Struct libbgs::numbers::QuadNum

pub struct QuadNum<const P: u128>(pub FpNum<P>, pub FpNum<P>);

An integer modulo P^2. An element $x$ is represented as $x = a_0 + a_1\sqrt{r}$, where $r$ is the fixed basis element. See Lubeck, Frank. (2003). “Standard generators of finite fields and their cyclic subgroups.” Journal of Symbolic Computation (117) 51-67. Note that the SylowDecomposable implementation for a QuadNum returns the decomposition for the subgroup with $p + 1$ elements, not the full group $\mathbb{F}_{p^2}^\times$. Also, <QuadNum<P> as GroupElem>::SIZE == P + 1, again refering to the subgroup. For these reasons, this API is likely to change in the future to bring the definitions of QuadNum<P> as GroupElem and the SylowDecomp instance in line with describing the full group.

Tuple Fields

0: FpNum<P>

The value $a_0$, when writing this QuadNum as $a_0 + a_1\sqrt{r}$.

1: FpNum<P>

The value $a_1$, when writing this QuadNum as $a_0 + a_1\sqrt{r}$.

Implementations

impl<const P: u128> QuadNum<P>

pub const R: FpNum<P> = _

The basis element for the numbers outside of the prime subfield.

pub const ZERO: QuadNum<P> = _

The constant zero.

pub fn is_zero(&self) -> bool

True if this number is zero; false otherwise.

pub fn steinitz(i: u128) -> QuadNum<P>

Returns the Steinitz element of $\mathbb{F}_{p^2}$ with index i.

pub fn int_sqrt_either(x: FpNum<P>) -> Either<QuadNum<P>, FpNum<P>>

Calculates the square root of an integer modulo P, casting to an FpNum<P> if x is a quadratic residue. Returns a Left QuadNum<P> if x is a quadratic nonresidue, or a Right FpNum<P> if x is a quadratic residue (including 0).

pub fn int_sqrt(x: FpNum<P>) -> QuadNum<P>

Calculates the square root af in integer modulo P.

Trait Implementations

impl<const P: u128> Add<QuadNum<P>> for QuadNum<P>

type Output = QuadNum<P>

The resulting type after applying the + operator.

fn add(self, other: Self) -> QuadNum<P>

Performs the + operation.

impl<const P: u128> AddAssign<QuadNum<P>> for QuadNum<P>

fn add_assign(&mut self, other: Self)

Performs the += operation.

impl<const P: u128> Clone for QuadNum<P>

fn clone(&self) -> QuadNum<P>

Returns a copy of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl<const P: u128> Debug for QuadNum<P>

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl<const P: u128> From<(u128, u128)> for QuadNum<P>

fn from(value: (u128, u128)) -> QuadNum<P>

Converts to this type from the input type.

impl<const P: u128> From<FpNum<P>> for QuadNum<P>

fn from(value: FpNum<P>) -> QuadNum<P>

Converts to this type from the input type.

impl<S, const P: u128> FromChi<S, P> for QuadNum<P>where QuadNum<P>: Factor<S>,

fn from_chi<const L: usize>( chi: &SylowElem<S, L, QuadNum<P>>, decomp: &SylowDecomp<S, L, QuadNum<P>> ) -> FpNum<P>

Returns $\chi + \chi^{-1}$.

fn from_chi_conj<const L: usize>( chi: &SylowElem<S, L, QuadNum<P>>, decomp: &SylowDecomp<S, L, QuadNum<P>> ) -> FpNum<P>

Returns $\chi - \chi^{-1}$.

impl<const P: u128> GroupElem for QuadNum<P>

const ONE: Self = _

The unique identity element of this group.

const SIZE: u128 = _

The size of the group this element belongs to.

fn multiply(&self, other: &QuadNum<P>) -> QuadNum<P>

Returns the product of two elements under the group binary operator. If you implement this trait, you must guarantee that the operation is associative; that is, a.multiply(b.multiply(c)) == a.multiply(b).multiply(c).

fn inverse(&self) -> QuadNum<P>

Returns the multiplicative inverse of this element. If you implement this trait, you must guarantee x.inverse().multiply(x) and x.multiply(x.inverse()) both evaluate to ONE.

const ONE_256: [Self; 256] = _

👎Deprecated: To be replaced by inline const expressions once stabilized.

! 256 copies of Self::ONE in an array.

fn pow(&self, n: u128) -> Self

Raises this element to the power of n.

fn order<S>(&self) -> u128where Self: Factor<S>,

Returns the order of this element, that is, the smallest positive power p for which a.pow(p).is_one() returns True.

impl<const P: u128> Hash for QuadNum<P>

fn hash<__H: Hasher>(&self, state: &mut __H)

Feeds this value into the given Hasher.

fn hash_slice<H>(data: &[Self], state: &mut H)where H: Hasher, Self: Sized,

Feeds a slice of this type into the given Hasher.

impl<const P: u128> Mul<QuadNum<P>> for QuadNum<P>

type Output = QuadNum<P>

The resulting type after applying the * operator.

fn mul(self, other: Self) -> QuadNum<P>

Performs the * operation.

impl<const P: u128> PartialEq<QuadNum<P>> for QuadNum<P>

fn eq(&self, other: &QuadNum<P>) -> bool

This method tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<const P: u128> PartialEq<u128> for QuadNum<P>

fn eq(&self, other: &u128) -> bool

This method tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<const P: u128> Sub<QuadNum<P>> for QuadNum<P>

type Output = QuadNum<P>

The resulting type after applying the - operator.

fn sub(self, other: Self) -> QuadNum<P>

Performs the - operation.

impl<S, const P: u128> SylowDecomposable<S> for QuadNum<P>where QuadNum<P>: Factor<S>,

fn find_sylow_generator(i: usize) -> QuadNum<P>

Finds a Sylow generator for the Sylow subgroup of prime power index i.

fn is_sylow_generator(candidate: &Self, d: (u128, usize)) -> Option<Self>

True if the given element is a generator of the Sylow subgroup of the prime power represented by d.

fn count_elements_of_order(ds: &[usize]) -> u128

Returns the number of elements of a particular order. The argument is the powers of the prime factors of the group’s order.

impl<const P: u128> Copy for QuadNum<P>

impl<const P: u128> Eq for QuadNum<P>

impl<const P: u128> StructuralEq for QuadNum<P>

impl<const P: u128> StructuralPartialEq for QuadNum<P>