fq_nmod_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 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.

_fq_nmod_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.

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

fq_nmod_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.

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

fq_nmod_ctx_init_modulus ctx modulus 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.

fq_nmod_ctx_clear ctx

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

fq_nmod_ctx_modulus ctx

Returns a pointer to the modulus in the context.

fq_nmod_ctx_degree ctx

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

fq_nmod_ctx_order f ctx

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

fq_nmod_ctx_fprint file ctx

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

fq_nmod_ctx_print ctx

Prints the context information to stdout.

fq_nmod_ctx_randtest ctx

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

fq_nmod_ctx_randtest_reducible ctx

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

fq_nmod_init rop ctx

Initialises the element rop, setting its value to \(0\). Currently, the behaviour is identical to fq_nmod_init2, as it also ensures rop has enough space for it to be an element of ctx, this may change in the future.

fq_nmod_init2 rop ctx

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

fq_nmod_clear rop ctx

Clears the element rop.

_fq_nmod_sparse_reduce R lenR ctx

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

_fq_nmod_dense_reduce R lenR ctx

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

_fq_nmod_reduce r lenR 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.

fq_nmod_reduce rop ctx

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

fq_nmod_add rop op1 op2 ctx

Sets rop to the sum of op1 and op2.

fq_nmod_sub rop op1 op2 ctx

Sets rop to the difference of op1 and op2.

fq_nmod_sub_one rop op1 ctx

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

fq_nmod_neg rop op ctx

Sets rop to the negative of op.

fq_nmod_mul rop op1 op2 ctx

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

fq_nmod_mul_fmpz rop op x ctx

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

fq_nmod_mul_si rop op x ctx

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

fq_nmod_mul_ui rop op x ctx

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

fq_nmod_sqr rop op ctx

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

_fq_nmod_inv rop op len ctx

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

fq_nmod_inv rop op ctx

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

fq_nmod_gcdinv f inv op 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.

_fq_nmod_pow rop op len e 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.

fq_nmod_pow rop op e ctx

Sets rop to 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_nmod_pow_ui rop op e ctx

Sets rop to 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_nmod_sqrt rop op1 ctx

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

fq_nmod_pth_root rop op1 ctx

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

fq_nmod_is_square op ctx

Return 1 if op is a square.

fq_nmod_fprint_pretty file op ctx

Prints a pretty representation of op to file.

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

fq_nmod_print_pretty op ctx

Prints a pretty representation of op to stdout.

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

fq_nmod_fprint file op ctx

Prints a representation of op to file.

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

fq_nmod_print op ctx

Prints a representation of op to stdout.

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

fq_nmod_get_str op ctx

Returns the plain FLINT string representation of the element op.

fq_nmod_get_str_pretty op ctx

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

fq_nmod_randtest rop state ctx

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

fq_nmod_randtest_not_zero rop state ctx

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

fq_nmod_randtest_dense rop state ctx

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

fq_nmod_rand rop state ctx

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

fq_nmod_rand_not_zero rop state ctx

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

fq_nmod_set rop op ctx

Sets rop to op.

fq_nmod_set_si rop x ctx

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

fq_nmod_set_ui rop x ctx

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

fq_nmod_set_fmpz rop x ctx

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

fq_nmod_swap op1 op2 ctx

Swaps the two elements op1 and op2.

fq_nmod_zero rop ctx

Sets rop to zero.

fq_nmod_one rop ctx

Sets rop to one, reduced in the given context.

fq_nmod_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_nmod_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_nmod_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_nmod_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_nmod_get_nmod_mat col a ctx

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

fq_nmod_set_nmod_mat a col ctx

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

fq_nmod_is_zero op ctx

Returns whether op is equal to zero.

fq_nmod_is_one op ctx

Returns whether op is equal to one.

fq_nmod_equal op1 op2 ctx

Returns whether op1 and op2 are equal.

fq_nmod_is_invertible op ctx

Returns whether op is an invertible element.

fq_nmod_is_invertible_f f op ctx

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

fq_nmod_cmp a b ctx

Return 1 (resp. -1, or 0) if a is after (resp. before, same as) b in some arbitrary but fixed total ordering of the elements.

_fq_nmod_trace rop op len ctx

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

fq_nmod_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_nmod_norm rop op len ctx

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

fq_nmod_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_nmod_frobenius rop op len e 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.

fq_nmod_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_nmod_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.

This function can also be used to check primitivity of a generator of a finite field whose defining polynomial is not primitive.

fq_nmod_is_primitive op ctx

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

fq_nmod_bit_pack f op bit_size ctx

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

fq_nmod_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.