fq.h – finite fields

Description.

Types, macros and constants

fq_ctx_struct

fq_ctx_t

Description.

fq_struct

fq_t

Description.

Context Management

void fq_ctx_init(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)

Initialises the context for prime~`p` and extension degree~`d`, with name var for the generator. By default, it will try use a Conway polynomial; if one is not available, a random irreducible polynomial will be used.

Assumes that \(p\) is a prime.

Assumes that the string var is a null-terminated string of length at least one.

int _fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)

Attempts to initialise the context for prime~`p` and extension degree~`d`, with name var for the generator using a Conway polynomial for the modulus.

Returns \(1\) if the Conway polynomial is in the database for the given size and the initialization is successful; otherwise, returns \(0\).

Assumes that \(p\) is a prime.

Assumes that the string var is a null-terminated string of length at least one.

void fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)

Initialises the context for prime~`p` and extension degree~`d`, with name var for the generator using a Conway polynomial for the modulus.

Assumes that \(p\) is a prime.

Assumes that the string var is a null-terminated string of length at least one.

void fq_ctx_init_modulus(fq_ctx_t ctx, fmpz_mod_poly_t modulus, const char *var)

Initialises the context for given modulus with name var for the generator.

Assumes that modulus is an irreducible polynomial over \(\mathbf{F}_{p}\).

Assumes that the string var is a null-terminated string of length at least one.

void fq_ctx_clear(fq_ctx_t ctx)

Clears all memory that has been allocated as part of the context.

const fmpz_mod_poly_struct* fq_ctx_modulus(const fq_ctx_t ctx)

Returns a pointer to the modulus in the context.

long fq_ctx_degree(const fq_ctx_t ctx)

Returns the degree of the field extension \([\mathbf{F}_{q} : \mathbf{F}_{p}]\), which is equal to \(\log_{p} q\).

fmpz * fq_ctx_prime(const fq_ctx_t ctx)

Returns a pointer to the prime \(p\) in the context.

void fq_ctx_order(fmpz_t f, const fq_ctx_t ctx)

Sets \(f\) to be the size of the finite field.

int fq_ctx_fprint(FILE * file, const fq_ctx_t ctx)

Prints the context information to {tt{file}}. Returns 1 for a success and a negative number for an error.

void fq_ctx_print(const fq_ctx_t ctx)

Prints the context information to {tt{stdout}}.

void fq_ctx_randtest(fq_ctx_t ctx)

Initializes ctx to a random finite field. Assumes that fq_ctx_init as not been called on ctx already.

void fq_ctx_randtest_reducible(fq_ctx_t ctx)

Initializes ctx to a random extension of a prime field. The modulus may or may not be irreducible. Assumes that fq_ctx_init has not been called on ctx already.

Memory management

void fq_init(fq_t rop, const fq_ctx_t ctx)

Initialises the element rop, setting its value to~`0`.

void fq_init2(fq_t rop, const fq_ctx_t ctx)

Initialises poly with at least enough space for it to be an element of ctx and sets it to~`0`.

void fq_clear(fq_t rop, const fq_ctx_t ctx)

Clears the element rop.

void _fq_sparse_reduce(fmpz *R, slong lenR, const fq_ctx_t ctx)

Reduces (R, lenR) modulo the polynomial \(f\) given by the modulus of ctx.

void _fq_dense_reduce(fmpz *R, slong lenR, const fq_ctx_t ctx)

Reduces (R, lenR) modulo the polynomial \(f\) given by the modulus of ctx using Newton division.

void _fq_reduce(fmpz *r, slong lenR, const fq_ctx_t ctx)

Reduces (R, lenR) modulo the polynomial \(f\) given by the modulus of ctx. Does either sparse or dense reduction based on ctx->sparse_modulus.

void fq_reduce(fq_t rop, const fq_ctx_t ctx)

Reduces the polynomial rop as an element of \(\mathbf{F}_p[X] / (f(X))\).

Basic arithmetic

void fq_add(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the sum of op1 and op2.

void fq_sub(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the difference of op1 and op2.

void fq_sub_one(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

Sets rop to the difference of op1 and \(1\).

void fq_neg(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the negative of op.

void fq_mul(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2, reducing the output in the given context.

void fq_mul_fmpz(fq_t rop, const fq_t op, const fmpz_t x, const fq_ctx_t ctx)

Sets rop to the product of op and \(x\), reducing the output in the given context.

void fq_mul_si(fq_t rop, const fq_t op, slong x, const fq_ctx_t ctx)

Sets rop to the product of op and \(x\), reducing the output in the given context.

void fq_mul_ui(fq_t rop, const fq_t op, ulong x, const fq_ctx_t ctx)

Sets rop to the product of op and \(x\), reducing the output in the given context.

void fq_sqr(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the square of op, reducing the output in the given context.

void fq_div(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the quotient of op1 and op2, reducing the output in the given context.

void _fq_inv(fmpz *rop, const fmpz *op, slong len, const fq_ctx_t ctx)

Sets (rop, d) to the inverse of the non-zero element (op, len).

void fq_inv(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the inverse of the non-zero element op.

void fq_gcdinv(fq_t f, fq_t inv, const fq_t op, const fq_ctx_t ctx)

Sets inv to be the inverse of op modulo the modulus of ctx. If op is not invertible, then f is set to a factor of the modulus; otherwise, it is set to one.

void _fq_pow(fmpz *rop, const fmpz *op, slong len, const fmpz_t e, const fq_ctx_t ctx)

Sets (rop, 2*d-1) to (op,len) raised to the power~`e`, reduced modulo \(f(X)\), the modulus of ctx.

Assumes that \(e \geq 0\) and that len is positive and at most~`d`.

Although we require that rop provides space for \(2d - 1\) coefficients, the output will be reduced modulo \(f(X)\), which is a polynomial of degree~`d`.

Does not support aliasing.

void fq_pow(fq_t rop, const fq_t op, const fmpz_t e, const fq_ctx_t ctx)

Sets rop the op raised to the power~`e`.

Currently assumes that \(e \geq 0\).

Note that for any input op, rop is set to~`1` whenever \(e = 0\).

void fq_pow_ui(fq_t rop, const fq_t op, const ulong e, const fq_ctx_t ctx)

Sets rop the op raised to the power~`e`.

Currently assumes that \(e \geq 0\).

Note that for any input op, rop is set to~`1` whenever \(e = 0\).

Roots

void fq_pth_root(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

Sets rop to a \(p^{th}\) root root of op1. Currently, this computes the root by raising op1 to \(p^{d-1}\) where \(d\) is the degree of the extension.

Output

int fq_fprint_pretty(FILE *file, const fq_t op, const fq_ctx_t ctx)

Prints a pretty representation of op to file.

In the current implementation, always returns~`1`. The return code is part of the function’s signature to allow for a later implementation to return the number of characters printed or a non-positive error code.

int fq_print_pretty(const fq_t op, const fq_ctx_t ctx)

Prints a pretty representation of op to stdout.

In the current implementation, always returns~`1`. The return code is part of the function’s signature to allow for a later implementation to return the number of characters printed or a non-positive error code.

void fq_fprint(FILE * file, const fq_t op, const fq_ctx_t ctx)

Prints a representation of op to file.

For further details on the representation used, see fmpz_mod_poly_fprint().

void fq_print(const fq_t op, const fq_ctx_t ctx)

Prints a representation of op to stdout.

For further details on the representation used, see fmpz_mod_poly_print().

char * fq_get_str(const fq_t op, const fq_ctx_t ctx)

Returns the plain FLINT string representation of the element op.

char * fq_get_str_pretty(const fq_t op, const fq_ctx_t ctx)

Returns a pretty representation of the element~``op`` using the null-terminated string x as the variable name.

Randomisation

void fq_randtest(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a random element of \(\mathbf{F}_q\).

void fq_randtest_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a random non-zero element of \(\mathbf{F}_q\).

void fq_randtest_dense(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a random element of \(\mathbf{F}_q\) which has an underlying polynomial with dense coefficients.

Assignments and conversions

void fq_set(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to op.

void fq_set_si(fq_t rop, const slong x, const fq_ctx_t ctx)

Sets rop to x, considered as an element of \(\mathbf{F}_p\).

void fq_set_ui(fq_t rop, const ulong x, const fq_ctx_t ctx)

Sets rop to x, considered as an element of \(\mathbf{F}_p\).

void fq_set_fmpz(fq_t rop, const fmpz_t x, const fq_ctx_t ctx)

Sets rop to x, considered as an element of \(\mathbf{F}_p\).

void fq_swap(fq_t op1, fq_t op2, const fq_ctx_t ctx)

Swaps the two elements op1 and op2.

void fq_zero(fq_t rop, const fq_ctx_t ctx)

Sets rop to zero.

void fq_one(fq_t rop, const fq_ctx_t ctx)

Sets rop to one, reduced in the given context.

void fq_gen(fq_t rop, const fq_ctx_t ctx)

Sets rop to a generator for the finite field. There is no guarantee this is a multiplicative generator of the finite field.

Comparison

int fq_is_zero(const fq_t op, const fq_ctx_t ctx)

Returns whether op is equal to zero.

int fq_is_one(const fq_t op, const fq_ctx_t ctx)

Returns whether op is equal to one.

int fq_equal(const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Returns whether op1 and op2 are equal.

int fq_is_invertible(const fq_t op, const fq_ctx_t ctx)

Returns whether op is an invertible element.

int fq_is_invertible_f(fq_t f, const fq_t op, const fq_ctx_t ctx)

Returns whether op is an invertible element. If it is not, then f is set of a factor of the modulus.

Special functions

void _fq_trace(fmpz_t rop, const fmpz *op, slong len, const fq_ctx_t ctx)

Sets rop to the trace of the non-zero element (op, len) in \(\mathbf{F}_{q}\).

void fq_trace(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the trace of op.

For an element \(a \in \mathbf{F}_q\), multiplication by \(a\) defines a \(\mathbf{F}_p\)-linear map on \(\mathbf{F}_q\). We define the trace of \(a\) as the trace of this map. Equivalently, if \(\Sigma\) generates \(\Gal(\mathbf{F}_q / \mathbf{F}_p)\) then the trace of \(a\) is equal to \(\sum_{i=0}^{d-1} \Sigma^i (a)\), where \(d = \log_{p} q\).

void _fq_norm(fmpz_t rop, const fmpz *op, slong len, const fq_ctx_t ctx)

Sets rop to the norm of the non-zero element (op, len) in \(\mathbf{F}_{q}\).

void fq_norm(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

Computes the norm of op.

For an element \(a \in \mathbf{F}_q\), multiplication by \(a\) defines a \(\mathbf{F}_p\)-linear map on \(\mathbf{F}_q\). We define the norm of \(a\) as the determinant of this map. Equivalently, if \(\Sigma\) generates \(\Gal(\mathbf{F}_q / \mathbf{F}_p)\) then the trace of \(a\) is equal to \(\prod_{i=0}^{d-1} \Sigma^i (a)\), where \(d = \text{dim}_{\mathbf{F}_p}(\mathbf{F}_q)\).

Algorithm selection is automatic depending on the input.

void _fq_frobenius(fmpz *rop, const fmpz *op, slong len, slong e, const fq_ctx_t ctx)

Sets (rop, 2d-1) to the image of (op, len) under the Frobenius operator raised to the e-th power, assuming that neither op nor e are zero.

void fq_frobenius(fq_t rop, const fq_t op, slong e, const fq_ctx_t ctx)

Evaluates the homomorphism \(\Sigma^e\) at op.

Recall that \(\mathbf{F}_q / \mathbf{F}_p\) is Galois with Galois group \(\langle \sigma \rangle\), which is also isomorphic to \(\mathbf{Z}/d\mathbf{Z}\), where \(\sigma \in \Gal(\mathbf{F}_q/\mathbf{F}_p)\) is the Frobenius element \(\sigma \colon x \mapsto x^p\).

int fq_multiplicative_order(fmpz_t ord, const fq_t op, const fq_ctx_t ctx)

Computes the order of op as an element of the multiplicative group of ctx.

Returns 0 if op is 0, otherwise it returns 1 if op is a generator of the multiplicative group, and -1 if it is not.

int fq_is_primitive(const fq_t op, const fq_ctx_t ctx)

Returns whether op is primitive, i.e., whether it is a generator of the multiplicative group of ctx.

Bit packing

void fq_bit_pack(fmpz_t f, const fq_t op, mp_bitcnt_t bit_size, const fq_ctx_t ctx)

Packs op into bitfields of size bit_size, writing the result to f.

void fq_bit_unpack(fq_t rop, const fmpz_t f, mp_bitcnt_t bit_size, const fq_ctx_t ctx)

Unpacks into rop the element with coefficients packed into fields of size bit_size as represented by the integer f.