fq_zech.h – finite fields (Zech logarithm representation)

We represent an element of the finite field as a power of a generator for the multiplicative group of the finite field. In particular, we use a root of \(f(x)\), where \(f(X) \in \mathbf{F}_p[X]\) is a monic, irreducible polynomial of degree \(n\), as a polynomial in \(\mathbf{F}_p[X]\) of degree less than \(n\). The underlying data structure is just an mp_limb_t.

The default choice for \(f(X)\) is the Conway polynomial for the pair \((p,n)\), enabled by Frank Lübeck’s data base of Conway polynomials using the _nmod_poly_conway() function. If a Conway polynomial is not available, then a random irreducible polynomial will be chosen for \(f(X)\). Additionally, the user is able to supply their own \(f(X)\).

We required that the order of the field fits inside of an mp_limb_t; however, it is recommended that \(p^n < 2^{20}\) due to the time and memory needed to compute the Zech logarithm table.

Types, macros and constants

type fq_zech_ctx_struct

type fq_zech_ctx_t

type fq_zech_struct

type fq_zech_t

Context Management

void fq_zech_ctx_init_ui(fq_zech_ctx_t ctx, ulong 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 primitive polynomial will be used.

Assumes that \(p\) is a prime and \(p^d < 2^{\mathtt{FLINT\_BITS}}\).

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

int _fq_zech_ctx_init_conway_ui(fq_zech_ctx_t ctx, ulong 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 and \(p^d < 2^\mathtt{FLINT\_BITS}\).

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

void fq_zech_ctx_init_conway_ui(fq_zech_ctx_t ctx, ulong 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 and \(p^d < 2^\mathtt{FLINT\_BITS}\).

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

void fq_zech_ctx_init_random_ui(fq_zech_ctx_t ctx, ulong p, slong d, const char *var)

Initialises the context for prime \(p\) and extension degree \(d\), with name var for the generator using a random primitive polynomial.

Assumes that \(p\) is a prime and \(p^d < 2^\mathtt{FLINT\_BITS}\).

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

void fq_zech_ctx_init_modulus(fq_zech_ctx_t ctx, const nmod_poly_t modulus, const char *var)

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

Assumes that modulus is an primitive polynomial over \(\mathbf{F}_{p}\). An exception is raised if a non-primitive modulus is detected.

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

int fq_zech_ctx_init_modulus_check(fq_zech_ctx_t ctx, const nmod_poly_t modulus, const char *var)

As per the previous function, but returns \(0\) if the modulus was not primitive and \(1\) if the context was successfully initialised with the given modulus. No exception is raised.

void fq_zech_ctx_init_fq_nmod_ctx(fq_zech_ctx_t ctx, fq_nmod_ctx_t ctxn)

Initializes the context ctx to be the Zech representation for the finite field given by ctxn.

int fq_zech_ctx_init_fq_nmod_ctx_check(fq_zech_ctx_t ctx, fq_nmod_ctx_t ctxn)

As per the previous function but returns \(0\) if a non-primitive modulus is detected. Returns \(0\) if the Zech representation was successfully initialised.

void fq_zech_ctx_init_randtest(fq_zech_ctx_t ctx, flint_rand_t state, int type)

Initialises ctx to a random finite field, where the prime and degree is set according to type. If type is \(0\) the prime and degree may be large, else if type is \(1\) the degree is small but the prime may be large, else if type is \(2\) the prime is small but the degree may be large, else if type is \(3\) both prime and degree are small.

void fq_zech_ctx_init_randtest_reducible(fq_zech_ctx_t ctx, flint_rand_t state, int type)

Since the Zech logarithm representation does not work with a non-irreducible modulus, this function does the same as fq_zech_ctx_init_randtest().

void fq_zech_ctx_clear(fq_zech_ctx_t ctx)

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

const nmod_poly_struct *fq_zech_ctx_modulus(const fq_zech_ctx_t ctx)

Returns a pointer to the modulus in the context.

slong fq_zech_ctx_degree(const fq_zech_ctx_t ctx)

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

ulong fq_zech_ctx_prime(const fq_zech_ctx_t ctx)

Returns the prime \(p\) of the context.

void fq_zech_ctx_order(fmpz_t f, const fq_zech_ctx_t ctx)

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

mp_limb_t fq_zech_ctx_order_ui(const fq_zech_ctx_t ctx)

Returns the size of the finite field.

int fq_zech_ctx_fprint(FILE *file, const fq_zech_ctx_t ctx)

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

void fq_zech_ctx_print(const fq_zech_ctx_t ctx)

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

Memory management

void fq_zech_init(fq_zech_t rop, const fq_zech_ctx_t ctx)

Initialises the element rop, setting its value to \(0\).

void fq_zech_init2(fq_zech_t rop, const fq_zech_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_zech_clear(fq_zech_t rop, const fq_zech_ctx_t ctx)

Clears the element rop.

void _fq_zech_sparse_reduce(mp_ptr R, slong lenR, const fq_zech_ctx_t ctx)

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

void _fq_zech_dense_reduce(mp_ptr R, slong lenR, const fq_zech_ctx_t ctx)

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

void _fq_zech_reduce(mp_ptr r, slong lenR, const fq_zech_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_zech_reduce(fq_zech_t rop, const fq_zech_ctx_t ctx)

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

Basic arithmetic

void fq_zech_add(fq_zech_t rop, const fq_zech_t op1, const fq_zech_t op2, const fq_zech_ctx_t ctx)

Sets rop to the sum of op1 and op2.

void fq_zech_sub(fq_zech_t rop, const fq_zech_t op1, const fq_zech_t op2, const fq_zech_ctx_t ctx)

Sets rop to the difference of op1 and op2.

void fq_zech_sub_one(fq_zech_t rop, const fq_zech_t op1, const fq_zech_ctx_t ctx)

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

void fq_zech_neg(fq_zech_t rop, const fq_zech_t op, const fq_zech_ctx_t ctx)

Sets rop to the negative of op.

void fq_zech_mul(fq_zech_t rop, const fq_zech_t op1, const fq_zech_t op2, const fq_zech_ctx_t ctx)

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

void fq_zech_mul_fmpz(fq_zech_t rop, const fq_zech_t op, const fmpz_t x, const fq_zech_ctx_t ctx)

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

void fq_zech_mul_si(fq_zech_t rop, const fq_zech_t op, slong x, const fq_zech_ctx_t ctx)

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

void fq_zech_mul_ui(fq_zech_t rop, const fq_zech_t op, ulong x, const fq_zech_ctx_t ctx)

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

void fq_zech_sqr(fq_zech_t rop, const fq_zech_t op, const fq_zech_ctx_t ctx)

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

void fq_zech_div(fq_zech_t rop, const fq_zech_t op1, const fq_zech_t op2, const fq_zech_ctx_t ctx)

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

void _fq_zech_inv(mp_ptr *rop, mp_srcptr *op, slong len, const fq_zech_ctx_t ctx)

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

void fq_zech_inv(fq_zech_t rop, const fq_zech_t op, const fq_zech_ctx_t ctx)

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

void fq_zech_gcdinv(fq_zech_t f, fq_zech_t inv, const fq_zech_t op, const fq_zech_ctx_t ctx)

Sets inv to be the inverse of op modulo the modulus of ctx and sets f to one. Since the modulus for ctx is always irreducible, op is always invertible.

void _fq_zech_pow(fmpz *rop, const fmpz *op, slong len, const fmpz_t e, const fmpz *a, const slong *j, slong lena, const fmpz_t p)

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_zech_pow(fq_zech_t rop, const fq_zech_t op, const fmpz_t e, const fq_zech_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_zech_pow_ui(fq_zech_t rop, const fq_zech_t op, const ulong e, const fq_zech_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

int fq_zech_sqrt(fq_zech_t rop, const fq_zech_t op1, const fq_zech_ctx_t ctx)

Sets rop to the square root of op1 if it is a square, and return \(1\), otherwise return \(0\).

void fq_zech_pth_root(fq_zech_t rop, const fq_zech_t op1, const fq_zech_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.

int fq_zech_is_square(const fq_zech_t op, const fq_zech_ctx_t ctx)

Return 1 if op is a square.

Output

int fq_zech_fprint_pretty(FILE *file, const fq_zech_t op, const fq_zech_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.

void fq_zech_print_pretty(const fq_zech_t op, const fq_zech_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.

int fq_zech_fprint(FILE *file, const fq_zech_t op, const fq_zech_ctx_t ctx)

Prints a representation of op to file.

void fq_zech_print(const fq_zech_t op, const fq_zech_ctx_t ctx)

Prints a representation of op to stdout.

char *fq_zech_get_str(const fq_zech_t op, const fq_zech_ctx_t ctx)

Returns the plain FLINT string representation of the element op.

char *fq_zech_get_str_pretty(const fq_zech_t op, const fq_zech_ctx_t ctx)

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

Randomisation

void fq_zech_randtest(fq_zech_t rop, flint_rand_t state, const fq_zech_ctx_t ctx)

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

void fq_zech_randtest_not_zero(fq_zech_t rop, flint_rand_t state, const fq_zech_ctx_t ctx)

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

void fq_zech_randtest_dense(fq_zech_t rop, flint_rand_t state, const fq_zech_ctx_t ctx)

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

void fq_zech_rand(fq_zech_t rop, flint_rand_t state, const fq_zech_ctx_t ctx)

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

void fq_zech_rand_not_zero(fq_zech_t rop, flint_rand_t state, const fq_zech_ctx_t ctx)

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

Assignments and conversions

void fq_zech_set(fq_zech_t rop, const fq_zech_t op, const fq_zech_ctx_t ctx)

Sets rop to op.

void fq_zech_set_si(fq_zech_t rop, const slong x, const fq_zech_ctx_t ctx)

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

void fq_zech_set_ui(fq_zech_t rop, const ulong x, const fq_zech_ctx_t ctx)

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

void fq_zech_set_fmpz(fq_zech_t rop, const fmpz_t x, const fq_zech_ctx_t ctx)

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

void fq_zech_swap(fq_zech_t op1, fq_zech_t op2, const fq_zech_ctx_t ctx)

Swaps the two elements op1 and op2.

void fq_zech_zero(fq_zech_t rop, const fq_zech_ctx_t ctx)

Sets rop to zero.

void fq_zech_one(fq_zech_t rop, const fq_zech_ctx_t ctx)

Sets rop to one, reduced in the given context.

void fq_zech_gen(fq_zech_t rop, const fq_zech_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.

int fq_zech_get_fmpz(fmpz_t rop, const fq_zech_t op, const fq_zech_ctx_t ctx)

If op has a lift to the integers, return \(1\) and set rop to the lift in \([0,p)\). Otherwise, return \(0\) and leave \(rop\) undefined.

void fq_zech_get_fq_nmod(fq_nmod_t rop, const fq_zech_t op, const fq_zech_ctx_t ctx)

Sets rop to the fq_nmod_t element corresponding to op.

void fq_zech_set_fq_nmod(fq_zech_t rop, const fq_nmod_t op, const fq_zech_ctx_t ctx)

Sets rop to the fq_zech_t element corresponding to op.

void fq_zech_get_nmod_poly(nmod_poly_t a, const fq_zech_t b, const fq_zech_ctx_t ctx)

Set a to a representative of b in ctx. The representatives are taken in \((\mathbb{Z}/p\mathbb{Z})[x]/h(x)\) where \(h(x)\) is the defining polynomial in ctx.

void fq_zech_set_nmod_poly(fq_zech_t a, const nmod_poly_t b, const fq_zech_ctx_t ctx)

Set a to the element in ctx with representative b. The representatives are taken in \((\mathbb{Z}/p\mathbb{Z})[x]/h(x)\) where \(h(x)\) is the defining polynomial in ctx.

void fq_zech_get_nmod_mat(nmod_mat_t col, const fq_zech_t a, const fq_zech_ctx_t ctx)

Convert a to a column vector of length degree(ctx).

void fq_zech_set_nmod_mat(fq_zech_t a, const nmod_mat_t col, const fq_zech_ctx_t ctx)

Convert a column vector col of length degree(ctx) to an element of ctx.

Comparison

int fq_zech_is_zero(const fq_zech_t op, const fq_zech_ctx_t ctx)

Returns whether op is equal to zero.

int fq_zech_is_one(const fq_zech_t op, const fq_zech_ctx_t ctx)

Returns whether op is equal to one.

int fq_zech_equal(const fq_zech_t op1, const fq_zech_t op2, const fq_zech_ctx_t ctx)

Returns whether op1 and op2 are equal.

int fq_zech_is_invertible(const fq_zech_t op, const fq_zech_ctx_t ctx)

Returns whether op is an invertible element.

int fq_zech_is_invertible_f(fq_zech_t f, const fq_zech_t op, const fq_zech_ctx_t ctx)

Returns whether op is an invertible element. If it is not, then f is set of a factor of the modulus. Since the modulus for an fq_zech_ctx_t is always irreducible, then any non-zero op will be invertible.

Special functions

void fq_zech_trace(fmpz_t rop, const fq_zech_t op, const fq_zech_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 \(\operatorname{Gal}(\mathbf{F}_q / \mathbf{F}_p)\) then the trace of \(a\) is equal to \(\sum_{i=0}^{d-1} \Sigma^i (a)\), where \(d = \log_{p} q\).

void fq_zech_norm(fmpz_t rop, const fq_zech_t op, const fq_zech_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 \(\operatorname{Gal}(\mathbf{F}_q / \mathbf{F}_p)\) 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_zech_frobenius(fq_zech_t rop, const fq_zech_t op, slong e, const fq_zech_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 \(\sigma \in \operatorname{Gal}(\mathbf{F}_q/\mathbf{F}_p)\) is the Frobenius element \(\sigma \colon x \mapsto x^p\).

int fq_zech_multiplicative_order(fmpz *ord, const fq_zech_t op, const fq_zech_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.

Note that ctx must already correspond to a finite field defined by a primitive polynomial and so this function cannot be used to check primitivity of the generator, but can be used to check that other elements are primitive.

int fq_zech_is_primitive(const fq_zech_t op, const fq_zech_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_zech_bit_pack(fmpz_t f, const fq_zech_t op, flint_bitcnt_t bit_size, const fq_zech_ctx_t ctx)

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

void fq_zech_bit_unpack(fq_zech_t rop, const fmpz_t f, flint_bitcnt_t bit_size, const fq_zech_ctx_t ctx)

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