# nmod_poly.h – univariate polynomials over integers mod n (word-size n)¶

Description.

## Types, macros and constants¶

nmod_poly_struct
nmod_poly_t

Description.

## Helper functions¶

int signed_mpn_sub_n(mp_ptr res, mp_srcptr op1, mp_srcptr op2, slong n)

If op1 >= op2 return 0 and set res to op1 - op2 else return 1 and set res to op2 - op1.

## 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 by n_preinvert_limb().

void nmod_poly_init_mod(nmod_poly_t poly, const nmod_t mod)

Initialises poly using an already initialised modulus mod.

void nmod_poly_init2(nmod_poly_t poly, mp_limb_t n, slong alloc)

Initialises poly. It will have coefficients modulo~n. Up to alloc coefficients may be stored in poly.

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 by n_preinvert_limb(). Up to alloc coefficients may be stored in poly.

void nmod_poly_realloc(nmod_poly_t poly, slong alloc)

Reallocates poly to the given length. If the current length is less than alloc, the polynomial is truncated and normalised. If alloc 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 least alloc 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.

## Assignment and basic manipulation¶

void nmod_poly_set(nmod_poly_t a, const nmod_poly_t b)

Sets a to a copy of b.

void nmod_poly_swap(nmod_poly_t poly1, nmod_poly_t poly2)

Efficiently swaps poly1 and poly2 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. If len is greater than the current length of poly, then nothing happens.

void nmod_poly_set_trunc(nmod_poly_t res, const nmod_poly_t poly, slong n)

Notionally truncate poly to length $$n$$ and set res 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 of input, which is of length len, but thinking of it as a polynomial of length~m, notionally zero-padded if necessary. The length~m must be non-negative, but there are no other restrictions. The polynomial output must have space for m coefficients. Supports aliasing of output and input, 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 of input, thinking of it as a polynomial of length~m, notionally zero-padded if necessary). The length~m must be non-negative, but there are no other restrictions. The output polynomial will be set to length~m and then normalised.

## 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 length len. It will generate up to max_attempts trinomials in attempt to find an irreducible one. If max_attempts is 0, then it will keep generating trinomials until an irreducible one is found. Returns $$1$$ if one is found and $$0$$ otherwise.

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 length len. It will generate up to max_attempts pentomials in attempt to find an irreducible one. If max_attempts is 0, then it will keep generating pentomials until an irreducible one is found. Returns $$1$$ if one is found and $$0$$ otherwise.

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

## Getting and setting coefficients¶

ulong nmod_poly_get_coeff_ui(const nmod_poly_t poly, slong j)

Returns the coefficient of poly at index~j, where coefficients are numbered with zero being the constant coefficient, and returns it as an ulong. If j refers to a coefficient beyond the end of poly, zero is returned.

void nmod_poly_set_coeff_ui(nmod_poly_t poly, slong j, ulong c)

Sets the coefficient of poly at index j, where coefficients are numbered with zero being the constant coefficient, to the value c reduced modulo the modulus of poly. If j refers to a coefficient beyond the current end of poly, 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 for nmod_poly_print(). The string must be freed by the user when finished. For this it is sufficient to call flint_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 for nmod_poly_print_pretty(). The string must be freed by the user when finished. For this it is sufficient to call flint_free().

It is assumed that the top coefficient is non-zero.

int nmod_poly_set_str(nmod_poly_t poly, const char * s)

Reads poly from a string s. The format is as described for nmod_poly_print(). If a polynomial in the correct format is read, a positive value is returned, otherwise a non-positive 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 non-positive value.

int nmod_poly_print_pretty(const nmod_poly_t a, const char * x)

Prints the polynomial to stdout using the string x to represent the indeterminate.

It is assumed that the top coefficient is non-zero.

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

int nmod_poly_fread(FILE * f, nmod_poly_t poly)

Reads poly from the file stream f. If this is a file that has just been written, the file should be closed then opened again. The format is as described for nmod_poly_print(). If a polynomial in the correct format is read, a positive value is returned, otherwise a non-positive 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 for nmod_poly_print(). If the polynomial is written correctly, a positive value is returned, otherwise a non-positive value is returned.

In case of success, returns a positive value. In case of failure, returns a non-positive 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 for nmod_poly_print_pretty(). If the polynomial is written correctly, a positive value is returned, otherwise a non-positive value is returned.

It is assumed that the top coefficient is non-zero.

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

int nmod_poly_read(nmod_poly_t poly)

Read poly from stdin. The format is as described for nmod_poly_print(). If a polynomial in the correct format is read, a positive value is returned, otherwise a non-positive 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_trunc(const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)

Notionally truncate poly1 and poly2 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.

## 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 by k coefficients. Assumes that res has space for len + k coefficients.

void nmod_poly_shift_left(nmod_poly_t res, const nmod_poly_t poly, slong k)

Sets res to poly shifted left by k 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 by k coefficients. It is assumed that k <= len and that res has space for at least len - k coefficients.

void nmod_poly_shift_right(nmod_poly_t res, const nmod_poly_t poly, slong k)

Sets res to poly shifted right by k coefficients, i.e.divide by $$x^k$$ and throws away the remainder. If k is greater than or equal to the length of poly, the result is the zero polynomial.

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 of poly1 and poly2.

void nmod_poly_add_series(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)

Notionally truncate poly1 and poly2 to length $$n$$ and set res 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 of poly1 and poly2.

void nmod_poly_sub_series(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)

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

void nmod_poly_neg(nmod_poly_t res, const nmod_poly_t poly)

Sets res to the negation of poly.

## Scalar multiplication and division¶

void nmod_poly_scalar_mul_nmod(nmod_poly_t res, const nmod_poly_t poly, ulong c)

Sets res to (poly, len) multiplied by~c, where~c is reduced modulo the modulus of poly.

void _nmod_poly_make_monic(mp_ptr output, mp_srcptr input, slong len, nmod_t mod)

Sets output to be the scalar multiple of input of length len > 0 that has leading coefficient one, if such a polynomial exists. If the leading coefficient of input is not invertible, output is set to the multiple of input whose leading coefficient is the greatest common divisor of the leading coefficient and the modulus of input.

void nmod_poly_make_monic(nmod_poly_t output, const nmod_poly_t input)

Sets output to be the scalar multiple of input with leading coefficient one, if such a polynomial exists. If input is zero an exception is raised. If the leading coefficient of input is not invertible, output is set to the multiple of input whose leading coefficient is the greatest common divisor of the leading coefficient and the modulus of input.

## Bit packing and unpacking¶

void _nmod_poly_bit_pack(mp_ptr res, mp_srcptr poly, slong len, flint_bitcnt_t bits)

Packs len coefficients of poly into fields of the given number of bits in the large integer res, i.e.evaluates poly at 2^bits and store the result in res. Assumes len > 0 and bits > 0. Also assumes that no coefficient of poly is bigger than bits/2 bits. We also assume bits < 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 integer mpn in bit fields of the given number of bits, reduces them modulo the given modulus, then stores them in the polynomial res. We assume len > 0 and 3 * 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 size bit_size, writing the result to f.

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 integer f.

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 requires b <= 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 requires b <= 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 requires FLINT_BITS < b <= 2 * FLINT_BITS (i.e. writes two words per coefficient).

void _nmod_poly_KS2_unpack3(mp_ptr res, mp_srcptr op, slong n, ulong b, ulong k)

Same as _nmod_poly_KS2_unpack, but requires 2 * FLINT_BITS < b < 3 * FLINT_BITS (i.e. writes three words per coefficient).

void _nmod_poly_KS2_unpack(mp_ptr res, mp_srcptr op, slong n, ulong b, ulong k)

Bit unpacking code used by KS2 and KS4 multiplication.

## 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 requires 0 < 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 requires FLINT_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 requires b == FLINT_BITS.

void _nmod_poly_KS2_recover_reduce3(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 requires 2 * FLINT_BITS < 2 * b <= 3 * FLINT_BITS.

void _nmod_poly_KS2_recover_reduce(mp_ptr res, slong s, mp_srcptr op1, mp_srcptr op2, slong n, ulong b, nmod_t mod)

Reduction code used by KS4 multiplication.

## 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). Assumes len1 >= 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 of poly1 and poly2.

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 lower trunc coefficients of the product of (poly1, len1) and (poly2, len2). Assumes that len1 >= len2 > 0 and trunc > 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 lower trunc coefficients of the product of poly1 and poly2.

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

void 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 and poly2 and writes the coefficients from start onwards into the high coefficients of res, 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 of in1 and in2 assuming the output coefficients are at most the given number of bits wide. If bits is set to $$0$$ an appropriate value is computed automatically. Assumes that len1 >= 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 of poly1 and poly2 assuming the output coefficients are at most the given number of bits wide. If bits 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 of op1 and op2. Assumes that len1 >= 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 of poly1 and poly2.

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 of op1 and op2. Assumes that len1 >= 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 of poly1 and poly2.

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 of in1 of length len1 times in2 of length len2. The output must have space for n coefficients. We assume that len1 >= len2 > 0 and that 0 < 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 of in1 of length len1 times in2 of length len2.

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 of poly1 of length len1 and poly2 of length len2. Assumes len1 >= 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 of poly1 and poly2.

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 first n coefficients of the product of poly1 of length len1 and poly2 of length len2. It is assumed that 0 < n <= len1 + len2 - 1 and that len1 >= 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 first trunc coefficients of the product of poly1 and poly2.

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 of poly1 of length len1 and poly2 of length len2, the other coefficients being arbitrary. It is assumed that len1 >= len2 > 0 and that 0 < 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 of poly1 and poly2, 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 of poly1 and poly2 upon polynomial division by f.

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

Aliasing of f and res 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 of poly1 and poly2 upon polynomial division by f.

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 of poly1 and poly2 upon polynomial division by f.

It is required that finv is the inverse of the reverse of f mod x^lenf. It is required that len1 + len2 - lenf > 0, which is equivalent to requiring that the result will actually be reduced. It is required that len1 < lenf and len2 < lenf. Otherwise, simply use _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 of poly1 and poly2 upon polynomial division by f. finv is the inverse of the reverse of f. It is required that poly1 and poly2 are reduced modulo f.

## Powering¶

void _nmod_poly_pow_binexp(mp_ptr res, mp_srcptr poly, slong len, ulong e, nmod_t mod)

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

void nmod_poly_pow_trunc_binexp(nmod_poly_t res, const nmod_poly_t poly, ulong e, slong trunc)

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

void _nmod_poly_pow_trunc(mp_ptr res, mp_srcptr poly, ulong e, slong trunc, nmod_t mod)

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

void nmod_poly_pow_trunc(nmod_poly_t res, const nmod_poly_t poly, ulong e, slong trunc)

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

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

void nmod_poly_powmod_ui_binexp(nmod_poly_t res, const nmod_poly_t poly, ulong e, const nmod_poly_t f)

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

void _nmod_poly_powmod_mpz_binexp(mp_ptr res, mp_srcptr poly, mpz_srcptr e, mp_srcptr f, slong lenf, nmod_t mod)

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

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

void nmod_poly_powmod_mpz_binexp(nmod_poly_t res, const nmod_poly_t poly, mpz_srcptr e, const nmod_poly_t f)

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

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

void nmod_poly_powmod_fmpz_binexp(nmod_poly_t res, const nmod_poly_t poly, fmpz_t e, const nmod_poly_t f)

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

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

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

void _nmod_poly_powmod_mpz_binexp_preinv(mp_ptr res, mp_srcptr poly, mpz_srcptr e, mp_srcptr f, slong lenf, mp_srcptr finv, slong lenfinv, nmod_t mod)

Sets res to poly raised to the power e modulo f, using binary exponentiation. We require e > 0. We require finv to be the inverse of the reverse of f. We require lenf > 1. It is assumed that poly is already reduced modulo f and zero-padded as necessary to have length exactly lenf - 1. The output res must have room for lenf - 1 coefficients.

void nmod_poly_powmod_mpz_binexp_preinv(nmod_poly_t res, const nmod_poly_t poly, mpz_srcptr e, const nmod_poly_t f, const nmod_poly_t finv)

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

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

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

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

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

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

void nmod_poly_powmod_x_ui_preinv(nmod_poly_t res, ulong e, const nmod_poly_t f, const nmod_poly_t finv)

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

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

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

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

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

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 to f^0, f^1, ..., f^(n-1) mod g. No aliasing is permitted between the entries of res 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^(n-1) mod g, where g has length glen and f is reduced mod g and has length flen (possibly zero spaced). Assumes res is an array of n arrays each with space for at least glen - 1 coefficients and that flen > 0. We require that ginv of length ginvlen is set to the power series inverse of the reverse of g.

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

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

## Division¶

void _nmod_poly_divrem_basecase(mp_ptr Q, mp_ptr R, mp_ptr W, 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 of NMOD_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_div_basecase(mp_ptr Q, mp_ptr W, mp_srcptr A, slong A_len, mp_srcptr B, slong B_len, nmod_t mod)

Notionally finds polynomials $$Q$$ and $$R$$ such that $$A = B Q + R$$ with $$\operatorname{len}(R) < \operatorname{len}(B)$$, but returns only Q. If $$\operatorname{len}(B) = 0$$ an exception is raised. We require that W is temporary space of NMOD_DIV_BC_ITCH(A_len, B_len, mod) coefficients.

void nmod_poly_div_basecase(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)

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

void _nmod_poly_divrem_divconquer_recursive(mp_ptr Q, mp_ptr BQ, mp_ptr W, mp_ptr V, mp_srcptr A, mp_srcptr B, slong lenB, nmod_t mod)

Computes $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R)$$ less than lenB, where A is of length 2 * lenB - 1 and B is of length lenB. Sets BQ to the low lenB - 1 coefficients of B * Q. We require that Q have space for lenB coefficients, that W be temporary space of size lenB - 1 and V be temporary space for a number of coefficients computed by NMOD_DIVREM_DC_ITCH(lenB, mod).

void _nmod_poly_divrem_divconquer(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, where A is of length lenA and B is of length lenB. We require that Q have space for lenA - lenB + 1 coefficients.

void nmod_poly_divrem_divconquer(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_divrem_q0(mp_ptr Q, mp_ptr R, mp_srcptr A, mp_srcptr B, slong lenA, nmod_t mod)

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

Requires that $$Q$$ and $$R$$ have space for $$1$$ and $$\operatorname{len}(B) - 1$$ coefficients, respectively.

Does not support aliasing or zero-padding.

void _nmod_poly_divrem_q1(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 $$\operatorname{len}(R) < \operatorname{len}(B)$$, where $$\operatorname{len}(A) = \operatorname{len}(B) + 1 \geq \operatorname{len}(B) > 0$$.

Requires that $$Q$$ and $$R$$ have space for $$\operatorname{len}(A) - \operatorname{len}(B) + 1$$ and $$\operatorname{len}(B) - 1$$ coefficients, respectively.

Does not support aliasing or zero-padding.

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, where A is of length lenA and B is of length lenB. We require that Q have space for lenA - 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_divconquer_recursive(mp_ptr Q, mp_ptr W, mp_ptr V, mp_srcptr A, mp_srcptr B, slong lenB, nmod_t mod)

Computes $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R)$$ less than lenB, where A is of length 2 * lenB - 1 and B is of length lenB. We require that Q have space for lenB coefficients and that W be temporary space of size lenB - 1 and V be temporary space for a number of coefficients computed by NMOD_DIV_DC_ITCH(lenB, mod).

void _nmod_poly_div_divconquer(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, where A is of length lenA and B is of length lenB, but returns only Q. We require that Q have space for lenA - lenB + 1 coefficients.

void nmod_poly_div_divconquer(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)

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

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, where A is of length lenA and B is of length lenB, but returns only Q. We require that Q have space for lenA - 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_basecase(mp_ptr R, mp_ptr W, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)
void nmod_poly_rem_basecase(nmod_poly_t R, const nmod_poly_t A, const nmod_poly_t B)
void _nmod_poly_rem_q1(mp_ptr R, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, nmod_t mod)

Notationally, computes $$Q$$ and $$R$$ such that $$A = BQ + R$$ with $$\operatorname{len}(R) < \operatorname{len}(B)$$, where $$\operatorname{len}(A) = \operatorname{len}(B) + 1 \geq \operatorname{len}(B) > 0$$, but returns only the remainder.

Requires that $$R$$ has space for $$\operatorname{len}(B) - 1$$ coefficients, respectively.

Does not support aliasing or zero-padding.

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_inv_series_basecase(mp_ptr Qinv, mp_srcptr Q, slong Qlen, slong n, nmod_t mod)

Given Q of length Qlen whose leading coefficient is invertible modulo the given modulus, finds a polynomial Qinv of length n such that the top n coefficients of the product Q * Qinv is $$x^{n - 1}$$. Requires that n > 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 least n find Qinv of length n such that the top n coefficients of the product Q * Qinv is $$x^{n - 1}$$. An exception is raised if n = 0 or if the length of Q is less than n. The leading coefficient of Q must be invertible modulo the modulus of Q. 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 length Qlen whose constant coefficient is invertible modulo the given modulus, find a polynomial Qinv of length n such that Q * Qinv is 1 modulo $$x^n$$. Requires n > 0. This function can be viewed as inverting a power series via Newton iteration.

void nmod_poly_inv_series_newton(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)

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

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

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

void nmod_poly_inv_series(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)

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

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 and B of length Alen and Blen, finds the polynomial Q of length n such that Q * B = A modulo $$x^n$$. We assume n > 0 and that the constant coefficient of B is invertible modulo the given modulus. The polynomial Q must have space for n 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 and B considered modulo n, finds the polynomial Q of length at most n such that Q * B = A modulo $$x^n$$. We assume n > 0 and that the constant coefficient of B is invertible modulo the modulus. An exception is raised if n == 0 or the constant coefficient of B 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 and B of length Alen and Blen, finds the polynomial Q of length n such that Q * B = A modulo $$x^n$$. We assume n > 0 and that the constant coefficient of B is invertible modulo the given modulus. The polynomial Q must have space for n 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 and B considered modulo n, finds the polynomial Q of length at most n such that Q * B = A modulo $$x^n$$. We assume n > 0 and that the constant coefficient of B is invertible modulo the modulus. An exception is raised if n == 0 or the constant coefficient of B is zero.

void _nmod_poly_div_newton(mp_ptr Q, mp_srcptr A, slong Alen, mp_srcptr B, slong Blen, nmod_t mod)

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

We require that $$Q$$ have space for lenA - lenB + 1 coefficients and assume that the leading coefficient of $$B$$ is a unit.

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

void nmod_poly_div_newton(nmod_poly_t Q, const nmod_poly_t A, const nmod_poly_t B)

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.

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(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, where A is of length lenA and B is of length lenB, but return only $$Q$$.

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

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

void 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(mp_ptr Q, mp_ptr R, mp_srcptr A, slong Alen, mp_srcptr B, slong Blen, nmod_t mod)

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

void nmod_poly_divrem_newton(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 $$\operatorname{len}(R) < \operatorname{len}(B)$$. The algorithm used is to call div_newton() and then multiply out and compute the remainder.

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

void 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, len-1) 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.

## 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 of x_prime to the derivative of x which is assumed to be of length len. It is assumed that len > 0.

void nmod_poly_derivative(nmod_poly_t x_prime, const nmod_poly_t x)

Sets x_prime to the derivative of x.

void _nmod_poly_integral(mp_ptr x_int, mp_srcptr x, slong len, nmod_t mod)

Set the first len coefficients of x_int to the integral of x which is assumed to be of length len - 1. The constant term of x_int is set to zero. It is assumed that len > 0. The result is only well-defined if the modulus is a prime number strictly larger than the degree of x. 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 of x with constant term zero. The result is only well-defined if the modulus is a prime number strictly larger than the degree of x.

## Evaluation¶

mp_limb_t _nmod_poly_evaluate_nmod(mp_srcptr poly, slong len, mp_limb_t c, nmod_t mod)

Evaluates poly at the value~c and reduces modulo the given modulus of poly. The value~c should be reduced modulo the modulus. The algorithm used is Horner’s method.

mp_limb_t nmod_poly_evaluate_nmod(nmod_poly_t poly, mp_limb_t c)

Evaluates poly at the value~c and reduces modulo the modulus of poly. The value~c 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 value c and stores the result in dest. The dimension and modulus of dest is assumed to be same as that of c. dest and c 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 value c and stores the result in dest. The dimension and modulus of dest is assumed to be same as that of c. dest and c may be aliased. Paterson-Stockmeyer 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 value c and stores the result in dest. The dimension and modulus of dest is assumed to be same as that of c. dest and c may be aliased. This function automatically switches between Horner’s method and the Paterson-Stockmeyer 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 the n values given in the vector xs, writing the output values to ys. The values in xs 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 the n values given in the vector xs, writing the output values to ys. The values in xs 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 the len values given by the precomputed subproduct tree tree.

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 the n values given in the vector xs, writing the output values to ys. The values in xs 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 the n values given in the vector xs, writing the output values to ys. The values in xs 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 the n values given in the vector xs, writing the output values to ys. The values in xs 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 the n values given in the vector xs, writing the output values to ys. The values in xs 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 most n that interpolates the n given evaluation points xs and values ys. If the interpolating polynomial is shorter than length n, the leading coefficients are set to zero.

The values in xs and ys should be reduced modulo the modulus, and all xs must be distinct. Aliasing between poly and xs or ys 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 length n that interpolates the n given evaluation points xs and values ys. The values in xs and ys should be reduced modulo the modulus, and all xs 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 tree tree and barycentric interpolation weights weights 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 length len1 with poly2 of length len2 and sets res to the result, i.e.evaluates poly1 at poly2. The algorithm used is Horner’s algorithm. We require that res have space for (len1 - 1)*(len2 - 1) + 1 coefficients. It is assumed that len1 > 0 and len2 > 0.

void nmod_poly_compose_horner(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)

Composes poly1 with poly2 and sets res to the result, i.e.evaluates poly1 at poly2. 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 length len1 with poly2 of length len2 and sets res to the result, i.e.evaluates poly1 at poly2. The algorithm used is the divide and conquer algorithm. We require that res have space for (len1 - 1)*(len2 - 1) + 1 coefficients. It is assumed that len1 > 0 and len2 > 0.

void nmod_poly_compose_divconquer(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)

Composes poly1 with poly2 and sets res to the result, i.e.evaluates poly1 at poly2. 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 length len1 with poly2 of length len2 and sets res to the result, i.e.evaluates poly1 at poly2. We require that res have space for (len1 - 1)*(len2 - 1) + 1 coefficients. It is assumed that len1 > 0 and len2 > 0.

void nmod_poly_compose(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2)

Composes poly1 with poly2 and sets res to the result, that is, evaluates poly1 at poly2.

## 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 $$x+c$$ in-place. 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 $$x+c$$ in-place. 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 $$x+c$$ in-place. 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 Brent-Kung 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 Brent-Kung 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 require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

The algorithm used is the Brent-Kung matrix algorithm.

void 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 require hinv to be the inverse of the reverse of h. The algorithm used is the Brent-Kung 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 in nmod_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 require ginv to be the inverse of the reverse of g and $$g$$ to be nonzero. f has to be reduced modulo g and of length one less than leng (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 require ginv to be the inverse of the reverse of g.

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 in nmod_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 require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

The algorithm used is the Brent-Kung matrix algorithm.

void 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 require hinv to be the inverse of the reverse of h. This version of Brent-Kung modular composition is particularly useful if one has to perform several modular composition of the form $$f(g)$$ modulo $$h$$ for fixed $$g$$ and $$h$$.

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 first l elements of polys. 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 require res to have enough memory allocated to hold l nmod_poly_struct’s. The entries of res need to be initialised and l needs to be less than len1 Furthermore, we require hinv to be the inverse of the reverse of h. The output is not allowed to be aliased with any of the inputs.

The algorithm used is the Brent-Kung matrix algorithm.

void 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 first n elements of polys. We require res to have enough memory allocated to hold n nmod_poly_struct. The entries of res need to be initialised and n needs to be less than len1. We require that $$h$$ is nonzero and that $$f_i$$ and $$g$$ have smaller degree than $$h$$. Furthermore, we require hinv to be the inverse of the reverse of h. No aliasing of res and polys is allowed. The algorithm used is the Brent-Kung 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 across flint_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 across flint_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 across flint_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 length lenB, where lenA >= 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 for lenB 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 quotients q 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], and M[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 length lenB, where lenA >= 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 for lenB 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 and T are computed such that S*A + T*B = G. The length of S will be at most lenB and the length of T will be at most lenA.

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

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

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

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

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

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 half-gcd algorithm.

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

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

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

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

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

Assumes that 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 half-gcd algorithm.

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

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

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

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

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

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

slong _nmod_poly_gcdinv(mp_ptr G, mp_ptr S, mp_srcptr A, slong lenA, mp_srcptr B, slong lenB, const nmod_t mod)

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

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

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_ptr A, mp_srcptr B, slong lenB, mp_srcptr P, slong lenP, const nmod_t mod)

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

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

Does not support aliasing.

Assumes that $$mod$$ 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.

## Power series composition¶

mp_limb_t _nmod_poly_discriminant(mp_srcptr poly, slong len, nmod_t mod)

Return the discriminant of (poly, len). Assumes len > 1.

mp_limb_t nmod_poly_discriminant(const nmod_poly_t f)

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

## Power series composition¶

void _nmod_poly_compose_series_horner(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong n)

Sets res to the composition of poly1 and poly2 modulo $$x^n$$, where the constant term of poly2 is required to be zero.

Assumes that len1, len2, n > 0, that len1, len2 <= n, and that (len1-1) * (len2-1) + 1 <= n, and that res has space for n coefficients. Does not support aliasing between any of the inputs and the output.

This implementation uses the Horner scheme.

void nmod_poly_compose_series_horner(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)

Sets res to the composition of poly1 and poly2 modulo $$x^n$$, where the constant term of poly2 is required to be zero.

This implementation uses the Horner scheme.

void _nmod_poly_compose_series_brent_kung(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong n)

Sets res to the composition of poly1 and poly2 modulo $$x^n$$, where the constant term of poly2 is required to be zero.

Assumes that len1, len2, n > 0, that len1, len2 <= n, and that\ (len1-1) * (len2-1) + 1 <= n, and that res has space for n coefficients. Does not support aliasing between any of the inputs and the output.

This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978].

void nmod_poly_compose_series_brent_kung(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong n)

Sets res to the composition of poly1 and poly2 modulo $$x^n$$, where the constant term of poly2 is required to be zero.

This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978].

void _nmod_poly_compose_series_divconquer(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong N, nmod_t mod)

Composes poly1 of length $$\ell_1$$ with poly2 of length $$\ell_2$$ modulo $$x^N$$ and sets res to the result, i.e.evaluates poly1 at poly2.

Writes $$\min\{(\ell_1 - 1)(\ell_2 - 2) + 1, N\}$$ coefficients to the vector res.

The algorithm used is the divide and conquer algorithm. It is assumed that $$0 < \ell_1$$ and $$0 < \ell_2 \leq N$$.

Does not support aliasing between the inputs and the output.

void nmod_poly_compose_series_divconquer(nmod_poly_t res, const nmod_poly_t poly1, const nmod_poly_t poly2, slong N)

Composes poly1 with poly2 modulo $$x^N$$ and sets res to the result, i.e.evaluates poly1 at poly2.

The algorithm used is the divide and conquer algorithm.

void _nmod_poly_compose_series(mp_ptr res, mp_srcptr poly1, slong len1, mp_srcptr poly2, slong len2, slong n)

Sets res to the composition of poly1 and poly2 modulo $$x^n$$, where the constant term of poly2 is required to be zero.

Assumes that len1, len2, n > 0, that len1, len2 <= n, and that\ (len1-1) * (len2-1) + 1 <= n, and that res has space for n coefficients. Does not support aliasing between any of the inputs and the output.

This implementation automatically switches between the Horner scheme and Brent-Kung algorithm 2.1 depending on the size of the inputs.

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 of poly1 and poly2 modulo $$x^n$$, where the constant term of poly2 is required to be zero.

This implementation automatically switches between the Horner scheme and Brent-Kung algorithm 2.1 depending on the size of the inputs.

## Power series reversion¶

void _nmod_poly_revert_series_lagrange(mp_ptr Qinv, mp_srcptr Q, slong n, nmod_t mod)

Sets Qinv to the compositional inverse or reversion of Q as a power series, i.e. computes $$Q^{-1}$$ such that $$Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n$$. The arguments must both have length n and may not be aliased.

It is required that $$Q_0 = 0$$ and that $$Q_1$$ as well as the integers $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation uses the Lagrange inversion formula.

void nmod_poly_revert_series_lagrange(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)

Sets Qinv to the compositional inverse or reversion of Q 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 $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation uses the Lagrange inversion formula.

void _nmod_poly_revert_series_lagrange_fast(mp_ptr Qinv, mp_srcptr Q, slong n, nmod_t mod)

Sets Qinv to the compositional inverse or reversion of Q as a power series, i.e. computes $$Q^{-1}$$ such that $$Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n$$. The arguments must both have length n and may not be aliased.

It is required that $$Q_0 = 0$$ and that $$Q_1$$ as well as the integers $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation uses a reduced-complexity implementation of the Lagrange inversion formula.

void nmod_poly_revert_series_lagrange_fast(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)

Sets Qinv to the compositional inverse or reversion of Q 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 $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation uses a reduced-complexity implementation of the Lagrange inversion formula.

void _nmod_poly_revert_series_newton(mp_ptr Qinv, mp_srcptr Q, slong n, nmod_t mod)

Sets Qinv to the compositional inverse or reversion of Q as a power series, i.e. computes $$Q^{-1}$$ such that $$Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n$$. The arguments must both have length n and may not be aliased.

It is required that $$Q_0 = 0$$ and that $$Q_1$$ as well as the integers $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation uses Newton iteration [BrentKung1978].

void nmod_poly_revert_series_newton(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)

Sets Qinv to the compositional inverse or reversion of Q 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 $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation uses Newton iteration [BrentKung1978].

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

Sets Qinv to the compositional inverse or reversion of Q as a power series, i.e. computes $$Q^{-1}$$ such that $$Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n$$. The arguments must both have length n and may not be aliased.

It is required that $$Q_0 = 0$$ and that $$Q_1$$ as well as the integers $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation automatically chooses between the Lagrange inversion formula and Newton iteration based on the size of the input.

void nmod_poly_revert_series(nmod_poly_t Qinv, const nmod_poly_t Q, slong n)

Sets Qinv to the compositional inverse or reversion of Q 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 $$1, 2, \ldots, n-1$$ are invertible modulo the modulus.

This implementation automatically chooses between the Lagrange inversion formula and Newton iteration based on the size of the input.

## 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 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 and that $$h$$ is zero-padded as necessary to length $$n$$. 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 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 and that $$h$$ is zero-padded as necessary to length $$n$$. 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_monomial_ui(mp_ptr g, mp_limb_t c, ulong r, slong n, nmod_t mod)

Set $$g = \log(1+cx^r) + O(x^n)$$. Assumes $$n > 0$$, $$r > 0$$, and that the coefficient is reduced by the modulus. Works efficiently in linear time.

void nmod_poly_log_series_monomial_ui(nmod_poly_t g, mp_limb_t c, ulong r, slong n)

Set $$g = \log(1+cx^r) + O(x^n)$$. Works efficiently in linear time.

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_monomial_ui(mp_ptr g, mp_limb_t c, ulong r, slong n, nmod_t mod)

Set $$g = \exp(cx^r) + O(x^n)$$. Assumes $$n > 0$$, $$r > 0$$, and that the coefficient is reduced by the modulus. Works efficiently in linear time.

void nmod_poly_exp_series_monomial_ui(nmod_poly_t g, mp_limb_t c, ulong r, slong n)

Set $$g = \exp(cx^r) + O(x^n)$$. Works efficiently in linear time.

void _nmod_poly_exp_series_basecase(mp_ptr g, mp_srcptr h, slong hlen, slong n, nmod_t mod)

Set $$g = \exp(h) + O(x^n)$$ using a simple $$O(n^2)$$ algorithm. Assumes $$n > 0$$ and $$\operatorname{hlen} > 0$$. Only the first $$\operatorname{hlen}$$ coefficients of $$h$$ will be read. Aliasing of $$f$$ and $$h$$ is allowed.

void nmod_poly_exp_series_basecase(nmod_poly_t g, const nmod_poly_t h, slong n)

Set $$g = \exp(h) + O(x^n)$$ using a simple $$O(n^2)$$ algorithm.

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 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 zero-padded 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 n, nmod_t mod)

Set $$g = \operatorname{atan}(h) + O(x^n)$$. Assumes $$n > 0$$ and that $$h$$ is zero-padded as necessary to length $$n$$. 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 n, nmod_t mod)

Set $$g = \operatorname{atanh}(h) + O(x^n)$$. Assumes $$n > 0$$ and that $$h$$ is zero-padded as necessary to length $$n$$. 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 n, nmod_t mod)

Set $$g = \operatorname{asin}(h) + O(x^n)$$. Assumes $$n > 0$$ and that $$h$$ is zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is allowed. The modulus must be less than $$n$$ and not equal to $$2$$.

void nmod_poly_asin_series(nmod_poly_t g, const nmod_poly_t h, slong n)

Set $$g = \operatorname{asin}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

void _nmod_poly_asinh_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)

Set $$g = \operatorname{asinh}(h) + O(x^n)$$. Assumes $$n > 0$$ and that $$h$$ is zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is allowed. The modulus must be less than $$n$$ and not equal to $$2$$.

void nmod_poly_asinh_series(nmod_poly_t g, const nmod_poly_t h, slong n)

Set $$g = \operatorname{asinh}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

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 zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is allowed. The modulus must be less than $$n$$ and not equal to $$2$$. 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 $$g = \operatorname{sin}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

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 zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is allowed. The modulus must be less than $$n$$ and not equal to $$2$$. 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 $$g = \operatorname{cos}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

void _nmod_poly_tan_series(mp_ptr g, mp_srcptr h, slong n, nmod_t mod)

Set $$g = \operatorname{tan}(h) + O(x^n)$$. Assumes $$n > 0$$ and that $$h$$ is zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is not allowed. The modulus must be less than $$n$$. 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 $$g = \operatorname{tan}(h) + O(x^n)$$. The modulus must be less than $$n$$.

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 zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is not allowed. The modulus must be less than $$n$$ and not equal to $$2$$. 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 $$g = \operatorname{sinh}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

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 zero-padded as necessary to length $$n$$. Aliasing of $$g$$ and $$h$$ is not allowed. The modulus must be less than $$n$$ and not equal to $$2$$. 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 $$g = \operatorname{cosh}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

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 zero-padded as necessary to length $$n$$. The modulus must be less than $$n$$ and not equal to $$2$$. 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 $$g = \operatorname{tanh}(h) + O(x^n)$$. The modulus must be less than $$n$$ and not equal to $$2$$.

## 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 $$(x - x_0)(x - x_1) \cdots (x - x_{n-1})$$, the roots $$x_i$$ being given by xs.

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 $$(x - x_0)(x - x_1) \cdots (x - x_{n-1})$$, the roots $$x_i$$ being given by xs.

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 to roots[0] to roots[deg(A) - 1] and return 1. Otherwise, return 0. It is assumed that A is nonzero and that the modulus of A is prime. This function uses Rabin’s probabilistic method via gcd’s with $$(x + \delta)^{\frac{p-1}{2}} - 1$$.

## 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

void _nmod_poly_tree_free(mp_ptr * tree, slong len)

Free the allocated space for the subproduct.

void _nmod_poly_tree_build(mp_ptr * tree, mp_srcptr roots, slong len, nmod_t mod)

Builds a subproduct tree in the preallocated space from the len monic linear factors $$(x-r_i)$$. The top level product is not computed.

## 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 by input and $$n$$ is given by deflation.

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 by input and $$n$$ is given by deflation. 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 if input is the zero polynomial and 1 of input 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 of moduli. The number of moduli, len, should be positive. A return of 0 indicates that the compilation failed and future calls to nmod_poly_multi_crt_precomp() will leave the output undefined. A return of 1 indicates that the compilation was successful, which occurs if and only if either (1) len == 1 and modulus + 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 to values + i modulo the moduli + i in nmod_poly_multi_crt_precompute(). The inputs values + 0, ..., values + len - 1 where len was used in nmod_poly_multi_crt_precompute() are expected to be valid and have modulus matching the modulus of the moduli used in nmod_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 in nmod_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 in outputs + 0, and outputs 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.

## Berlekamp-Massey 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

$U x^n + V (a_1 x^{n-1} + a_{n-1} x + \cdots + a_n) = R, \quad \deg(U) < \deg(V) \le n/2, \quad \deg(R) < n/2.$

The polynomials $$V$$ and $$R$$ may be obtained with nmod_berlekamp_massey_V_poly() and nmod_berlekamp_massey_R_poly(). This class differs from fmpz_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 $$n-1$$ 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 characteristic p 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 $$V$$ and 0 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 be 0. As another example, suppose the object is emptied, the points $$1, 1, 2, 3$$ are added, then reduce is called. This reduce should return 1 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 of 0.

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 be NULL if nmod_berlekamp_massey_point_count() returns 0.

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`.