fq_zech_ctx_init ctx p d 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.

_fq_zech_ctx_init_conway ctx p d 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.

fq_zech_ctx_init_conway ctx p d 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.

fq_zech_ctx_init_random ctx p d 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.

fq_zech_ctx_init_modulus ctx modulus 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.

fq_zech_ctx_init_modulus_check ctx modulus 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.

fq_zech_ctx_init_fq_nmod_ctx ctx ctxn

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

fq_zech_ctx_init_fq_nmod_ctx_check ctx 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.

fq_zech_ctx_clear ctx

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

fq_zech_ctx_modulus ctx

Returns a pointer to the modulus in the context.

fq_zech_ctx_degree ctx

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

fq_zech_ctx_order f ctx

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

fq_zech_ctx_order_ui ctx

Returns the size of the finite field.

fq_zech_ctx_fprint file ctx

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

fq_zech_ctx_print ctx

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

fq_zech_ctx_randtest ctx

Initializes ctx to a random finite field. Assumes that fq_zech_ctx_init has not been called on ctx already.

fq_zech_ctx_randtest_reducible ctx

Since the Zech logarithm representation does not work with a non-irreducible modulus, does the same as fq_zech_ctx_randtest.

fq_zech_init rop ctx

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

fq_zech_init2 rop ctx

Initialises poly with at least enough space for it to be an element of ctx and sets it to \(0\).

fq_zech_clear rop ctx

Clears the element rop.

fq_zech_reduce rop ctx

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

fq_zech_add rop op1 op2 ctx

Sets rop to the sum of op1 and op2.

fq_zech_sub rop op1 op2 ctx

Sets rop to the difference of op1 and op2.

fq_zech_sub_one rop op1 ctx

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

fq_zech_neg rop op ctx

Sets rop to the negative of op.

fq_zech_mul rop op1 op2 ctx

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

fq_zech_mul_fmpz rop op x ctx

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

fq_zech_mul_si rop op x ctx

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

fq_zech_mul_ui rop op x ctx

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

fq_zech_sqr rop op ctx

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

fq_zech_div rop op1 op2 ctx

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

fq_zech_inv rop op ctx

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

fq_zech_gcdinv f inv op 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.

fq_zech_pow rop op e 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 \(e = 0\).

fq_zech_pow_ui rop op e 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 \(e = 0\).

fq_zech_sqrt rop op1 ctx

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

fq_zech_pth_root rop op1 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.

fq_zech_is_square op ctx

Return 1 if op is a square.

fq_zech_fprint_pretty file op 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.

fq_zech_print_pretty op 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.

fq_zech_fprint file op ctx

Prints a representation of op to file.

fq_zech_print op ctx

Prints a representation of op to stdout.

fq_zech_get_str op ctx

Returns the plain FLINT string representation of the element op.

fq_zech_get_str_pretty op ctx

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

fq_zech_randtest rop state ctx

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

fq_zech_randtest_not_zero rop state ctx

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

fq_zech_rand rop state ctx

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

fq_zech_rand_not_zero rop state ctx

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

fq_zech_set rop op ctx

Sets rop to op.

fq_zech_set_si rop x ctx

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

fq_zech_set_ui rop x ctx

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

fq_zech_set_fmpz rop x ctx

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

fq_zech_swap op1 op2 ctx

Swaps the two elements op1 and op2.

fq_zech_zero rop ctx

Sets rop to zero.

fq_zech_one rop ctx

Sets rop to one, reduced in the given context.

fq_zech_gen rop ctx

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

fq_zech_get_fmpz rop op 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.

fq_zech_get_fq_nmod rop op ctx

Sets rop to the fq_nmod_t element corresponding to op.

fq_zech_set_fq_nmod rop op ctx

Sets rop to the fq_zech_t element corresponding to op.

fq_zech_get_nmod_poly a b 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.

fq_zech_set_nmod_poly a b 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.

fq_zech_get_nmod_mat col a ctx

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

fq_zech_set_nmod_mat a col ctx

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

fq_zech_is_zero op ctx

Returns whether op is equal to zero.

fq_zech_is_one op ctx

Returns whether op is equal to one.

fq_zech_equal op1 op2 ctx

Returns whether op1 and op2 are equal.

fq_zech_is_invertible op ctx

Returns whether op is an invertible element.

fq_zech_is_invertible_f f op 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.

fq_zech_trace rop op 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).

fq_zech_norm rop op 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.

fq_zech_frobenius rop op e 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 \(\sigma \colon x \mapsto x^p\).

fq_zech_multiplicative_order ord op 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.

fq_zech_bit_pack f op bit_size ctx

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

fq_zech_bit_unpack rop f bit_size ctx

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