\) to the quotient. Otherwise, returns \(0\) and sets \((Q, lenA - lenB + 1)\) to zero. We require that \(lenA >= lenB > 0\).

fmpz_mod_poly_divides_classical Q A B ctx

Returns \(1\) if \(B\) divides \(A\) and sets \(Q\) to the quotient. Otherwise returns \(0\) and sets \(Q\) to zero.

_fmpz_mod_poly_divides Q A lenA B lenB mod

Returns \(1\) if \((B, lenB)\) divides \((A, lenA)\) and sets \((Q, lenA - lenB + 1)\) to the quotient. Otherwise, returns \(0\) and sets \((Q, lenA - lenB + 1)\) to zero. We require that \(lenA >= lenB > 0\).

fmpz_mod_poly_divides Q A B ctx

Returns \(1\) if \(B\) divides \(A\) and sets \(Q\) to the quotient. Otherwise returns \(0\) and sets \(Q\) to zero.

_fmpz_mod_poly_inv_series Qinv Q n cinv p

Sets (Qinv, n) to the inverse of (Q, n) modulo \(x^n\), where \(n \geq 1\), assuming that the bottom coefficient of \(Q\) is invertible modulo \(p\) and that its inverse is cinv.

fmpz_mod_poly_inv_series Qinv Q n ctx

Sets Qinv to the inverse of Q modulo \(x^n\), where \(n \geq 1\), assuming that the bottom coefficient of \(Q\) is a unit.

fmpz_mod_poly_inv_series_f f Qinv Q n ctx

Either sets \(f\) to a nontrivial factor of \(p\) with the value of Qinv undefined, or sets Qinv to the inverse of Q modulo \(x^n\), where \(n \geq 1\).

_fmpz_mod_poly_div_series Q A Alen B Blen p n

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 modulo \(p\).

fmpz_mod_poly_div_series Q A B n 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 a unit.

fmpz_mod_poly_make_monic res poly ctx

If poly is non-zero, sets res to poly divided by its leading coefficient. This assumes that the leading coefficient of poly is invertible modulo \(p\).

Otherwise, if poly is zero, sets res to zero.

fmpz_mod_poly_make_monic_f f res poly ctx

Either set \(f\) to \(1\) and res to poly divided by its leading coefficient or set \(f\) to a nontrivial factor of \(p\) and leave res undefined.

_fmpz_mod_poly_gcd G A lenA B lenB invB p

Sets \(G\) to the greatest common divisor of \((A, \operatorname{len}(A))\) and \((B, \operatorname{len}(B))\) and returns its length.

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

Assumes that invB is the inverse of the leading coefficients of \(B\) modulo the prime number \(p\).

fmpz_mod_poly_gcd G A B ctx

Sets \(G\) to the greatest common divisor of \(A\) and \(B\).

In general, the greatest common divisor is defined in the polynomial ring \((\mathbf{Z}/(p \mathbf{Z}))[X]\) if and only if \(p\) is a prime number. Thus, this function assumes that \(p\) is prime.

_fmpz_mod_poly_gcd_euclidean_f f G A lenA B lenB p

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 \(f \in (1,p)\) to a non-trivial factor of \(p\) 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.

Does not support aliasing of any of the input arguments with any of the output argument.

fmpz_mod_poly_gcd_euclidean_f f G A B ctx

Either sets \(f = 1\) and \(G\) to the greatest common divisor of \(A\) and \(B\), or \( \in (1,p)\) to a non-trivial factor of \(p\).

In general, the greatest common divisor is defined in the polynomial ring \((\mathbf{Z}/(p \mathbf{Z}))[X]\) if and only if \(p\) is a prime number.

_fmpz_mod_poly_gcd_f f G A lenA B lenB p

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 \(f \in (1,p)\) to a non-trivial factor of \(p\) 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.

Does not support aliasing of any of the input arguments with any of the output arguments.

fmpz_mod_poly_gcd_f f G A B ctx

Either sets \(f = 1\) and \(G\) to the greatest common divisor of \(A\) and \(B\), or \(f \in (1,p)\) to a non-trivial factor of \(p\).

In general, the greatest common divisor is defined in the polynomial ring \((\mathbf{Z}/(p \mathbf{Z}))[X]\) if and only if \(p\) is a prime number.

_fmpz_mod_poly_hgcd M lenM A lenA B lenB a lena b lenb mod

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 \(\operatorname{len}(a)\).

_fmpz_mod_poly_xgcd_euclidean_f f G S T A lenA B lenB invB p

If \(f\) returns with the value \(1\) then the function operates as per _fmpz_mod_poly_xgcd_euclidean, otherwise \(f\) is set to a nontrivial factor of \(p\).

fmpz_mod_poly_xgcd_euclidean_f f G S T A B ctx

If \(f\) returns with the value \(1\) then the function operates as per fmpz_mod_poly_xgcd_euclidean, otherwise \(f\) is set to a nontrivial factor of \(p\).

_fmpz_mod_poly_xgcd G S T A lenA B lenB invB p

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.

fmpz_mod_poly_xgcd G S T A B 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.

fmpz_mod_poly_xgcd_f f G S T A B ctx

If \(f\) returns with the value \(1\) then the function operates as per fmpz_mod_poly_xgcd, otherwise \(f\) is set to a nontrivial factor of \(p\).

_fmpz_mod_poly_gcdinv_euclidean G S A lenA B lenB p

Computes (G, lenA), (S, lenB-1) such that \(G \cong S A \pmod{B}\), returning the actual length of \(G\).

Assumes that \(0 < \operatorname{len}(A) < \operatorname{len}(B)\).

fmpz_mod_poly_gcdinv_euclidean G S A B ctx

Computes polynomials \(G\) and \(S\), both reduced modulo~`B`, such that \(G \cong S A \pmod{B}\), where \(B\) is assumed to have \(\operatorname{len}(B) \geq 2\).

In the case that \(A = 0 \pmod{B}\), returns \(G = S = 0\).

_fmpz_mod_poly_gcdinv_euclidean_f f G S A lenA B lenB p

If \(f\) returns with value \(1\) then the function operates as per _fmpz_mod_poly_gcdinv_euclidean, otherwise \(f\) is set to a nontrivial factor of \(p\).

fmpz_mod_poly_gcdinv_euclidean_f f G S A B ctx

If \(f\) returns with value \(1\) then the function operates as per fmpz_mod_poly_gcdinv_euclidean, otherwise \(f\) is set to a nontrivial factor of the modulus of \(A\).

_fmpz_mod_poly_gcdinv G S A lenA B lenB p

Computes (G, lenA), (S, lenB-1) such that \(G \cong S A \pmod{B}\), returning the actual length of \(G\).

Assumes that \(0 < \operatorname{len}(A) < \operatorname{len}(B)\).

_fmpz_mod_poly_gcdinv_f f G S A lenA B lenB p

If \(f\) returns with value \(1\) then the function operates as per _fmpz_mod_poly_gcdinv, otherwise \(f\) will be set to a nontrivial factor of \(p\).

fmpz_mod_poly_gcdinv G S A B ctx

Computes polynomials \(G\) and \(S\), both reduced modulo~`B`, such that \(G \cong S A \pmod{B}\), where \(B\) is assumed to have \(\operatorname{len}(B) \geq 2\).

In the case that \(A = 0 \pmod{B}\), returns \(G = S = 0\).

fmpz_mod_poly_gcdinv_f f G S A B ctx

If \(f\) returns with value \(1\) then the function operates as per fmpz_mod_poly_gcdinv, otherwise \(f\) will be set to a nontrivial factor of \(p\).

_fmpz_mod_poly_invmod A B lenB P lenP p

Attempts to set (A, lenP-1) to the inverse of (B, lenB) modulo the polynomial (P, lenP). Returns \(1\) if (B, lenB) is invertible and \(0\) otherwise.

Assumes that \(0 < \operatorname{len}(B) < \operatorname{len}(P)\), and hence also \(\operatorname{len}(P) \geq 2\), but supports zero-padding in (B, lenB).

Does not support aliasing.

Assumes that \(p\) is a prime number.

_fmpz_mod_poly_invmod_f f A B lenB P lenP p

If \(f\) returns with the value \(1\), then the function operates as per _fmpz_mod_poly_invmod. Otherwise \(f\) is set to a nontrivial factor of \(p\).

fmpz_mod_poly_invmod A B P ctx

Attempts to set \(A\) to the inverse of \(B\) modulo \(P\) in the polynomial ring \((\mathbf{Z}/p\mathbf{Z})[X]\), where we assume that \(p\) is a prime number.

If \(\deg(P) < 2\), raises an exception.

If the greatest common divisor of \(B\) and \(P\) is~`1`, returns~`1` and sets \(A\) to the inverse of \(B\). Otherwise, returns~`0` and the value of \(A\) on exit is undefined.

fmpz_mod_poly_invmod_f f A B P ctx

If \(f\) returns with the value \(1\), then the function operates as per fmpz_mod_poly_invmod. Otherwise \(f\) is set to a nontrivial factor of \(p\).

_fmpz_mod_poly_minpoly_bm poly seq len p

Sets poly to the coefficients of a minimal generating polynomial for sequence (seq, len) modulo \(p\).

The return value equals the length of poly.

It is assumed that \(p\) is prime and poly has space for at least \(len+1\) coefficients. No aliasing between inputs and outputs is allowed.

fmpz_mod_poly_minpoly_bm poly seq len ctx

Sets poly to a minimal generating polynomial for sequence seq of length len.

Assumes that the modulus is prime.

This version uses the Berlekamp-Massey algorithm, whose running time is proportional to len times the size of the minimal generator.

_fmpz_mod_poly_minpoly_hgcd poly seq len p

Sets poly to the coefficients of a minimal generating polynomial for sequence (seq, len) modulo \(p\).

The return value equals the length of poly.

It is assumed that \(p\) is prime and poly has space for at least \(len+1\) coefficients. No aliasing between inputs and outputs is allowed.

fmpz_mod_poly_minpoly_hgcd poly seq len ctx

Sets poly to a minimal generating polynomial for sequence seq of length len.

Assumes that the modulus is prime.

This version uses the HGCD algorithm, whose running time is \(O(n \log^2 n)\) field operations, regardless of the actual size of the minimal generator.

_fmpz_mod_poly_minpoly poly seq len p

Sets poly to the coefficients of a minimal generating polynomial for sequence (seq, len) modulo \(p\).

The return value equals the length of poly.

It is assumed that \(p\) is prime and poly has space for at least \(len+1\) coefficients. No aliasing between inputs and outputs is allowed.

fmpz_mod_poly_minpoly poly seq len ctx

Sets poly to a minimal generating polynomial for sequence seq of length len.

A minimal generating polynomial is a monic polynomial \(f = x^d + c_{d-1}x^{d-1} + \cdots + c_1 x + c_0\), of minimal degree \(d\), that annihilates any consecutive \(d+1\) terms in seq. That is, for any \(i < len - d\),

\(seq_i = -\sum_{j=0}^{d-1} seq_{i+j}*f_j.\)

Assumes that the modulus is prime.

This version automatically chooses the fastest underlying implementation based on len and the size of the modulus.

_fmpz_mod_poly_resultant_euclidean res poly1 len1 poly2 len2 mod

Sets \(r\) to the resultant of (poly1, len1) and (poly2, len2) using the Euclidean algorithm.

Assumes that len1 >= len2 > 0.

Assumes that the modulus is prime.

fmpz_mod_poly_resultant_euclidean r f g ctx

Computes the resultant of \(f\) and \(g\) using the Euclidean algorithm.

For two non-zero polynomials \(f(x) = a_m x^m + \dotsb + a_0\) and \(g(x) = b_n x^n + \dotsb + b_0\) of degrees \(m\) and \(n\), the resultant is defined to be

\[`\] \[a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).\]

For convenience, we define the resultant to be equal to zero if either of the two polynomials is zero.

_fmpz_mod_poly_resultant_hgcd res A lenA B lenB mod

Sets res to the resultant of (A, lenA) and (B, lenB) using the half-gcd algorithm.

This algorithm computes the half-gcd as per _fmpz_mod_poly_gcd_hgcd but additionally updates the resultant every time a division occurs. The half-gcd algorithm computes the GCD recursively. Given inputs \(a\) and \(b\) it lets m = len(a)/2 and (recursively) performs all quotients in the Euclidean algorithm which do not require the low \(m\) coefficients of \(a\) and \(b\).

This performs quotients in exactly the same order as the ordinary Euclidean algorithm except that the low \(m\) coefficients of the polynomials in the remainder sequence are not computed. A correction step after hgcd has been called computes these low \(m\) coefficients (by matrix multiplication by a transformation matrix also computed by hgcd).

This means that from the point of view of the resultant, all but the last quotient performed by a recursive call to hgcd is an ordinary quotient as per the usual Euclidean algorithm. However, the final quotient may give a remainder of less than \(m + 1\) coefficients, which won't be corrected until the hgcd correction step is performed afterwards.

To compute the adjustments to the resultant coming from this corrected quotient, we save the relevant information in an nmod_poly_res_t struct at the time the quotient is performed so that when the correction step is performed later, the adjustments to the resultant can be computed at that time also.

The only time an adjustment to the resultant is not required after a call to hgcd is if hgcd does nothing (the remainder may already have had less than \(m + 1\) coefficients when hgcd was called).

Assumes that lenA >= lenB > 0.

Assumes that the modulus is prime.

fmpz_mod_poly_resultant_hgcd res f g ctx

Computes the resultant of \(f\) and \(g\) using the half-gcd algorithm.

For two non-zero polynomials \(f(x) = a_m x^m + \dotsb + a_0\) and \(g(x) = b_n x^n + \dotsb + b_0\) of degrees \(m\) and \(n\), the resultant is defined to be

\[`\] \[a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).\]

For convenience, we define the resultant to be equal to zero if either of the two polynomials is zero.

_fmpz_mod_poly_resultant res poly1 len1 poly2 len2 mod

Returns the resultant of (poly1, len1) and (poly2, len2).

Assumes that len1 >= len2 > 0.

Assumes that the modulus is prime.

fmpz_mod_poly_resultant res f g ctx

Computes the resultant of $f$ and $g$.

For two non-zero polynomials \(f(x) = a_m x^m + \dotsb + a_0\) and \(g(x) = b_n x^n + \dotsb + b_0\) of degrees \(m\) and \(n\), the resultant is defined to be

\[`\] \[a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).\]

For convenience, we define the resultant to be equal to zero if either of the two polynomials is zero.

_fmpz_mod_poly_discriminant d poly len mod

Set \(d\) to the discriminant of (poly, len). Assumes len > 1.

fmpz_mod_poly_discriminant d f ctx

Set \(d\) to the discriminant of \(f\). We normalise the discriminant so that (operatorname{disc}(f) = (-1)^(n(n-1)2) operatorname{res}(f, f') operatorname{lc}(f)^(n - m - 2)), where n = len(f) and m = len(f'). Thus (operatorname{disc}(f) = operatorname{lc}(f)^(2n - 2) prod_{i < j} (r_i - r_j)^2), where \(\operatorname{lc}(f)\) is the leading coefficient of \(f\) and \(r_i\) are the roots of \(f\).

_fmpz_mod_poly_derivative res poly len p

Sets (res, len - 1) to the derivative of (poly, len). Also handles the cases where len is \(0\) or \(1\) correctly. Supports aliasing of res and poly.

fmpz_mod_poly_derivative res poly ctx

Sets res to the derivative of poly.