# 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 ulong.

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 ulong; 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.

ulong 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(nn_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(nn_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(nn_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(nn_ptr *rop, nn_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.