fq_default.h

Initialise ctx as a context for \mathbb{F}_{p^d}, for a prime p and d \ge 1, with the name var for the generator (used for printing). The defining polynomial is a Conway polynomial if (p, d) is in FLINT's table; otherwise it is chosen as in the underlying module. fq_default_ctx_init chooses the representation by the rules above; fq_default_ctx_init_type uses the given type, one of the five type codes, or 0 for the automatic choice.

Initialise ctx as a context for \mathbb{F}_p[x]/(f), where f is modulus, an irreducible polynomial of degree d \ge 1 over \mathbb{F}_p, and p is the modulus of mod_ctx, respectively of the nmod_poly. The generator, printed as var, is the class of x. The representation is chosen as by fq_default_ctx_init, or given by type (0 for the automatic choice).

Frees the memory used by ctx. Elements, polynomials and matrices over the field must be cleared first.

Returns the representation used by ctx as one of the type codes FQ_DEFAULT_FQ_ZECH (1), FQ_DEFAULT_FQ_NMOD (2), FQ_DEFAULT_FQ (3), FQ_DEFAULT_NMOD (4) and FQ_DEFAULT_FMPZ_MOD (5).

Returns a pointer to the context of the underlying representation: an fq_zech_ctx_struct *, fq_nmod_ctx_struct *, fq_ctx_struct *, nmod_t * or fmpz_mod_ctx_struct *, according to fq_default_ctx_type. Together with the corresponding member of fq_default_struct, this gives direct access to the functions of the underlying module.

Returns the degree d = [\mathbb{F}_q : \mathbb{F}_p].

Sets prime to the characteristic p.

Sets f to the number of elements q = p^d.

Sets p to the defining polynomial of the field. The fmpz_mod_poly must have been initialised with a context whose modulus is the characteristic. For a prime field the result is x - a, where a is the generator.

Print a description of the field to file, respectively to stdout: the characteristic, and for extension fields the degree and the defining polynomial, preceded by Zech Representation: for the Zech representation. fq_default_ctx_fprint returns a positive value on success.

Initialises ctx as a random finite field in a random one of the five representations, for testing: up to 4-bit primes and degree 3 for Zech logarithms, up to 30-bit primes and degree 15 for fq_nmod and nmod, and up to 63-bit primes for fq (degree up to 7) and fmpz_mod.

Initialise rop for the field ctx and set it to zero. fq_default_init2 also allocates space for d coefficients in the polynomial representations, so that later operations need not allocate.

Frees the memory used by rop.

Sets rop to op.

Swaps op1 and op2 efficiently.

Set rop to 0, respectively 1.

Sets rop to the generator of the field: the class of x in \mathbb{F}_p[x]/(f), a root of the defining polynomial f. It generates \mathbb{F}_q as an algebra over \mathbb{F}_p; it generates the multiplicative group if f is primitive, which is the case for Conway polynomials and always in the Zech representation.

Set rop to the image of the integer x in the prime field \mathbb{F}_p \subseteq \mathbb{F}_q.

If op lies in the prime field \mathbb{F}_p, sets z to its representative in [0, p) and returns 1; otherwise returns 0 and leaves z unchanged.

Sets c to the coefficient of x^n in the representative of op as a polynomial of degree less than d in the generator, or to zero if n \ge d. For a prime field, the representative is a constant.

Convert between op and its representative of degree less than d in \mathbb{F}_p[x]. fq_default_set_nmod_poly accepts a polynomial of any degree and reduces it modulo the defining polynomial. The modulus of poly must be p; it is not checked. Only for the types FQ_DEFAULT_FQ_ZECH, FQ_DEFAULT_FQ_NMOD and FQ_DEFAULT_NMOD; the other types throw FLINT_ERROR.

Convert between op and its representative in \mathbb{F}_p[x], as the nmod_poly versions but for all representations. The modulus of the fmpz_mod_poly context must be p; it is not checked.

Convert between an element and an integer polynomial. fq_default_get_fmpz_poly sets a to the representative of b of degree less than d, with coefficients in [0, p). fq_default_set_fmpz_poly reduces the coefficients of b modulo p and the polynomial modulo the defining polynomial.

Return whether op is 0, respectively 1.

Returns whether op1 and op2 are equal.

Returns whether op is invertible, that is, nonzero.

Set rop to \mathit{op1} + \mathit{op2}, respectively \mathit{op1} - \mathit{op2}.

Sets rop to \mathit{op1} - 1.

Sets rop to -\mathit{op1}.

Set rop to \mathit{op1} \cdot \mathit{op2}, respectively \mathit{op}^2.

Set rop to op times the image of the integer x in \mathbb{F}_p.

Sets rop to \mathit{op1}^{-1}. Throws FLINT_ERROR if op1 is zero.

Sets rop to \mathit{op1} / \mathit{op2}. Throws FLINT_ERROR if op2 is zero. The inputs are not modified, although they are not declared const.

Set rop to \mathit{op}^e, with 0^0 = 1. In fq_default_pow the exponent may be negative if op is nonzero; 0^e with e < 0 throws FLINT_ERROR.

If op1 is a square, sets rop to a square root of it and returns 1. Otherwise returns 0, and the value of rop is unspecified.

Returns whether op is a square. If p = 2, every element is a square.

Sets rop to the unique p-th root of op1, which is \mathit{op1}^{p^{d-1}}, the image under the inverse of the Frobenius map.

Sets rop to \sigma^e(\mathit{op}) = \mathit{op}^{p^e}. Any e is allowed, including negative values; it is taken modulo d.

Set rop to the norm N(\mathit{op}), respectively the trace \operatorname{Tr}(\mathit{op}), as an integer in [0, p).

Set rop to a uniformly random element, respectively a uniformly random nonzero element, of the field.

Set rop to a random element, respectively a random nonzero element, for testing. In the polynomial representations, representatives with many zero coefficients are more likely than with uniform sampling.

Print op to file, respectively stdout, in the plain format of the representation: the format of nmod_poly_print (length, modulus, coefficients from the constant term up) for fq_nmod, that of fmpz_poly_print (length, coefficients) for fq, the discrete logarithm k of \mathit{op} = g^k for the Zech representation (with q - 1 for zero), and the residue for prime fields. fq_default_fprint returns a positive value on success.

Print op to file, respectively stdout, in a readable form using the variable name of the context: a polynomial such as a+3 in the polynomial representations, a power such as a^36 in the Zech representation (0 for zero), and the residue in prime fields. fq_default_fprint_pretty returns a positive value on success.

Return the plain, respectively readable, representation of op as a newly allocated string, which the caller must free with flint_free.