nmod_poly.h – univariate polynomials over integers mod n (wordsize n)¶
The nmod_poly_t
data type represents elements of
\(\mathbb{Z}/n\mathbb{Z}[x]\) for a fixed modulus \(n\). The nmod_poly
module provides routines for memory management, basic arithmetic and
some higher level functions such as GCD, etc.
Each coefficient of an nmod_poly_t
is of type mp_limb_t
and represents an integer reduced modulo the fixed modulus \(n\).
Unless otherwise specified, all functions in this section permit aliasing between their input arguments and between their input and output arguments.
The nmod_poly_t
type is a typedef for an array of length 1 of
nmod_poly_struct
’s. This permits passing parameters of type
nmod_poly_t
by reference.
In reality one never deals directly with the struct
and simply
deals with objects of type nmod_poly_t
. For simplicity we will
think of an nmod_poly_t
as a struct
, though in practice to
access fields of this struct
, one needs to dereference first,
e.g.to access the length
field of an nmod_poly_t
called
poly1
one writes poly1>length
.
An nmod_poly_t
is said to be normalised if either length
is zero, or if the leading coefficient of the polynomial is nonzero.
All nmod_poly
functions expect their inputs to be normalised and
for all coefficients to be reduced modulo \(n\) and unless otherwise
specified they produce output that is normalised with coefficients
reduced modulo \(n\).
It is recommended that users do not access the fields of an
nmod_poly_t
or its coefficient data directly, but make use of
the functions designed for this purpose, detailed below.
Functions in nmod_poly
do all the memory management for the user.
One does not need to specify the maximum length in advance before
using a polynomial object. FLINT reallocates space automatically as
the computation proceeds, if more space is required.
Simple example¶
The following example computes the square of the polynomial \(5x^3 + 6\) in \(\mathbb{Z}/7\mathbb{Z}[x]\).
#include "nmod_poly.h"
int main()
{
nmod_poly_t x, y;
nmod_poly_init(x, 7);
nmod_poly_init(y, 7);
nmod_poly_set_coeff_ui(x, 3, 5);
nmod_poly_set_coeff_ui(x, 0, 6);
nmod_poly_mul(y, x, x);
nmod_poly_print(x); flint_printf("\n");
nmod_poly_print(y); flint_printf("\n");
nmod_poly_clear(x);
nmod_poly_clear(y);
}
The output is:
4 7 6 0 0 5
7 7 1 0 0 4 0 0 4
Types, macros and constants¶

type nmod_poly_struct¶

type nmod_poly_t¶
Helper functions¶
Memory management¶

void nmod_poly_init(nmod_poly_t poly, mp_limb_t n)¶
Initialises
poly
. It will have coefficients modulo \(n\).

void nmod_poly_init_preinv(nmod_poly_t poly, mp_limb_t n, mp_limb_t ninv)¶
Initialises
poly
. It will have coefficients modulo \(n\). The caller supplies a precomputed inverse limb generated byn_preinvert_limb()
.

void nmod_poly_init_mod(nmod_poly_t poly, const nmod_t mod)¶
Initialises
poly
using an already initialised modulusmod
.

void nmod_poly_init2(nmod_poly_t poly, mp_limb_t n, slong alloc)¶
Initialises
poly
. It will have coefficients modulo \(n\). Up toalloc
coefficients may be stored inpoly
.

void nmod_poly_init2_preinv(nmod_poly_t poly, mp_limb_t n, mp_limb_t ninv, slong alloc)¶
Initialises
poly
. It will have coefficients modulo \(n\). The caller supplies a precomputed inverse limb generated byn_preinvert_limb()
. Up toalloc
coefficients may be stored inpoly
.

void nmod_poly_realloc(nmod_poly_t poly, slong alloc)¶
Reallocates
poly
to the given length. If the current length is less thanalloc
, the polynomial is truncated and normalised. Ifalloc
is zero, the polynomial is cleared.

void nmod_poly_clear(nmod_poly_t poly)¶
Clears the polynomial and releases any memory it used. The polynomial cannot be used again until it is initialised.

void nmod_poly_fit_length(nmod_poly_t poly, slong alloc)¶
Ensures
poly
has space for at leastalloc
coefficients. This function only ever grows the allocated space, so no data loss can occur.

void _nmod_poly_normalise(nmod_poly_t poly)¶
Internal function for normalising a polynomial so that the top coefficient, if there is one at all, is not zero.
Polynomial properties¶

slong nmod_poly_length(const nmod_poly_t poly)¶
Returns the length of the polynomial
poly
. The zero polynomial has length zero.

slong nmod_poly_degree(const nmod_poly_t poly)¶
Returns the degree of the polynomial
poly
. The zero polynomial is deemed to have degree \(1\).

mp_limb_t nmod_poly_modulus(const nmod_poly_t poly)¶
Returns the modulus of the polynomial
poly
. This will be a positive integer.

flint_bitcnt_t nmod_poly_max_bits(const nmod_poly_t poly)¶
Returns the maximum number of bits of any coefficient of
poly
.

int nmod_poly_is_unit(const nmod_poly_t poly)¶
Returns \(1\) if the polynomial is a nonzero constant (in the case of prime modulus, this is equivalent to being a unit), otherwise \(0\).

int nmod_poly_is_monic(const nmod_poly_t poly)¶
Returns \(1\) if the polynomial is monic, i.e. nonzero with leading coefficient \(1\), otherwise \(0\).
Assignment and basic manipulation¶

void nmod_poly_set(nmod_poly_t a, const nmod_poly_t b)¶
Sets
a
to a copy ofb
.

void nmod_poly_swap(nmod_poly_t poly1, nmod_poly_t poly2)¶
Efficiently swaps
poly1
andpoly2
by swapping pointers internally.

void nmod_poly_zero(nmod_poly_t res)¶
Sets
res
to the zero polynomial.

void nmod_poly_truncate(nmod_poly_t poly, slong len)¶
Truncates
poly
to the given length and normalises it. Iflen
is greater than the current length ofpoly
, then nothing happens.

void nmod_poly_set_trunc(nmod_poly_t res, const nmod_poly_t poly, slong len)¶
Notionally truncate
poly
to lengthlen
and setres
to the result. The result is normalised.

void _nmod_poly_reverse(mp_ptr output, mp_srcptr input, slong len, slong m)¶
Sets
output
to the reverse ofinput
, which is of lengthlen
, but thinking of it as a polynomial of lengthm
, notionally zeropadded if necessary. The lengthm
must be nonnegative, but there are no other restrictions. The polynomialoutput
must have space form
coefficients. Supports aliasing ofoutput
andinput
, but the behaviour is undefined in case of partial overlap.

void nmod_poly_reverse(nmod_poly_t output, const nmod_poly_t input, slong m)¶
Sets
output
to the reverse ofinput
, thinking of it as a polynomial of lengthm
, notionally zeropadded if necessary). The lengthm
must be nonnegative, but there are no other restrictions. The output polynomial will be set to lengthm
and then normalised.
Randomization¶

void nmod_poly_randtest(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random polynomial with length up to
len
.

void nmod_poly_randtest_irreducible(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random irreducible polynomial with length up to
len
.

void nmod_poly_randtest_monic(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random monic polynomial with length
len
.

void nmod_poly_randtest_monic_irreducible(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random monic irreducible polynomial with length
len
.

void nmod_poly_randtest_monic_primitive(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random monic irreducible primitive polynomial with length
len
.

void nmod_poly_randtest_trinomial(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random monic trinomial of length
len
.

int nmod_poly_randtest_trinomial_irreducible(nmod_poly_t poly, flint_rand_t state, slong len, slong max_attempts)¶
Attempts to set
poly
to a monic irreducible trinomial of lengthlen
. It will generate up tomax_attempts
trinomials in attempt to find an irreducible one. Ifmax_attempts
is0
, then it will keep generating trinomials until an irreducible one is found. Returns \(1\) if one is found and \(0\) otherwise.

void nmod_poly_randtest_pentomial(nmod_poly_t poly, flint_rand_t state, slong len)¶
Generates a random monic pentomial of length
len
.

int nmod_poly_randtest_pentomial_irreducible(nmod_poly_t poly, flint_rand_t state, slong len, slong max_attempts)¶
Attempts to set
poly
to a monic irreducible pentomial of lengthlen
. It will generate up tomax_attempts
pentomials in attempt to find an irreducible one. Ifmax_attempts
is0
, then it will keep generating pentomials until an irreducible one is found. Returns \(1\) if one is found and \(0\) otherwise.

void nmod_poly_randtest_sparse_irreducible(nmod_poly_t poly, flint_rand_t state, slong len)¶
Attempts to set
poly
to a sparse, monic irreducible polynomial with lengthlen
. It attempts to find an irreducible trinomial. If that does not succeed, it attempts to find a irreducible pentomial. If that fails, thenpoly
is just set to a random monic irreducible polynomial.
Getting and setting coefficients¶

ulong nmod_poly_get_coeff_ui(const nmod_poly_t poly, slong j)¶
Returns the coefficient of
poly
at indexj
, where coefficients are numbered with zero being the constant coefficient, and returns it as anulong
. Ifj
refers to a coefficient beyond the end ofpoly
, zero is returned.

void nmod_poly_set_coeff_ui(nmod_poly_t poly, slong j, ulong c)¶
Sets the coefficient of
poly
at indexj
, where coefficients are numbered with zero being the constant coefficient, to the valuec
reduced modulo the modulus ofpoly
. Ifj
refers to a coefficient beyond the current end ofpoly
, the polynomial is first resized, with intervening coefficients being set to zero.
Input and output¶

char *nmod_poly_get_str(const nmod_poly_t poly)¶
Writes
poly
to a string representation. The format is as described fornmod_poly_print()
. The string must be freed by the user when finished. For this it is sufficient to callflint_free()
.

char *nmod_poly_get_str_pretty(const nmod_poly_t poly, const char *x)¶
Writes
poly
to a pretty string representation. The format is as described fornmod_poly_print_pretty()
. The string must be freed by the user when finished. For this it is sufficient to callflint_free()
.It is assumed that the top coefficient is nonzero.

int nmod_poly_set_str(nmod_poly_t poly, const char *s)¶
Reads
poly
from a strings
. The format is as described fornmod_poly_print()
. If a polynomial in the correct format is read, a positive value is returned, otherwise a nonpositive value is returned.

int nmod_poly_print(const nmod_poly_t a)¶
Prints the polynomial to
stdout
. The length is printed, followed by a space, then the modulus. If the length is zero this is all that is printed, otherwise two spaces followed by a space separated list of coefficients is printed, beginning with the constant coefficient.In case of success, returns a positive value. In case of failure, returns a nonpositive value.

int nmod_poly_print_pretty(const nmod_poly_t a, const char *x)¶
Prints the polynomial to
stdout
using the stringx
to represent the indeterminate.It is assumed that the top coefficient is nonzero.
In case of success, returns a positive value. In case of failure, returns a nonpositive value.

int nmod_poly_fread(FILE *f, nmod_poly_t poly)¶
Reads
poly
from the file streamf
. If this is a file that has just been written, the file should be closed then opened again. The format is as described fornmod_poly_print()
. If a polynomial in the correct format is read, a positive value is returned, otherwise a nonpositive value is returned.

int nmod_poly_fprint(FILE *f, const nmod_poly_t poly)¶
Writes a polynomial to the file stream
f
. If this is a file then the file should be closed and reopened before being read. The format is as described fornmod_poly_print()
. If the polynomial is written correctly, a positive value is returned, otherwise a nonpositive value is returned.In case of success, returns a positive value. In case of failure, returns a nonpositive value.

int nmod_poly_fprint_pretty(FILE *f, const nmod_poly_t poly, const char *x)¶
Writes a polynomial to the file stream
f
. If this is a file then the file should be closed and reopened before being read. The format is as described fornmod_poly_print_pretty()
. If the polynomial is written correctly, a positive value is returned, otherwise a nonpositive value is returned.It is assumed that the top coefficient is nonzero.
In case of success, returns a positive value. In case of failure, returns a nonpositive value.

int nmod_poly_read(nmod_poly_t poly)¶
Read
poly
fromstdin
. The format is as described fornmod_poly_print()
. If a polynomial in the correct format is read, a positive value is returned, otherwise a nonpositive value is returned.
Comparison¶

int nmod_poly_equal(const nmod_poly_t a, const nmod_poly_t b)¶
Returns \(1\) if the polynomials are equal, otherwise \(0\).

int nmod_poly_equal_nmod(const nmod_poly_t poly, ulong cst)¶
Returns \(1\) if the polynomial
poly
is constant, equal tocst
, otherwise \(0\).cst
is assumed to be already reduced, i.e. less than the modulus ofpoly
.

int nmod_poly_equal_ui(const nmod_poly_t poly, ulong cst)¶
Returns \(1\) if the polynomial
poly
is constant and equal tocst
up to reduction modulo the modulus ofpoly
, otherwise returns \(0\).

int nmod_poly_equal_trunc(const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)¶
Notionally truncate
poly1
andpoly2
to length \(n\) and return \(1\) if the truncations are equal, otherwise return \(0\).

int nmod_poly_is_zero(const nmod_poly_t poly)¶
Returns \(1\) if the polynomial
poly
is the zero polynomial, otherwise returns \(0\).

int nmod_poly_is_one(const nmod_poly_t poly)¶
Returns \(1\) if the polynomial
poly
is the constant polynomial 1, otherwise returns \(0\).

int nmod_poly_is_gen(const nmod_poly_t poly)¶
Returns \(1\) if the polynomial is the generating indeterminate (i.e. has degree \(1\), constant coefficient \(0\), and leading coefficient \(1\)), otherwise returns \(0\).
Shifting¶

void _nmod_poly_shift_left(mp_ptr res, mp_srcptr poly, slong len, slong k)¶
Sets
(res, len + k)
to(poly, len)
shifted left byk
coefficients. Assumes thatres
has space forlen + k
coefficients.

void nmod_poly_shift_left(nmod_poly_t res, const nmod_poly_t poly, slong k)¶
Sets
res
topoly
shifted left byk
coefficients, i.e. multiplied by \(x^k\).

void _nmod_poly_shift_right(mp_ptr res, mp_srcptr poly, slong len, slong k)¶
Sets
(res, len  k)
to(poly, len)
shifted left byk
coefficients. It is assumed thatk <= len
and thatres
has space for at leastlen  k
coefficients.

void nmod_poly_shift_right(nmod_poly_t res, const nmod_poly_t poly, slong k)¶
Sets
res
topoly
shifted right byk
coefficients, i.e. divide by \(x^k\) and throw away the remainder. Ifk
is greater than or equal to the length ofpoly
, the result is the zero polynomial.
Addition and subtraction¶

void _nmod_poly_add(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Sets
res
to the sum of(poly1, len1)
and(poly2, len2)
. There are no restrictions on the lengths.

void nmod_poly_add(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Sets
res
to the sum ofpoly1
andpoly2
.

void nmod_poly_add_series(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)¶
Notionally truncate
poly1
andpoly2
to length \(n\) and setres
to the sum.

void _nmod_poly_sub(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Sets
res
to the difference of(poly1, len1)
and(poly2, len2)
. There are no restrictions on the lengths.

void nmod_poly_sub(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Sets
res
to the difference ofpoly1
andpoly2
.

void nmod_poly_sub_series(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)¶
Notionally truncate
poly1
andpoly2
to length \(n\) and setres
to the difference.

void nmod_poly_neg(nmod_poly_t res, const nmod_poly_t poly)¶
Sets
res
to the negation ofpoly
.
Scalar multiplication and division¶

void nmod_poly_scalar_mul_nmod(nmod_poly_t res, const nmod_poly_t poly, ulong c)¶
Sets
res
topoly
multiplied by \(c\). The element \(c\) is assumed to be less than the modulus ofpoly
.

void nmod_poly_scalar_addmul_nmod(nmod_poly_t res, const nmod_poly_t poly, ulong c)¶
Adds
poly
multiplied by \(c\) tores
. The element \(c\) is assumed to be less than the modulus ofpoly
.

void _nmod_poly_make_monic(mp_ptr output, mp_srcptr input, slong len, nmod_t mod)¶
Sets
output
to be the scalar multiple ofinput
of lengthlen > 0
that has leading coefficient one, if such a polynomial exists. If the leading coefficient ofinput
is not invertible,output
is set to the multiple ofinput
whose leading coefficient is the greatest common divisor of the leading coefficient and the modulus ofinput
.

void nmod_poly_make_monic(nmod_poly_t output, const nmod_poly_t input)¶
Sets
output
to be the scalar multiple ofinput
with leading coefficient one, if such a polynomial exists. Ifinput
is zero an exception is raised. If the leading coefficient ofinput
is not invertible,output
is set to the multiple ofinput
whose leading coefficient is the greatest common divisor of the leading coefficient and the modulus ofinput
.
Bit packing and unpacking¶

void _nmod_poly_bit_pack(mp_ptr res, mp_srcptr poly, slong len, flint_bitcnt_t bits)¶
Packs
len
coefficients ofpoly
into fields of the given number of bits in the large integerres
, i.e. evaluatespoly
at2^bits
and store the result inres
. Assumeslen > 0
andbits > 0
. Also assumes that no coefficient ofpoly
is bigger thanbits/2
bits. We also assumebits < 3 * FLINT_BITS
.

void _nmod_poly_bit_unpack(mp_ptr res, slong len, mp_srcptr mpn, ulong bits, nmod_t mod)¶
Unpacks
len
coefficients stored in the big integermpn
in bit fields of the given number of bits, reduces them modulo the given modulus, then stores them in the polynomialres
. We assumelen > 0
and3 * FLINT_BITS > bits > 0
. There are no restrictions on the size of the actual coefficients as stored within the bitfields.

void nmod_poly_bit_pack(fmpz_t f, const nmod_poly_t poly, flint_bitcnt_t bit_size)¶
Packs
poly
into bitfields of sizebit_size
, writing the result tof
.

void nmod_poly_bit_unpack(nmod_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)¶
Unpacks the polynomial from fields of size
bit_size
as represented by the integerf
.

void _nmod_poly_KS2_pack1(mp_ptr res, mp_srcptr op, slong n, slong s, ulong b, ulong k, slong r)¶
Same as
_nmod_poly_KS2_pack
, but requiresb <= FLINT_BITS
.

void _nmod_poly_KS2_pack(mp_ptr res, mp_srcptr op, slong n, slong s, ulong b, ulong k, slong r)¶
Bit packing routine used by KS2 and KS4 multiplication.

void _nmod_poly_KS2_unpack1(mp_ptr res, mp_srcptr op, slong n, ulong b, ulong k)¶
Same as
_nmod_poly_KS2_unpack
, but requiresb <= FLINT_BITS
(i.e. writes one word per coefficient).

void _nmod_poly_KS2_unpack2(mp_ptr res, mp_srcptr op, slong n, ulong b, ulong k)¶
Same as
_nmod_poly_KS2_unpack
, but requiresFLINT_BITS < b <= 2 * FLINT_BITS
(i.e. writes two words per coefficient).
KS2/KS4 Reduction¶

void _nmod_poly_KS2_reduce(mp_ptr res, slong s, mp_srcptr op, slong n, ulong w, nmod_t mod)¶
Reduction code used by KS2 and KS4 multiplication.

void _nmod_poly_KS2_recover_reduce1(mp_ptr res, slong s, mp_srcptr op1, mp_srcptr op2, slong n, ulong b, nmod_t mod)¶
Same as
_nmod_poly_KS2_recover_reduce
, but requires0 < 2 * b <= FLINT_BITS
.

void _nmod_poly_KS2_recover_reduce2(mp_ptr res, slong s, mp_srcptr op1, mp_srcptr op2, slong n, ulong b, nmod_t mod)¶
Same as
_nmod_poly_KS2_recover_reduce
, but requiresFLINT_BITS < 2 * b < 2*FLINT_BITS
.

void _nmod_poly_KS2_recover_reduce2b(mp_ptr res, slong s, mp_srcptr op1, mp_srcptr op2, slong n, ulong b, nmod_t mod)¶
Same as
_nmod_poly_KS2_recover_reduce
, but requiresb == FLINT_BITS
.
Multiplication¶

void _nmod_poly_mul_classical(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Sets
(res, len1 + len2  1)
to the product of(poly1, len1)
and(poly2, len2)
. Assumeslen1 >= len2 > 0
. Aliasing of inputs and output is not permitted.

void nmod_poly_mul_classical(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Sets
res
to the product ofpoly1
andpoly2
.

void _nmod_poly_mullow_classical(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong trunc, nmod_t mod)¶
Sets
res
to the lowertrunc
coefficients of the product of(poly1, len1)
and(poly2, len2)
. Assumes thatlen1 >= len2 > 0
andtrunc > 0
. Aliasing of inputs and output is not permitted.

void nmod_poly_mullow_classical(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong trunc)¶
Sets
res
to the lowertrunc
coefficients of the product ofpoly1
andpoly2
.

void _nmod_poly_mulhigh_classical(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong start, nmod_t mod)¶
Computes the product of
(poly1, len1)
and(poly2, len2)
and writes the coefficients fromstart
onwards into the high coefficients ofres
, the remaining coefficients being arbitrary but reduced. Assumes thatlen1 >= len2 > 0
. Aliasing of inputs and output is not permitted.

void nmod_poly_mulhigh_classical(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong start)¶
Computes the product of
poly1
andpoly2
and writes the coefficients fromstart
onwards into the high coefficients ofres
, the remaining coefficients being arbitrary but reduced.

void _nmod_poly_mul_KS(mp_ptr out, mp_srcptr in1, slong len1, mp_srcptr in2, slong len2, flint_bitcnt_t bits, nmod_t mod)¶
Sets
res
to the product ofin1
andin2
assuming the output coefficients are at most the given number of bits wide. Ifbits
is set to \(0\) an appropriate value is computed automatically. Assumes thatlen1 >= len2 > 0
.

void nmod_poly_mul_KS(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, flint_bitcnt_t bits)¶
Sets
res
to the product ofpoly1
andpoly2
assuming the output coefficients are at most the given number of bits wide. Ifbits
is set to \(0\) an appropriate value is computed automatically.

void _nmod_poly_mul_KS2(mp_ptr res, mp_srcptr op1, slong n1, mp_srcptr op2, slong n2, nmod_t mod)¶
Sets
res
to the product ofop1
andop2
. Assumes thatlen1 >= len2 > 0
.

void nmod_poly_mul_KS2(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Sets
res
to the product ofpoly1
andpoly2
.

void _nmod_poly_mul_KS4(mp_ptr res, mp_srcptr op1, slong n1, mp_srcptr op2, slong n2, nmod_t mod)¶
Sets
res
to the product ofop1
andop2
. Assumes thatlen1 >= len2 > 0
.

void nmod_poly_mul_KS4(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Sets
res
to the product ofpoly1
andpoly2
.

void _nmod_poly_mullow_KS(mp_ptr out, mp_srcptr in1, slong len1, mp_srcptr in2, slong len2, flint_bitcnt_t bits, slong n, nmod_t mod)¶
Sets
out
to the low \(n\) coefficients ofin1
of lengthlen1
timesin2
of lengthlen2
. The output must have space forn
coefficients. We assume thatlen1 >= len2 > 0
and that0 < n <= len1 + len2  1
.

void nmod_poly_mullow_KS(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, flint_bitcnt_t bits, slong n)¶
Set
res
to the low \(n\) coefficients ofin1
of lengthlen1
timesin2
of lengthlen2
.

void _nmod_poly_mul(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Sets
res
to the product ofpoly1
of lengthlen1
andpoly2
of lengthlen2
. Assumeslen1 >= len2 > 0
. No aliasing is permitted between the inputs and the output.

void nmod_poly_mul(nmod_poly_t res, const nmod_poly_t poly, const nmod_poly_t poly2)¶
Sets
res
to the product ofpoly1
andpoly2
.

void _nmod_poly_mullow(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong n, nmod_t mod)¶
Sets
res
to the firstn
coefficients of the product ofpoly1
of lengthlen1
andpoly2
of lengthlen2
. It is assumed that0 < n <= len1 + len2  1
and thatlen1 >= len2 > 0
. No aliasing of inputs and output is permitted.

void nmod_poly_mullow(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong trunc)¶
Sets
res
to the firsttrunc
coefficients of the product ofpoly1
andpoly2
.

void _nmod_poly_mulhigh(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong n, nmod_t mod)¶
Sets all but the low \(n\) coefficients of
res
to the corresponding coefficients of the product ofpoly1
of lengthlen1
andpoly2
of lengthlen2
, the other coefficients being arbitrary. It is assumed thatlen1 >= len2 > 0
and that0 < n <= len1 + len2  1
. Aliasing of inputs and output is not permitted.

void nmod_poly_mulhigh(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)¶
Sets all but the low \(n\) coefficients of
res
to the corresponding coefficients of the product ofpoly1
andpoly2
, the remaining coefficients being arbitrary.

void _nmod_poly_mulmod(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, mp_srcptr f, slong lenf, nmod_t mod)¶
Sets
res
to the remainder of the product ofpoly1
andpoly2
upon polynomial division byf
.It is required that
len1 + len2  lenf > 0
, which is equivalent to requiring that the result will actually be reduced. Otherwise, simply use_nmod_poly_mul
instead.Aliasing of
f
andres
is not permitted.

void nmod_poly_mulmod(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, const nmod_poly_t f)¶
Sets
res
to the remainder of the product ofpoly1
andpoly2
upon polynomial division byf
.

void _nmod_poly_mulmod_preinv(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, mp_srcptr f, slong lenf, mp_srcptr finv, slong lenfinv, nmod_t mod)¶
Sets
res
to the remainder of the product ofpoly1
andpoly2
upon polynomial division byf
.It is required that
finv
is the inverse of the reverse off
modx^lenf
. It is required thatlen1 + len2  lenf > 0
, which is equivalent to requiring that the result will actually be reduced. It is required thatlen1 < lenf
andlen2 < lenf
. Otherwise, simply use_nmod_poly_mul
instead.Aliasing of
`res
with any of the inputs is not permitted.

void nmod_poly_mulmod_preinv(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, const nmod_poly_t f, const nmod_poly_t finv)¶
Sets
res
to the remainder of the product ofpoly1
andpoly2
upon polynomial division byf
.finv
is the inverse of the reverse off
. It is required thatpoly1
andpoly2
are reduced modulof
.
Powering¶

void _nmod_poly_pow_binexp(mp_ptr res, mp_srcptr poly, slong len, ulong e, nmod_t mod)¶
Raises
poly
of lengthlen
to the powere
and setsres
to the result. We require thatres
has enough space for(len  1)*e + 1
coefficients. Assumes thatlen > 0
,e > 1
. Aliasing is not permitted. Uses the binary exponentiation method.

void nmod_poly_pow_binexp(nmod_poly_t res, const nmod_poly_t poly, ulong e)¶
Raises
poly
to the powere
and setsres
to the result. Uses the binary exponentiation method.

void _nmod_poly_pow(mp_ptr res, mp_srcptr poly, slong len, ulong e, nmod_t mod)¶
Raises
poly
of lengthlen
to the powere
and setsres
to the result. We require thatres
has enough space for(len  1)*e + 1
coefficients. Assumes thatlen > 0
,e > 1
. Aliasing is not permitted.

void nmod_poly_pow(nmod_poly_t res, const nmod_poly_t poly, ulong e)¶
Raises
poly
to the powere
and setsres
to the result.

void _nmod_poly_pow_trunc_binexp(mp_ptr res, mp_srcptr poly, ulong e, slong trunc, nmod_t mod)¶
Sets
res
to the lowtrunc
coefficients ofpoly
(assumed to be zero padded if necessary to lengthtrunc
) to the powere
. This is equivalent to doing a powering followed by a truncation. We require thatres
has enough space fortrunc
coefficients, thattrunc > 0
and thate > 1
. Aliasing is not permitted. Uses the binary exponentiation method.

void nmod_poly_pow_trunc_binexp(nmod_poly_t res, const nmod_poly_t poly, ulong e, slong trunc)¶
Sets
res
to the lowtrunc
coefficients ofpoly
to the powere
. This is equivalent to doing a powering followed by a truncation. Uses the binary exponentiation method.

void _nmod_poly_pow_trunc(mp_ptr res, mp_srcptr poly, ulong e, slong trunc, nmod_t mod)¶
Sets
res
to the lowtrunc
coefficients ofpoly
(assumed to be zero padded if necessary to lengthtrunc
) to the powere
. This is equivalent to doing a powering followed by a truncation. We require thatres
has enough space fortrunc
coefficients, thattrunc > 0
and thate > 1
. Aliasing is not permitted.

void nmod_poly_pow_trunc(nmod_poly_t res, const nmod_poly_t poly, ulong e, slong trunc)¶
Sets
res
to the lowtrunc
coefficients ofpoly
to the powere
. This is equivalent to doing a powering followed by a truncation.

void _nmod_poly_powmod_ui_binexp(mp_ptr res, mp_srcptr poly, ulong e, mp_srcptr f, slong lenf, nmod_t mod)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree > 0
.We require
lenf > 1
. It is assumed thatpoly
is already reduced modulof
and zeropadded as necessary to have length exactlylenf  1
. The outputres
must have room forlenf  1
coefficients.

void nmod_poly_powmod_ui_binexp(nmod_poly_t res, const nmod_poly_t poly, ulong e, const nmod_poly_t f)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree >= 0
.

void _nmod_poly_powmod_fmpz_binexp(mp_ptr res, mp_srcptr poly, fmpz_t e, mp_srcptr f, slong lenf, nmod_t mod)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree > 0
.We require
lenf > 1
. It is assumed thatpoly
is already reduced modulof
and zeropadded as necessary to have length exactlylenf  1
. The outputres
must have room forlenf  1
coefficients.

void nmod_poly_powmod_fmpz_binexp(nmod_poly_t res, const nmod_poly_t poly, fmpz_t e, const nmod_poly_t f)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree >= 0
.

void _nmod_poly_powmod_ui_binexp_preinv(mp_ptr res, mp_srcptr poly, ulong e, mp_srcptr f, slong lenf, mp_srcptr finv, slong lenfinv, nmod_t mod)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree > 0
. We requirefinv
to be the inverse of the reverse off
.We require
lenf > 1
. It is assumed thatpoly
is already reduced modulof
and zeropadded as necessary to have length exactlylenf  1
. The outputres
must have room forlenf  1
coefficients.

void nmod_poly_powmod_ui_binexp_preinv(nmod_poly_t res, const nmod_poly_t poly, ulong e, const nmod_poly_t f, const nmod_poly_t finv)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree >= 0
. We requirefinv
to be the inverse of the reverse off
.

void _nmod_poly_powmod_fmpz_binexp_preinv(mp_ptr res, mp_srcptr poly, fmpz_t e, mp_srcptr f, slong lenf, mp_srcptr finv, slong lenfinv, nmod_t mod)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree > 0
. We requirefinv
to be the inverse of the reverse off
.We require
lenf > 1
. It is assumed thatpoly
is already reduced modulof
and zeropadded as necessary to have length exactlylenf  1
. The outputres
must have room forlenf  1
coefficients.

void nmod_poly_powmod_fmpz_binexp_preinv(nmod_poly_t res, const nmod_poly_t poly, fmpz_t e, const nmod_poly_t f, const nmod_poly_t finv)¶
Sets
res
topoly
raised to the powere
modulof
, using binary exponentiation. We requiree >= 0
. We requirefinv
to be the inverse of the reverse off
.

void _nmod_poly_powmod_x_ui_preinv(mp_ptr res, ulong e, mp_srcptr f, slong lenf, mp_srcptr finv, slong lenfinv, nmod_t mod)¶
Sets
res
tox
raised to the powere
modulof
, using sliding window exponentiation. We requiree > 0
. We requirefinv
to be the inverse of the reverse off
.We require
lenf > 2
. The outputres
must have room forlenf  1
coefficients.

void nmod_poly_powmod_x_ui_preinv(nmod_poly_t res, ulong e, const nmod_poly_t f, const nmod_poly_t finv)¶
Sets
res
tox
raised to the powere
modulof
, using sliding window exponentiation. We requiree >= 0
. We requirefinv
to be the inverse of the reverse off
.

void _nmod_poly_powmod_x_fmpz_preinv(mp_ptr res, fmpz_t e, mp_srcptr f, slong lenf, mp_srcptr finv, slong lenfinv, nmod_t mod)¶
Sets
res
tox
raised to the powere
modulof
, using sliding window exponentiation. We requiree > 0
. We requirefinv
to be the inverse of the reverse off
.We require
lenf > 2
. The outputres
must have room forlenf  1
coefficients.

void nmod_poly_powmod_x_fmpz_preinv(nmod_poly_t res, fmpz_t e, const nmod_poly_t f, const nmod_poly_t finv)¶
Sets
res
tox
raised to the powere
modulof
, using sliding window exponentiation. We requiree >= 0
. We requirefinv
to be the inverse of the reverse off
.

void _nmod_poly_powers_mod_preinv_naive(mp_ptr *res, mp_srcptr f, slong flen, slong n, mp_srcptr g, slong glen, mp_srcptr ginv, slong ginvlen, const nmod_t mod)¶
Compute
f^0, f^1, ..., f^(n1) mod g
, whereg
has lengthglen
andf
is reduced modg
and has lengthflen
(possibly zero spaced). Assumesres
is an array ofn
arrays each with space for at leastglen  1
coefficients and thatflen > 0
. We require thatginv
of lengthginvlen
is set to the power series inverse of the reverse ofg
.

void nmod_poly_powers_mod_naive(nmod_poly_struct *res, const nmod_poly_t f, slong n, const nmod_poly_t g)¶
Set the entries of the array
res
tof^0, f^1, ..., f^(n1) mod g
. No aliasing is permitted between the entries ofres
and either of the inputs.

void _nmod_poly_powers_mod_preinv_threaded_pool(mp_ptr *res, mp_srcptr f, slong flen, slong n, mp_srcptr g, slong glen, mp_srcptr ginv, slong ginvlen, const nmod_t mod, thread_pool_handle *threads, slong num_threads)¶
Compute
f^0, f^1, ..., f^(n1) mod g
, whereg
has lengthglen
andf
is reduced modg
and has lengthflen
(possibly zero spaced). Assumesres
is an array ofn
arrays each with space for at leastglen  1
coefficients and thatflen > 0
. We require thatginv
of lengthginvlen
is set to the power series inverse of the reverse ofg
.

void _nmod_poly_powers_mod_preinv_threaded(mp_ptr *res, mp_srcptr f, slong flen, slong n, mp_srcptr g, slong glen, mp_srcptr ginv, slong ginvlen, const nmod_t mod)¶
Compute
f^0, f^1, ..., f^(n1) mod g
, whereg
has lengthglen
andf
is reduced modg
and has lengthflen
(possibly zero spaced). Assumesres
is an array ofn
arrays each with space for at leastglen  1
coefficients and thatflen > 0
. We require thatginv
of lengthginvlen
is set to the power series inverse of the reverse ofg
.

void nmod_poly_powers_mod_bsgs(nmod_poly_struct *res, const nmod_poly_t f, slong n, const nmod_poly_t g)¶
Set the entries of the array
res
tof^0, f^1, ..., f^(n1) mod g
. No aliasing is permitted between the entries ofres
and either of the inputs.
Division¶

void _nmod_poly_divrem_basecase(mp_ptr Q, mp_ptr R, mp_srcptr A, slong A_len, mp_srcptr B, slong B_len, nmod_t mod)¶
Finds \(Q\) and \(R\) such that \(A = B Q + R\) with \(\operatorname{len}(R) < \operatorname{len}(B)\). If \(\operatorname{len}(B) = 0\) an exception is raised. We require that
W
is temporary space ofNMOD_DIVREM_BC_ITCH(A_len, B_len, mod)
coefficients.

void nmod_poly_divrem_basecase(nmod_poly_t Q, nmod_poly_t R, const nmod_poly_t A, const nmod_poly_t B)¶
Finds \(Q\) and \(R\) such that \(A = B Q + R\) with \(\operatorname{len}(R) < \operatorname{len}(B)\). If \(\operatorname{len}(B) = 0\) an exception is raised.

void _nmod_poly_divrem(mp_ptr Q, mp_ptr R, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Computes \(Q\) and \(R\) such that \(A = BQ + R\) with \(\operatorname{len}(R)\) less than
lenB
, whereA
is of lengthlenA
andB
is of lengthlenB
. We require thatQ
have space forlenA  lenB + 1
coefficients.

void nmod_poly_divrem(nmod_poly_t Q, nmod_poly_t R, const nmod_poly_t A, const nmod_poly_t B)¶
Computes \(Q\) and \(R\) such that \(A = BQ + R\) with \(\operatorname{len}(R) < \operatorname{len}(B)\).

void _nmod_poly_div(mp_ptr Q, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Notionally computes polynomials \(Q\) and \(R\) such that \(A = BQ + R\) with \(\operatorname{len}(R)\) less than
lenB
, whereA
is of lengthlenA
andB
is of lengthlenB
, but returns onlyQ
. We require thatQ
have space forlenA  lenB + 1
coefficients.

void nmod_poly_div(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)¶
Computes the quotient \(Q\) on polynomial division of \(A\) and \(B\).

void _nmod_poly_rem(mp_ptr R, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Computes the remainder \(R\) on polynomial division of \(A\) by \(B\).

void nmod_poly_rem(nmod_poly_t R, const nmod_poly_t A, const nmod_poly_t B)¶
Computes the remainder \(R\) on polynomial division of \(A\) by \(B\).

void _nmod_poly_divexact(mp_ptr Q, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶

void nmod_poly_divexact(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)¶
Computes the quotient \(Q\) of \(A\) and \(B\) assuming that the division is exact.

void _nmod_poly_inv_series_basecase(mp_ptr Qinv, mp_srcptr Q, slong Qlen, slong n, nmod_t mod)¶
Given
Q
of lengthQlen
whose leading coefficient is invertible modulo the given modulus, finds a polynomialQinv
of lengthn
such that the topn
coefficients of the productQ * Qinv
is \(x^{n  1}\). Requires thatn > 0
. This function can be viewed as inverting a power series.

void nmod_poly_inv_series_basecase(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)¶
Given
Q
of length at leastn
findQinv
of lengthn
such that the topn
coefficients of the productQ * Qinv
is \(x^{n  1}\). An exception is raised ifn = 0
or if the length ofQ
is less thann
. The leading coefficient ofQ
must be invertible modulo the modulus ofQ
. This function can be viewed as inverting a power series.

void _nmod_poly_inv_series_newton(mp_ptr Qinv, mp_srcptr Q, slong Qlen, slong n, nmod_t mod)¶
Given
Q
of lengthQlen
whose constant coefficient is invertible modulo the given modulus, find a polynomialQinv
of lengthn
such thatQ * Qinv
is1
modulo \(x^n\). Requiresn > 0
. This function can be viewed as inverting a power series via Newton iteration.

void nmod_poly_inv_series_newton(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)¶
Given
Q
findQinv
such thatQ * Qinv
is1
modulo \(x^n\). The constant coefficient ofQ
must be invertible modulo the modulus ofQ
. An exception is raised if this is not the case or ifn = 0
. This function can be viewed as inverting a power series via Newton iteration.

void _nmod_poly_inv_series(mp_ptr Qinv, mp_srcptr Q, slong Qlen, slong n, nmod_t mod)¶
Given
Q
of lengthQlenn
whose constant coefficient is invertible modulo the given modulus, find a polynomialQinv
of lengthn
such thatQ * Qinv
is1
modulo \(x^n\). Requiresn > 0
. This function can be viewed as inverting a power series.

void nmod_poly_inv_series(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)¶
Given
Q
findQinv
such thatQ * Qinv
is1
modulo \(x^n\). The constant coefficient ofQ
must be invertible modulo the modulus ofQ
. An exception is raised if this is not the case or ifn = 0
. This function can be viewed as inverting a power series.

void _nmod_poly_div_series_basecase(mp_ptr Q, mp_srcptr A, slong Alen, mp_srcptr B, slong Blen, slong n, nmod_t mod)¶
Given polynomials
A
andB
of lengthAlen
andBlen
, finds the polynomialQ
of lengthn
such thatQ * B = A
modulo \(x^n\). We assumen > 0
and that the constant coefficient ofB
is invertible modulo the given modulus. The polynomialQ
must have space forn
coefficients.

void nmod_poly_div_series_basecase(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B, slong n)¶
Given polynomials
A
andB
considered modulon
, finds the polynomialQ
of length at mostn
such thatQ * B = A
modulo \(x^n\). We assumen > 0
and that the constant coefficient ofB
is invertible modulo the modulus. An exception is raised ifn == 0
or the constant coefficient ofB
is zero.

void _nmod_poly_div_series(mp_ptr Q, mp_srcptr A, slong Alen, mp_srcptr B, slong Blen, slong n, nmod_t mod)¶
Given polynomials
A
andB
of lengthAlen
andBlen
, finds the polynomialQ
of lengthn
such thatQ * B = A
modulo \(x^n\). We assumen > 0
and that the constant coefficient ofB
is invertible modulo the given modulus. The polynomialQ
must have space forn
coefficients.

void nmod_poly_div_series(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B, slong n)¶
Given polynomials
A
andB
considered modulon
, finds the polynomialQ
of length at mostn
such thatQ * B = A
modulo \(x^n\). We assumen > 0
and that the constant coefficient ofB
is invertible modulo the modulus. An exception is raised ifn == 0
or the constant coefficient ofB
is zero.

void _nmod_poly_div_newton_n_preinv(mp_ptr Q, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, mp_srcptr Binv, slong lenBinv, nmod_t mod)¶
Notionally computes polynomials \(Q\) and \(R\) such that \(A = BQ + R\) with \(\operatorname{len}(R)\) less than
lenB
, whereA
is of lengthlenA
andB
is of lengthlenB
, 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 nmod_poly_div_newton_n_preinv(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B, const nmod_poly_t Binv)¶
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 _nmod_poly_divrem_newton_n_preinv(mp_ptr Q, mp_ptr R, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, mp_srcptr Binv, slong lenBinv, nmod_t mod)¶
Computes \(Q\) and \(R\) such that \(A = BQ + R\) with \(\operatorname{len}(R)\) less than
lenB
, where \(A\) is of lengthlenA
and \(B\) is of lengthlenB
. We require that \(Q\) have space forlenA  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 calldiv_newton_n_preinv()
and then multiply out and compute the remainder.

void nmod_poly_divrem_newton_n_preinv(nmod_poly_t Q, nmod_poly_t R, const nmod_poly_t A, const nmod_poly_t B, const nmod_poly_t Binv)¶
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.

mp_limb_t _nmod_poly_div_root(mp_ptr Q, mp_srcptr A, slong len, mp_limb_t c, nmod_t mod)¶
Sets
(Q, len1)
to the quotient of(A, len)
on division by \((x  c)\), and returns the remainder, equal to the value of \(A\) evaluated at \(c\). \(A\) and \(Q\) are allowed to be the same, but may not overlap partially in any other way.

mp_limb_t nmod_poly_div_root(nmod_poly_t Q, const nmod_poly_t A, mp_limb_t c)¶
Sets \(Q\) to the quotient of \(A\) on division by \((x  c)\), and returns the remainder, equal to the value of \(A\) evaluated at \(c\).
Divisibility testing¶

int _nmod_poly_divides_classical(mp_ptr Q, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t 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 \(lenA >= lenB > 0\).

int nmod_poly_divides_classical(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)¶
Returns \(1\) if \(B\) divides \(A\) and sets \(Q\) to the quotient. Otherwise returns \(0\) and sets \(Q\) to zero.

int _nmod_poly_divides(mp_ptr Q, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t 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 \(lenA >= lenB > 0\).

int nmod_poly_divides(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)¶
Returns \(1\) if \(B\) divides \(A\) and sets \(Q\) to the quotient. Otherwise returns \(0\) and sets \(Q\) to zero.

ulong nmod_poly_remove(nmod_poly_t f, const nmod_poly_t p)¶
Removes the highest possible power of
p
fromf
and returns the exponent.
Derivative and integral¶

void _nmod_poly_derivative(mp_ptr x_prime, mp_srcptr x, slong len, nmod_t mod)¶
Sets the first
len  1
coefficients ofx_prime
to the derivative ofx
which is assumed to be of lengthlen
. It is assumed thatlen > 0
.

void nmod_poly_derivative(nmod_poly_t x_prime, const nmod_poly_t x)¶
Sets
x_prime
to the derivative ofx
.

void _nmod_poly_integral(mp_ptr x_int, mp_srcptr x, slong len, nmod_t mod)¶
Set the first
len
coefficients ofx_int
to the integral ofx
which is assumed to be of lengthlen  1
. The constant term ofx_int
is set to zero. It is assumed thatlen > 0
. The result is only welldefined if the modulus is a prime number strictly larger than the degree ofx
. Supports aliasing between the two polynomials.

void nmod_poly_integral(nmod_poly_t x_int, const nmod_poly_t x)¶
Set
x_int
to the indefinite integral ofx
with constant term zero. The result is only welldefined if the modulus is a prime number strictly larger than the degree ofx
.
Evaluation¶

mp_limb_t _nmod_poly_evaluate_nmod(mp_srcptr poly, slong len, mp_limb_t c, nmod_t mod)¶
Evaluates
poly
at the valuec
and reduces modulo the given modulus ofpoly
. The valuec
should be reduced modulo the modulus. The algorithm used is Horner’s method.

mp_limb_t nmod_poly_evaluate_nmod(const nmod_poly_t poly, mp_limb_t c)¶
Evaluates
poly
at the valuec
and reduces modulo the modulus ofpoly
. The valuec
should be reduced modulo the modulus. The algorithm used is Horner’s method.

void nmod_poly_evaluate_mat_horner(nmod_mat_t dest, const nmod_poly_t poly, const nmod_mat_t c)¶
Evaluates
poly
with matrix as an argument at the valuec
and stores the result indest
. The dimension and modulus ofdest
is assumed to be same as that ofc
.dest
andc
may be aliased. Horner’s Method is used to compute the result.

void nmod_poly_evaluate_mat_paterson_stockmeyer(nmod_mat_t dest, const nmod_poly_t poly, const nmod_mat_t c)¶
Evaluates
poly
with matrix as an argument at the valuec
and stores the result indest
. The dimension and modulus ofdest
is assumed to be same as that ofc
.dest
andc
may be aliased. PatersonStockmeyer algorithm is used to compute the result. The algorithm is described in [Paterson1973].

void nmod_poly_evaluate_mat(nmod_mat_t dest, const nmod_poly_t poly, const nmod_mat_t c)¶
Evaluates
poly
with matrix as an argument at the valuec
and stores the result indest
. The dimension and modulus ofdest
is assumed to be same as that ofc
.dest
andc
may be aliased. This function automatically switches between Horner’s method and the PatersonStockmeyer algorithm.
Multipoint evaluation¶

void _nmod_poly_evaluate_nmod_vec_iter(mp_ptr ys, mp_srcptr poly, slong len, mp_srcptr xs, slong n, nmod_t mod)¶
Evaluates (
coeffs
,len
) at then
values given in the vectorxs
, writing the output values toys
. The values inxs
should be reduced modulo the modulus.Uses Horner’s method iteratively.

void nmod_poly_evaluate_nmod_vec_iter(mp_ptr ys, const nmod_poly_t poly, mp_srcptr xs, slong n)¶
Evaluates
poly
at then
values given in the vectorxs
, writing the output values toys
. The values inxs
should be reduced modulo the modulus.Uses Horner’s method iteratively.

void _nmod_poly_evaluate_nmod_vec_fast_precomp(mp_ptr vs, mp_srcptr poly, slong plen, const mp_ptr *tree, slong len, nmod_t mod)¶
Evaluates (
poly
,plen
) at thelen
values given by the precomputed subproduct treetree
.

void _nmod_poly_evaluate_nmod_vec_fast(mp_ptr ys, mp_srcptr poly, slong len, mp_srcptr xs, slong n, nmod_t mod)¶
Evaluates (
coeffs
,len
) at then
values given in the vectorxs
, writing the output values toys
. The values inxs
should be reduced modulo the modulus.Uses fast multipoint evaluation, building a temporary subproduct tree.

void nmod_poly_evaluate_nmod_vec_fast(mp_ptr ys, const nmod_poly_t poly, mp_srcptr xs, slong n)¶
Evaluates
poly
at then
values given in the vectorxs
, writing the output values toys
. The values inxs
should be reduced modulo the modulus.Uses fast multipoint evaluation, building a temporary subproduct tree.

void _nmod_poly_evaluate_nmod_vec(mp_ptr ys, mp_srcptr poly, slong len, mp_srcptr xs, slong n, nmod_t mod)¶
Evaluates (
poly
,len
) at then
values given in the vectorxs
, writing the output values toys
. The values inxs
should be reduced modulo the modulus.

void nmod_poly_evaluate_nmod_vec(mp_ptr ys, const nmod_poly_t poly, mp_srcptr xs, slong n)¶
Evaluates
poly
at then
values given in the vectorxs
, writing the output values toys
. The values inxs
should be reduced modulo the modulus.
Interpolation¶

void _nmod_poly_interpolate_nmod_vec(mp_ptr poly, mp_srcptr xs, mp_srcptr ys, slong n, nmod_t mod)¶
Sets
poly
to the unique polynomial of length at mostn
that interpolates then
given evaluation pointsxs
and valuesys
. If the interpolating polynomial is shorter than lengthn
, the leading coefficients are set to zero.The values in
xs
andys
should be reduced modulo the modulus, and allxs
must be distinct. Aliasing betweenpoly
andxs
orys
is not allowed.

void nmod_poly_interpolate_nmod_vec(nmod_poly_t poly, mp_srcptr xs, mp_srcptr ys, slong n)¶
Sets
poly
to the unique polynomial of lengthn
that interpolates then
given evaluation pointsxs
and valuesys
. The values inxs
andys
should be reduced modulo the modulus, and allxs
must be distinct.

void _nmod_poly_interpolation_weights(mp_ptr w, const mp_ptr *tree, slong len, nmod_t mod)¶
Sets
w
to the barycentric interpolation weights for fast Lagrange interpolation with respect to a given subproduct tree.

void _nmod_poly_interpolate_nmod_vec_fast_precomp(mp_ptr poly, mp_srcptr ys, const mp_ptr *tree, mp_srcptr weights, slong len, nmod_t mod)¶
Performs interpolation using the fast Lagrange interpolation algorithm, generating a temporary subproduct tree.
The function values are given as
ys
. The function takes a precomputed subproduct treetree
and barycentric interpolation weightsweights
corresponding to the roots.

void _nmod_poly_interpolate_nmod_vec_fast(mp_ptr poly, mp_srcptr xs, mp_srcptr ys, slong n, nmod_t mod)¶
Performs interpolation using the fast Lagrange interpolation algorithm, generating a temporary subproduct tree.

void nmod_poly_interpolate_nmod_vec_fast(nmod_poly_t poly, mp_srcptr xs, mp_srcptr ys, slong n)¶
Performs interpolation using the fast Lagrange interpolation algorithm, generating a temporary subproduct tree.

void _nmod_poly_interpolate_nmod_vec_newton(mp_ptr poly, mp_srcptr xs, mp_srcptr ys, slong n, nmod_t mod)¶
Forms the interpolating polynomial in the Newton basis using the method of divided differences and then converts it to monomial form.

void nmod_poly_interpolate_nmod_vec_newton(nmod_poly_t poly, mp_srcptr xs, mp_srcptr ys, slong n)¶
Forms the interpolating polynomial in the Newton basis using the method of divided differences and then converts it to monomial form.

void _nmod_poly_interpolate_nmod_vec_barycentric(mp_ptr poly, mp_srcptr xs, mp_srcptr ys, slong n, nmod_t mod)¶
Forms the interpolating polynomial using a naive implementation of the barycentric form of Lagrange interpolation.

void nmod_poly_interpolate_nmod_vec_barycentric(nmod_poly_t poly, mp_srcptr xs, mp_srcptr ys, slong n)¶
Forms the interpolating polynomial using a naive implementation of the barycentric form of Lagrange interpolation.
Composition¶

void _nmod_poly_compose_horner(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Composes
poly1
of lengthlen1
withpoly2
of lengthlen2
and setsres
to the result, i.e. evaluatespoly1
atpoly2
. The algorithm used is Horner’s algorithm. We require thatres
have space for(len1  1)*(len2  1) + 1
coefficients. It is assumed thatlen1 > 0
andlen2 > 0
.

void nmod_poly_compose_horner(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Composes
poly1
withpoly2
and setsres
to the result, i.e. evaluatespoly1
atpoly2
. The algorithm used is Horner’s algorithm.

void _nmod_poly_compose_divconquer(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Composes
poly1
of lengthlen1
withpoly2
of lengthlen2
and setsres
to the result, i.e. evaluatespoly1
atpoly2
. The algorithm used is the divide and conquer algorithm. We require thatres
have space for(len1  1)*(len2  1) + 1
coefficients. It is assumed thatlen1 > 0
andlen2 > 0
.

void nmod_poly_compose_divconquer(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Composes
poly1
withpoly2
and setsres
to the result, i.e. evaluatespoly1
atpoly2
. The algorithm used is the divide and conquer algorithm.

void _nmod_poly_compose(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Composes
poly1
of lengthlen1
withpoly2
of lengthlen2
and setsres
to the result, i.e. evaluatespoly1
atpoly2
. We require thatres
have space for(len1  1)*(len2  1) + 1
coefficients. It is assumed thatlen1 > 0
andlen2 > 0
.

void nmod_poly_compose(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)¶
Composes
poly1
withpoly2
and setsres
to the result, that is, evaluatespoly1
atpoly2
.
Taylor shift¶

void _nmod_poly_taylor_shift_horner(mp_ptr poly, mp_limb_t c, slong len, nmod_t mod)¶
Performs the Taylor shift composing
poly
by \(x+c\) inplace. Uses an efficient version Horner’s rule.

void nmod_poly_taylor_shift_horner(nmod_poly_t g, const nmod_poly_t f, mp_limb_t c)¶
Performs the Taylor shift composing
f
by \(x+c\).

void _nmod_poly_taylor_shift_convolution(mp_ptr poly, mp_limb_t c, slong len, nmod_t mod)¶
Performs the Taylor shift composing
poly
by \(x+c\) inplace. Writes the composition as a single convolution with cost \(O(M(n))\). We require that the modulus is a prime at least as large as the length.

void nmod_poly_taylor_shift_convolution(nmod_poly_t g, const nmod_poly_t f, mp_limb_t c)¶
Performs the Taylor shift composing
f
by \(x+c\). Writes the composition as a single convolution with cost \(O(M(n))\). We require that the modulus is a prime at least as large as the length.

void _nmod_poly_taylor_shift(mp_ptr poly, mp_limb_t c, slong len, nmod_t mod)¶
Performs the Taylor shift composing
poly
by \(x+c\) inplace. We require that the modulus is a prime.

void nmod_poly_taylor_shift(nmod_poly_t g, const nmod_poly_t f, mp_limb_t c)¶
Performs the Taylor shift composing
f
by \(x+c\). We require that the modulus is a prime.
Modular composition¶

void _nmod_poly_compose_mod_horner(mp_ptr res, mp_srcptr f, slong lenf, mp_srcptr g, mp_srcptr h, slong lenh, nmod_t mod)¶
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 nmod_poly_compose_mod_horner(nmod_poly_t res, const nmod_poly_t f, const nmod_poly_t g, const nmod_poly_t h)¶
Sets
res
to the composition \(f(g)\) modulo \(h\). We require that \(h\) is nonzero. The algorithm used is Horner’s rule.

void _nmod_poly_compose_mod_brent_kung(mp_ptr res, mp_srcptr f, slong lenf, mp_srcptr g, mp_srcptr h, slong lenh, nmod_t mod)¶
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 BrentKung matrix algorithm.

void nmod_poly_compose_mod_brent_kung(nmod_poly_t res, const nmod_poly_t f, const nmod_poly_t g, const nmod_poly_t h)¶
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 BrentKung matrix algorithm.

void _nmod_poly_compose_mod_brent_kung_preinv(mp_ptr res, mp_srcptr f, slong lenf, mp_srcptr g, mp_srcptr h, slong lenh, mp_srcptr hinv, slong lenhinv, nmod_t mod)¶
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 requirehinv
to be the inverse of the reverse ofh
. The output is not allowed to be aliased with any of the inputs.The algorithm used is the BrentKung matrix algorithm.

void nmod_poly_compose_mod_brent_kung_preinv(nmod_poly_t res, const nmod_poly_t f, const nmod_poly_t g, const nmod_poly_t h, const nmod_poly_t hinv)¶
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 requirehinv
to be the inverse of the reverse ofh
. The algorithm used is the BrentKung matrix algorithm.

void _nmod_poly_reduce_matrix_mod_poly(nmod_mat_t A, const nmod_mat_t B, const nmod_poly_t f)¶
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 _nmod_poly_precompute_matrix_worker(void *arg_ptr)¶
Worker function version of
_nmod_poly_precompute_matrix
. Input/output is stored innmod_poly_matrix_precompute_arg_t
.

void _nmod_poly_precompute_matrix(nmod_mat_t A, mp_srcptr f, mp_srcptr g, slong leng, mp_srcptr ginv, slong lenginv, nmod_t mod)¶
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 requireginv
to be the inverse of the reverse ofg
and \(g\) to be nonzero.f
has to be reduced modulog
and of length one less thanleng
(possibly with zero padding).

void nmod_poly_precompute_matrix(nmod_mat_t A, const nmod_poly_t f, const nmod_poly_t g, const nmod_poly_t ginv)¶
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 requireginv
to be the inverse of the reverse ofg
.

void _nmod_poly_compose_mod_brent_kung_precomp_preinv_worker(void *arg_ptr)¶
Worker function version of
_nmod_poly_compose_mod_brent_kung_precomp_preinv
. Input/output is stored innmod_poly_compose_mod_precomp_preinv_arg_t
.

void _nmod_poly_compose_mod_brent_kung_precomp_preinv(mp_ptr res, mp_srcptr f, slong lenf, const nmod_mat_t A, mp_srcptr h, slong lenh, mp_srcptr hinv, slong lenhinv, nmod_t mod)¶
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 requirehinv
to be the inverse of the reverse ofh
. The output is not allowed to be aliased with any of the inputs.The algorithm used is the BrentKung matrix algorithm.

void nmod_poly_compose_mod_brent_kung_precomp_preinv(nmod_poly_t res, const nmod_poly_t f, const nmod_mat_t A, const nmod_poly_t h, const nmod_poly_t hinv)¶
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 requirehinv
to be the inverse of the reverse ofh
. This version of BrentKung modular composition is particularly useful if one has to perform several modular composition of the form \(f(g)\) modulo \(h\) for fixed \(g\) and \(h\).

void _nmod_poly_compose_mod_brent_kung_vec_preinv(nmod_poly_struct *res, const nmod_poly_struct *polys, slong len1, slong l, mp_srcptr g, slong leng, mp_srcptr h, slong lenh, mp_srcptr hinv, slong lenhinv, nmod_t mod)¶
Sets
res
to the composition \(f_i(g)\) modulo \(h\) for \(1\leq i \leq l\), where \(f_i\) are the firstl
elements ofpolys
. We require that \(h\) is nonzero and that the length of \(g\) is less than the length of \(h\). We also require that the length of \(f_i\) is less than the length of \(h\). We requireres
to have enough memory allocated to holdl
nmod_poly_struct
’s. The entries ofres
need to be initialised andl
needs to be less thanlen1
Furthermore, we requirehinv
to be the inverse of the reverse ofh
. The output is not allowed to be aliased with any of the inputs.The algorithm used is the BrentKung matrix algorithm.

void nmod_poly_compose_mod_brent_kung_vec_preinv(nmod_poly_struct *res, const nmod_poly_struct *polys, slong len1, slong n, const nmod_poly_t g, const nmod_poly_t h, const nmod_poly_t hinv)¶
Sets
res
to the composition \(f_i(g)\) modulo \(h\) for \(1\leq i \leq n\) where \(f_i\) are the firstn
elements ofpolys
. We requireres
to have enough memory allocated to holdn
nmod_poly_struct
. The entries ofres
need to be initialised andn
needs to be less thanlen1
. We require that \(h\) is nonzero and that \(f_i\) and \(g\) have smaller degree than \(h\). Furthermore, we requirehinv
to be the inverse of the reverse ofh
. No aliasing ofres
andpolys
is allowed. The algorithm used is the BrentKung matrix algorithm.

void _nmod_poly_compose_mod_brent_kung_vec_preinv_threaded_pool(nmod_poly_struct *res, const nmod_poly_struct *polys, slong lenpolys, slong l, mp_srcptr g, slong glen, mp_srcptr poly, slong len, mp_srcptr polyinv, slong leninv, nmod_t mod, thread_pool_handle *threads, slong num_threads)¶
Multithreaded version of
_nmod_poly_compose_mod_brent_kung_vec_preinv()
. Distributing the Horner evaluations acrossflint_get_num_threads()
threads.

void nmod_poly_compose_mod_brent_kung_vec_preinv_threaded_pool(nmod_poly_struct *res, const nmod_poly_struct *polys, slong len1, slong n, const nmod_poly_t g, const nmod_poly_t poly, const nmod_poly_t polyinv, thread_pool_handle *threads, slong num_threads)¶
Multithreaded version of
nmod_poly_compose_mod_brent_kung_vec_preinv()
. Distributing the Horner evaluations acrossflint_get_num_threads()
threads.

void nmod_poly_compose_mod_brent_kung_vec_preinv_threaded(nmod_poly_struct *res, const nmod_poly_struct *polys, slong len1, slong n, const nmod_poly_t g, const nmod_poly_t poly, const nmod_poly_t polyinv)¶
Multithreaded version of
nmod_poly_compose_mod_brent_kung_vec_preinv()
. Distributing the Horner evaluations acrossflint_get_num_threads()
threads.

void _nmod_poly_compose_mod(mp_ptr res, mp_srcptr f, slong lenf, mp_srcptr g, mp_srcptr h, slong lenh, nmod_t mod)¶
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 nmod_poly_compose_mod(nmod_poly_t res, const nmod_poly_t f, const nmod_poly_t g, const nmod_poly_t h)¶
Sets
res
to the composition \(f(g)\) modulo \(h\). We require that \(h\) is nonzero.
Greatest common divisor¶

slong _nmod_poly_gcd_euclidean(mp_ptr G, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Computes the GCD of \(A\) of length
lenA
and \(B\) of lengthlenB
, wherelenA >= lenB > 0
. 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 forlenB
coefficients.

void nmod_poly_gcd_euclidean(nmod_poly_t G, const nmod_poly_t A, const nmod_poly_t B)¶
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.

slong _nmod_poly_hgcd(mp_ptr *M, slong *lenM, mp_ptr A, slong *lenA, mp_ptr B, slong *lenB, mp_srcptr a, slong lena, mp_srcptr b, slong lenb, nmod_t mod)¶
Computes the HGCD of \(a\) and \(b\), that is, a matrix \(M\), a sign \(\sigma\) and two polynomials \(A\) and \(B\) such that
\[(A,B)^t = M^{1} (a,b)^t, \sigma = \det(M),\]and \(A\) and \(B\) are consecutive remainders in the Euclidean remainder sequence for the division of \(a\) by \(b\) satisfying deg(A) ge frac{deg(a)}{2} > deg(B). Furthermore, \(M\) will be the product of
[[q 1][1 0]]
for the quotientsq
generated by such a remainder sequence. Assumes that \(\operatorname{len}(a) > \operatorname{len}(b) > 0\), i.e. \(\deg(a) > \).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]
, andM[3]
each point to a vector of size at least \(\operatorname{len}(a)\).

slong _nmod_poly_gcd_hgcd(mp_ptr G, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Computes the monic GCD of \(A\) and \(B\), assuming that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\).
Assumes that \(G\) has space for \(\operatorname{len}(B)\) coefficients and returns the length of \(G\) on output.

void nmod_poly_gcd_hgcd(nmod_poly_t G, const nmod_poly_t A, const nmod_poly_t B)¶
Computes the monic GCD of \(A\) and \(B\) using the HGCD algorithm.
As a special case, the GCD of two zero polynomials is defined to be the zero polynomial.
The time complexity of the algorithm is \(\mathcal{O}(n \log^2 n)\). For further details, see [ThullYap1990].

slong _nmod_poly_gcd(mp_ptr G, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Computes the GCD of \(A\) of length
lenA
and \(B\) of lengthlenB
, wherelenA >= lenB > 0
. 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 forlenB
coefficients.

void nmod_poly_gcd(nmod_poly_t G, const nmod_poly_t A, const nmod_poly_t B)¶
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.

slong _nmod_poly_xgcd_euclidean(mp_ptr G, mp_ptr S, mp_ptr T, mp_srcptr A, slong A_len, mp_srcptr B, slong B_len, nmod_t mod)¶
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 nmod_poly_xgcd_euclidean(nmod_poly_t G, nmod_poly_t S, nmod_poly_t T, const nmod_poly_t A, const nmod_poly_t B)¶
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
andT
are computed such thatS*A + T*B = G
. The length ofS
will be at mostlenB
and the length ofT
will be at mostlenA
.

slong _nmod_poly_xgcd_hgcd(mp_ptr G, mp_ptr S, mp_ptr T, mp_srcptr A, slong A_len, mp_srcptr B, slong B_len, nmod_t mod)¶
Computes the GCD of \(A\) and \(B\), where \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\), together with cofactors \(S\) and \(T\) such that \(S A + T B = G\). Returns the length of \(G\).
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, \(\operatorname{len}(S) \leq \operatorname{len}(B)  \operatorname{len}(G)\) and \(\operatorname{len}(T) \leq \operatorname{len}(A)  \operatorname{len}(G)\).
Both \(S\) and \(T\) must have space for at least \(2\) coefficients.
No aliasing of input and output operands is permitted.

void nmod_poly_xgcd_hgcd(nmod_poly_t G, nmod_poly_t S, nmod_poly_t T, const nmod_poly_t A, const nmod_poly_t B)¶
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
andT
are computed such thatS*A + T*B = G
. The length ofS
will be at mostlenB
and the length ofT
will be at mostlenA
.

slong _nmod_poly_xgcd(mp_ptr G, mp_ptr S, mp_ptr T, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)¶
Computes the GCD of \(A\) and \(B\), where \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\), together with cofactors \(S\) and \(T\) such that \(S A + T B = G\). Returns the length of \(G\).
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, \(\operatorname{len}(S) \leq \operatorname{len}(B)  \operatorname{len}(G)\) and \(\operatorname{len}(T) \leq \operatorname{len}(A)  \operatorname{len}(G)\).
No aliasing of input and output operands is permitted.

void nmod_poly_xgcd(nmod_poly_t G, nmod_poly_t S, nmod_poly_t T, const nmod_poly_t A, const nmod_poly_t B)¶
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.
The polynomials
S
andT
are set such thatS*A + T*B = G
. The length ofS
will be at mostlenB
and the length ofT
will be at mostlenA
.

mp_limb_t _nmod_poly_resultant_euclidean(mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Returns the resultant of
(poly1, len1)
and(poly2, len2)
using the Euclidean algorithm.Assumes that
len1 >= len2 > 0
.Assumes that the modulus is prime.

mp_limb_t nmod_poly_resultant_euclidean(const nmod_poly_t f, const nmod_poly_t g)¶
Computes the resultant of \(f\) and \(g\) using the Euclidean algorithm.
For two nonzero 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.

mp_limb_t _nmod_poly_resultant_hgcd(mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Returns the resultant of
(poly1, len1)
and(poly2, len2)
using the halfgcd algorithm.This algorithm computes the halfgcd as per
_nmod_poly_gcd_hgcd()
but additionally updates the resultant every time a division occurs. The halfgcd algorithm computes the GCD recursively. Given inputs \(a\) and \(b\) it letsm = 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
len1 >= len2 > 0
.Assumes that the modulus is prime.

mp_limb_t nmod_poly_resultant_hgcd(const nmod_poly_t f, const nmod_poly_t g)¶
Computes the resultant of \(f\) and \(g\) using the halfgcd algorithm.
For two nonzero 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.

mp_limb_t _nmod_poly_resultant(mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, nmod_t mod)¶
Returns the resultant of
(poly1, len1)
and(poly2, len2)
.Assumes that
len1 >= len2 > 0
.Assumes that the modulus is prime.

mp_limb_t nmod_poly_resultant(const nmod_poly_t f, const nmod_poly_t g)¶
Computes the resultant of \(f\) and \(g\).
For two nonzero 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.

slong _nmod_poly_gcdinv(mp_limb_t *G, mp_limb_t *S, const mp_limb_t *A, slong lenA, const mp_limb_t *B, slong lenB, const nmod_t mod)¶
Computes
(G, lenA)
,(S, lenB1)
such that \(G \cong S A \pmod{B}\), returning the actual length of \(G\).Assumes that \(0 < \operatorname{len}(A) < \operatorname{len}(B)\).

void nmod_poly_gcdinv(nmod_poly_t G, nmod_poly_t S, const nmod_poly_t A, const nmod_poly_t B)¶
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 \(G = S = 0\).

int _nmod_poly_invmod(mp_limb_t *A, const mp_limb_t *B, slong lenB, const mp_limb_t *P, slong lenP, const nmod_t mod)¶
Attempts to set
(A, lenP1)
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 zeropadding in
(B, lenB)
.Does not support aliasing.
Assumes that \(mod\) is a prime number.

int nmod_poly_invmod(nmod_poly_t A, const nmod_poly_t B, const nmod_poly_t P)¶
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 \(\operatorname{len}(P) < 2\), 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.
Discriminant¶

mp_limb_t _nmod_poly_discriminant(mp_srcptr poly, slong len, nmod_t mod)¶
Return the discriminant of
(poly, len)
. Assumeslen > 1
.

mp_limb_t nmod_poly_discriminant(const nmod_poly_t f)¶
Return the discriminant of \(f\). We normalise the discriminant so that \(\operatorname{disc}(f) = (1)^{n(n1)/2} \operatorname{res}(f, f') / \operatorname{lc}(f)^{n  m  2}\), where
n = len(f)
andm = 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 \(f\).
Power series composition¶

void _nmod_poly_compose_series(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong n, nmod_t mod)¶
Sets
res
to the composition ofpoly1
andpoly2
modulo \(x^n\), where the constant term ofpoly2
is required to be zero.Assumes that
len1, len2, n > 0
, thatlen1, len2 <= n
, and that(len11) * (len21) + 1 <= n
, and thatres
has space forn
coefficients. Does not support aliasing between any of the inputs and the output.Wraps
_gr_poly_compose_series()
which chooses automatically between various algorithms.

void nmod_poly_compose_series(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)¶
Sets
res
to the composition ofpoly1
andpoly2
modulo \(x^n\), where the constant term ofpoly2
is required to be zero.
Power series reversion¶

void _nmod_poly_revert_series(mp_ptr Qinv, mp_srcptr Q, slong Qlen, slong n, nmod_t mod)¶

void nmod_poly_revert_series(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)¶
Sets
Qinv
to the compositional inverse or reversion ofQ
as a power series, i.e. computes \(Q^{1}\) such that \(Q(Q^{1}(x)) = Q^{1}(Q(x)) = x \bmod x^n\).It is required that \(Q_0 = 0\) and that \(Q_1\) as well as the integers \(1, 2, \ldots, n1\) are invertible modulo the modulus.
Wraps
_gr_poly_revert_series()
which chooses automatically between various algorithms.
Square roots¶
The series expansions for \(\sqrt{h}\) and \(1/\sqrt{h}\) are defined
by means of the generalised binomial theorem
h^r = (1+y)^r =
\sum_{k=0}^{\infty} {r \choose k} y^k.
It is assumed that \(h\) has constant term \(1\) and that the coefficients
\(2^{k}\) exist in the coefficient ring (i.e. \(2\) must be invertible).

void _nmod_poly_invsqrt_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set the first \(n\) terms of \(g\) to the series expansion of \(1/\sqrt{h}\). It is assumed that \(n > 0\), that \(h\) has constant term 1. Aliasing is not permitted.

void nmod_poly_invsqrt_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g\) to the series expansion of \(1/\sqrt{h}\) to order \(O(x^n)\). It is assumed that \(h\) has constant term 1.

void _nmod_poly_sqrt_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set the first \(n\) terms of \(g\) to the series expansion of \(\sqrt{h}\). It is assumed that \(n > 0\), that \(h\) has constant term 1. Aliasing is not permitted.

void nmod_poly_sqrt_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g\) to the series expansion of \(\sqrt{h}\) to order \(O(x^n)\). It is assumed that \(h\) has constant term 1.

int _nmod_poly_sqrt(mp_ptr s, mp_srcptr p, slong n, nmod_t mod)¶
If
(p, n)
is a perfect square, sets(s, n / 2 + 1)
to a square root of \(p\) and returns 1. Otherwise returns 0.

int nmod_poly_sqrt(nmod_poly_t s, const nmod_poly_t p)¶
If \(p\) is a perfect square, sets \(s\) to a square root of \(p\) and returns 1. Otherwise returns 0.
Power sums¶

void _nmod_poly_power_sums_naive(mp_ptr res, mp_srcptr poly, slong len, slong n, nmod_t mod)¶
Compute the (truncated) power sums series of the polynomial
(poly,len)
up to length \(n\) using Newton identities.

void nmod_poly_power_sums_naive(nmod_poly_t res, const nmod_poly_t poly, slong n)¶
Compute the (truncated) power sum series of the polynomial
poly
up to length \(n\) using Newton identities.

void _nmod_poly_power_sums_schoenhage(mp_ptr res, mp_srcptr poly, slong len, slong n, nmod_t mod)¶
Compute the (truncated) power sums series of the polynomial
(poly,len)
up to length \(n\) using a series expansion (a formula due to Schoenhage).

void nmod_poly_power_sums_schoenhage(nmod_poly_t res, const nmod_poly_t poly, slong n)¶
Compute the (truncated) power sums series of the polynomial
poly
up to length \(n\) using a series expansion (a formula due to Schoenhage).

void _nmod_poly_power_sums(mp_ptr res, mp_srcptr poly, slong len, slong n, nmod_t mod)¶
Compute the (truncated) power sums series of the polynomial
(poly,len)
up to length \(n\).

void nmod_poly_power_sums(nmod_poly_t res, const nmod_poly_t poly, slong n)¶
Compute the (truncated) power sums series of the polynomial
poly
up to length \(n\).

void _nmod_poly_power_sums_to_poly_naive(mp_ptr res, mp_srcptr poly, slong len, nmod_t mod)¶
Compute the (monic) polynomial given by its power sums series
(poly,len)
using Newton identities.

void nmod_poly_power_sums_to_poly_naive(nmod_poly_t res, const nmod_poly_t Q)¶
Compute the (monic) polynomial given by its power sums series
Q
using Newton identities.

void _nmod_poly_power_sums_to_poly_schoenhage(mp_ptr res, mp_srcptr poly, slong len, nmod_t mod)¶
Compute the (monic) polynomial given by its power sums series
(poly,len)
using series expansion (a formula due to Schoenhage).

void nmod_poly_power_sums_to_poly_schoenhage(nmod_poly_t res, const nmod_poly_t Q)¶
Compute the (monic) polynomial given by its power sums series
Q
using series expansion (a formula due to Schoenhage).

void _nmod_poly_power_sums_to_poly(mp_ptr res, mp_srcptr poly, slong len, nmod_t mod)¶
Compute the (monic) polynomial given by its power sums series
(poly,len)
.

void nmod_poly_power_sums_to_poly(nmod_poly_t res, const nmod_poly_t Q)¶
Compute the (monic) polynomial given by its power sums series
Q
.
Transcendental functions¶
The elementary transcendental functions of a formal power series \(h\) are defined as
\(\exp(h(x)) = \sum_{k=0}^{\infty} \frac{(h(x))^k}{k!}\)
\(\log(h(x)) = \int_0^x \frac{h'(t)}{h(t)} dt\)
\(\operatorname{atan}(h(x)) = \int_0^x\frac{h'(t)}{1+(h(t))^2} dt\)
\(\operatorname{atanh}(h(x)) = \int_0^x\frac{h'(t)}{1(h(t))^2} dt\)
\(\operatorname{asin}(h(x)) = \int_0^x\frac{h'(t)}{\sqrt{1(h(t))^2}} dt\)
\(\operatorname{asinh}(h(x)) = \int_0^x\frac{h'(t)}{\sqrt{1+(h(t))^2}} dt\)
The functions sin, cos, tan, etc. are defined using standard inverse or functional relations. The logarithm function assumes that \(h\) has constant term \(1\). All other functions assume that \(h\) has constant term \(0\). All functions assume that the coefficient \(1/k\) or \(1/k!\) exists for all indices \(k\). When computing to order \(O(x^n)\), the modulus \(p\) must therefore be a prime satisfying \(p \ge n\). Further, we always require that \(p > 2\) in order to be able to multiply by \(1/2\) for internal purposes. If the input does not satisfy all these conditions, results are undefined. Except where otherwise noted, functions are implemented with optimal (up to constants) complexity \(O(M(n))\), where \(M(n)\) is the cost of polynomial multiplication.

void _nmod_poly_log_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(g = \log(h) + O(x^n)\). Assumes \(n > 0\) and
hlen > 0
. Aliasing of \(g\) and \(h\) is allowed.

void nmod_poly_log_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \log(h) + O(x^n)\). The case \(h = 1+cx^r\) is automatically detected and handled efficiently.

void _nmod_poly_exp_series(mp_ptr f, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(f = \exp(h) + O(x^n)\) where
h
is a polynomial. Assume \(n > 0\). Aliasing of \(g\) and \(h\) is not allowed.Uses Newton iteration (an improved version of the algorithm in [HanZim2004]). For small \(n\), falls back to the basecase algorithm.

void _nmod_poly_exp_expinv_series(mp_ptr f, mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(f = \exp(h) + O(x^n)\) and \(g = \exp(h) + O(x^n)\), more efficiently for large \(n\) than performing a separate inversion to obtain \(g\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Aliasing is not allowed.
Uses Newton iteration (the version given in [HanZim2004]). For small \(n\), falls back to the basecase algorithm.

void nmod_poly_exp_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \exp(h) + O(x^n)\). The case \(h = cx^r\) is automatically detected and handled efficiently. Otherwise this function automatically uses the basecase algorithm for small \(n\) and Newton iteration otherwise.

void _nmod_poly_atan_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(g = \operatorname{atan}(h) + O(x^n)\). Assumes \(n > 0\). Aliasing of \(g\) and \(h\) is allowed.

void nmod_poly_atan_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{atan}(h) + O(x^n)\).

void _nmod_poly_atanh_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(g = \operatorname{atanh}(h) + O(x^n)\). Assumes \(n > 0\). Aliasing of \(g\) and \(h\) is allowed.

void nmod_poly_atanh_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{atanh}(h) + O(x^n)\).

void _nmod_poly_asin_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(g = \operatorname{asin}(h) + O(x^n)\). Assumes \(n > 0\). Aliasing of \(g\) and \(h\) is allowed.

void nmod_poly_asin_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{asin}(h) + O(x^n)\).

void _nmod_poly_asinh_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(g = \operatorname{asinh}(h) + O(x^n)\). Assumes \(n > 0\). Aliasing of \(g\) and \(h\) is allowed.

void nmod_poly_asinh_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{asinh}(h) + O(x^n)\).

void _nmod_poly_sin_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)¶
Set \(g = \operatorname{sin}(h) + O(x^n)\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Aliasing of \(g\) and \(h\) is allowed. The value is computed using the identity \(\sin(x) = 2 \tan(x/2)) / (1 + \tan^2(x/2)).\)

void nmod_poly_sin_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{sin}(h) + O(x^n)\).

void _nmod_poly_cos_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)¶
Set \(g = \operatorname{cos}(h) + O(x^n)\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Aliasing of \(g\) and \(h\) is allowed. The value is computed using the identity \(\cos(x) = (1\tan^2(x/2)) / (1 + \tan^2(x/2)).\)

void nmod_poly_cos_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{cos}(h) + O(x^n)\).

void _nmod_poly_tan_series(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)¶
Set \(g = \operatorname{tan}(h) + O(x^n)\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Aliasing of \(g\) and \(h\) is not allowed. Uses Newton iteration to invert the atan function.

void nmod_poly_tan_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{tan}(h) + O(x^n)\).

void _nmod_poly_sinh_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)¶
Set \(g = \operatorname{sinh}(h) + O(x^n)\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Aliasing of \(g\) and \(h\) is not allowed. Uses the identity \(\sinh(x) = (e^x  e^{x})/2\).

void nmod_poly_sinh_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{sinh}(h) + O(x^n)\).

void _nmod_poly_cosh_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)¶
Set \(g = \operatorname{cos}(h) + O(x^n)\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Aliasing of \(g\) and \(h\) is not allowed. Uses the identity \(\cosh(x) = (e^x + e^{x})/2\).

void nmod_poly_cosh_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{cosh}(h) + O(x^n)\).

void _nmod_poly_tanh_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)¶
Set \(g = \operatorname{tanh}(h) + O(x^n)\). Assumes \(n > 0\) and that \(h\) is zeropadded as necessary to length \(n\). Uses the identity \(\tanh(x) = (e^{2x}1)/(e^{2x}+1)\).

void nmod_poly_tanh_series(nmod_poly_t g, const nmod_poly_t h, slong n)¶
Set \(g = \operatorname{tanh}(h) + O(x^n)\).
Special polynomials¶

int _nmod_poly_conway(mp_ptr op, ulong prime, slong deg)¶
Sets
op
to the coefficients to the Conway polynomial \(C_{p, d}\), where \(p\) isprime
and \(d\) isdeg
. This is done by checking against Frank Lübeck’s database [Lüb2004], which has been compressed in FLINT. Returns \(1\) in case of success and returns \(0\) in case of failure.

ulong _nmod_poly_conway_rand(slong *degree, flint_rand_t state, int type)¶
Returns a pseudorandom prime and sets
degree
that when put into_nmod_poly_conway()
will always succeed.Here,
type
can be the following values:0
for which there is a bijection between the image of this function and the database of Conway polynomials,1
returns a random prime found in the database and setsdegree
to some degree less than \(15\) along with some prime found in the database,2
returns a random prime less than \(2^{10}\) and setsdegree
to some random degree found in the database,3
returns a random prime less than \(2^{10}\) and setsdegree
to some random degree less than \(15\).
Products¶

void _nmod_poly_product_roots_nmod_vec(mp_ptr poly, mp_srcptr xs, slong n, nmod_t mod)¶
Sets
(poly, n + 1)
to the monic polynomial which is the product of \((x  x_0)(x  x_1) \cdots (x  x_{n1})\), the roots \(x_i\) being given byxs
.Aliasing of the input and output is not allowed.

void nmod_poly_product_roots_nmod_vec(nmod_poly_t poly, mp_srcptr xs, slong n)¶
Sets
poly
to the monic polynomial which is the product of \((x  x_0)(x  x_1) \cdots (x  x_{n1})\), the roots \(x_i\) being given byxs
.

int nmod_poly_find_distinct_nonzero_roots(mp_limb_t *roots, const nmod_poly_t A)¶
If
A
has \(\deg(A)\) distinct nonzero roots in \(\mathbb{F}_p\), write these roots out toroots[0]
toroots[deg(A)  1]
and return1
. Otherwise, return0
. It is assumed thatA
is nonzero and that the modulus ofA
is prime. This function uses Rabin’s probabilistic method via gcd’s with \((x + \delta)^{\frac{p1}{2}}  1\).
Subproduct trees¶

mp_ptr *_nmod_poly_tree_alloc(slong len)¶
Allocates space for a subproduct tree of the given length, having linear factors at the lowest level.
Entry \(i\) in the tree is a pointer to a single array of limbs, capable of storing \(\lfloor n / 2^i \rfloor\) subproducts of degree \(2^i\) adjacently, plus a trailing entry if \(n / 2^i\) is not an integer.
For example, a tree of length 7 built from monic linear factors has the following structure, where spaces have been inserted for illustrative purposes:
X1 X1 X1 X1 X1 X1 X1 XX1 XX1 XX1 X1 XXXX1 XX1 X1 XXXXXXX1
Inflation and deflation¶

void nmod_poly_inflate(nmod_poly_t result, const nmod_poly_t input, ulong inflation)¶
Sets
result
to the inflated polynomial \(p(x^n)\) where \(p\) is given byinput
and \(n\) is given bydeflation
.

void nmod_poly_deflate(nmod_poly_t result, const nmod_poly_t input, ulong deflation)¶
Sets
result
to the deflated polynomial \(p(x^{1/n})\) where \(p\) is given byinput
and \(n\) is given bydeflation
. Requires \(n > 0\).

ulong nmod_poly_deflation(const nmod_poly_t input)¶
Returns the largest integer by which
input
can be deflated. As special cases, returns 0 ifinput
is the zero polynomial and 1 ofinput
is a constant polynomial.
Chinese Remaindering¶
In all of these functions the moduli (mod.n) of all of the
nmod_poly
’s involved is assumed to match and be prime.

void nmod_poly_multi_crt_init(nmod_poly_multi_crt_t CRT)¶
Initialize
CRT
for Chinese remaindering.

int nmod_poly_multi_crt_precompute(nmod_poly_multi_crt_t CRT, const nmod_poly_struct *moduli, slong len)¶

int nmod_poly_multi_crt_precompute_p(nmod_poly_multi_crt_t CRT, const nmod_poly_struct *const *moduli, slong len)¶
Configure
CRT
for repeated Chinese remaindering ofmoduli
. The number of moduli,len
, should be positive. A return of0
indicates that the compilation failed and future calls tonmod_poly_multi_crt_precomp()
will leave the output undefined. A return of1
indicates that the compilation was successful, which occurs if and only if either (1)len == 1
andmodulus + 0
is nonzero, or (2) all of the moduli have positive degree and are pairwise relatively prime.

void nmod_poly_multi_crt_precomp(nmod_poly_t output, const nmod_poly_multi_crt_t CRT, const nmod_poly_struct *values)¶

void nmod_poly_multi_crt_precomp_p(nmod_poly_t output, const nmod_poly_multi_crt_t CRT, const nmod_poly_struct *const *values)¶
Set
output
to the polynomial of lowest possible degree that is congruent tovalues + i
modulo themoduli + i
innmod_poly_multi_crt_precompute()
. The inputsvalues + 0, ..., values + len  1
wherelen
was used innmod_poly_multi_crt_precompute()
are expected to be valid and have modulus matching the modulus of the moduli used innmod_poly_multi_crt_precompute()
.

int nmod_poly_multi_crt(nmod_poly_t output, const nmod_poly_struct *moduli, const nmod_poly_struct *values, slong len)¶
Perform the same operation as
nmod_poly_multi_crt_precomp()
while internally constructing and destroying the precomputed data. All of the remarks innmod_poly_multi_crt_precompute()
apply.

void nmod_poly_multi_crt_clear(nmod_poly_multi_crt_t CRT)¶
Free all space used by
CRT
.

slong _nmod_poly_multi_crt_local_size(const nmod_poly_multi_crt_t CRT)¶
Return the required length of the output for
_nmod_poly_multi_crt_run()
.

void _nmod_poly_multi_crt_run(nmod_poly_struct *outputs, const nmod_poly_multi_crt_t CRT, const nmod_poly_struct *inputs)¶

void _nmod_poly_multi_crt_run_p(nmod_poly_struct *outputs, const nmod_poly_multi_crt_t CRT, const nmod_poly_struct *const *inputs)¶
Perform the same operation as
nmod_poly_multi_crt_precomp()
using supplied temporary space. The actual output is placed inoutputs + 0
, andoutputs
should contain space for all temporaries and should be at least as long as_nmod_poly_multi_crt_local_size(CRT)
. Of course the moduli of these temporaries should match the modulus of the inputs.
BerlekampMassey Algorithm¶
The nmod_berlekamp_massey_t manages an unlimited stream of points \(a_1, a_2, \dots.\) At any point in time, after, say, \(n\) points have been added, a call to
nmod_berlekamp_massey_reduce()
will calculate the polynomials \(U\), \(V\) and \(R\) in the extended euclidean remainder sequence with\[U x^n + V (a_1 x^{n1} + a_{n1} x + \cdots + a_n) = R, \quad \deg(U) < \deg(V) \le n/2, \quad \deg(R) < n/2.\]The polynomials \(V\) and \(R\) may be obtained with
nmod_berlekamp_massey_V_poly()
andnmod_berlekamp_massey_R_poly()
. This class differs fromfmpz_mod_poly_minpoly()
in the following respect. Let \(v_i\) denote the coefficient of \(x^i\) in \(V\).fmpz_mod_poly_minpoly()
will return a polynomial \(V\) of lowest degree that annihilates the whole sequence \(a_1, \dots, a_n\) as\[\sum_{i} v_i a_{j + i} = 0, \quad 1 \le j \le n  \deg(V).\]The cost is that a polynomial of degree \(n1\) might be returned and the return is not generally uniquely determined by the input sequence. For the nmod_berlekamp_massey_t we have
\[\sum_{i,j} v_i a_{j+i} x^{j} = U + \frac{R}{x^n}\text{,}\]and it can be seen that \(\sum_{i} v_i a_{j + i}\) is zero for \(1 \le j < n  \deg(R)\). Thus whether or not \(V\) has annihilated the whole sequence may be checked by comparing the degrees of \(V\) and \(R\).

void nmod_berlekamp_massey_init(nmod_berlekamp_massey_t B, mp_limb_t p)¶
Initialize
B
in characteristicp
with an empty stream.

void nmod_berlekamp_massey_clear(nmod_berlekamp_massey_t B)¶
Free any space used by
B
.

void nmod_berlekamp_massey_start_over(nmod_berlekamp_massey_t B)¶
Empty the stream of points in
B
.

void nmod_berlekamp_massey_set_prime(nmod_berlekamp_massey_t B, mp_limb_t p)¶
Set the characteristic of the field and empty the stream of points in
B
.

void nmod_berlekamp_massey_add_points(nmod_berlekamp_massey_t B, const mp_limb_t *a, slong count)¶

void nmod_berlekamp_massey_add_zeros(nmod_berlekamp_massey_t B, slong count)¶

void nmod_berlekamp_massey_add_point(nmod_berlekamp_massey_t B, mp_limb_t a)¶
Add point(s) to the stream processed by
B
. The addition of any number of points will not update the \(V\) and \(R\) polynomial.

int nmod_berlekamp_massey_reduce(nmod_berlekamp_massey_t B)¶
Ensure that the polynomials \(V\) and \(R\) are up to date. The return value is
1
if this function changed \(V\) and0
otherwise. For example, if this function is called twice in a row without adding any points in between, the return of the second call should be0
. As another example, suppose the object is emptied, the points \(1, 1, 2, 3\) are added, then reduce is called. This reduce should return1
with \(\deg(R) < \deg(V) = 2\) because the Fibonacci sequence has been recognized. The further addition of the two points \(5, 8\) and a reduce will result in a return value of0
.

slong nmod_berlekamp_massey_point_count(const nmod_berlekamp_massey_t B)¶
Return the number of points stored in
B
.

const mp_limb_t *nmod_berlekamp_massey_points(const nmod_berlekamp_massey_t B)¶
Return a pointer to the array of points stored in
B
. This may beNULL
ifnmod_berlekamp_massey_point_count()
returns0
.

const nmod_poly_struct *nmod_berlekamp_massey_V_poly(const nmod_berlekamp_massey_t B)¶
Return the polynomial \(V\) in
B
.

const nmod_poly_struct *nmod_berlekamp_massey_R_poly(const nmod_berlekamp_massey_t B)¶
Return the polynomial \(R\) in
B
.