Struct curve25519_dalek::edwards::EdwardsBasepointTable [−][src]
A precomputed table of multiples of a basepoint, for accelerating
fixed-base scalar multiplication. One table, for the Ed25519
basepoint, is provided in the constants
module.
The basepoint tables are reasonably large, so they should probably be boxed.
The sizes for the tables and the number of additions required for one scalar multiplication are as follows:
EdwardsBasepointTableRadix16
: 30KB, 64A (this is the default size, and is used for [ED25519_BASEPOINT_TABLE
])EdwardsBasepointTableRadix64
: 120KB, 43AEdwardsBasepointTableRadix128
: 240KB, 37AEdwardsBasepointTableRadix256
: 480KB, 33A
Why 33 additions for radix-256?
Normally, the radix-256 tables would allow for only 32 additions per scalar
multiplication. However, due to the fact that standardised definitions of
legacy protocols—such as x25519—require allowing unreduced 255-bit scalar
invariants, when converting such an unreduced scalar’s representation to
radix-\(2^{8}\), we cannot guarantee the carry bit will fit in the last
coefficient (the coefficients are i8
s). When, \(w\), the power-of-2 of
the radix, is \(w < 8\), we can fold the final carry onto the last
coefficient, \(d\), because \(d < 2^{w/2}\), so
$$
d + carry \cdot 2^{w} = d + 1 \cdot 2^{w} < 2^{w+1} < 2^{8}
$$
When \(w = 8\), we can’t fit \(carry \cdot 2^{w}\) into an i8
, so we
add the carry bit onto an additional coefficient.
Implementations
impl EdwardsBasepointTable
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pub fn create(basepoint: &EdwardsPoint) -> EdwardsBasepointTable
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Create a table of precomputed multiples of basepoint
.
pub fn basepoint_mul(&self, scalar: &Scalar) -> EdwardsPoint
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The computation uses Pippenger’s algorithm, as described on page 13 of the Ed25519 paper. Write the scalar \(a\) in radix \(16\) with coefficients in \([-8,8)\), i.e., $$ a = a_0 + a_1 16^1 + \cdots + a_{63} 16^{63}, $$ with \(-8 \leq a_i < 8\), \(-8 \leq a_{63} \leq 8\). Then $$ a B = a_0 B + a_1 16^1 B + \cdots + a_{63} 16^{63} B. $$ Grouping even and odd coefficients gives $$ \begin{aligned} a B = \quad a_0 16^0 B +& a_2 16^2 B + \cdots + a_{62} 16^{62} B \\ + a_1 16^1 B +& a_3 16^3 B + \cdots + a_{63} 16^{63} B \\ = \quad(a_0 16^0 B +& a_2 16^2 B + \cdots + a_{62} 16^{62} B) \\ + 16(a_1 16^0 B +& a_3 16^2 B + \cdots + a_{63} 16^{62} B). \\ \end{aligned} $$ For each \(i = 0 \ldots 31\), we create a lookup table of $$ [16^{2i} B, \ldots, 8\cdot16^{2i} B], $$ and use it to select \( x \cdot 16^{2i} \cdot B \) in constant time.
The radix-\(16\) representation requires that the scalar is bounded by \(2^{255}\), which is always the case.
pub fn basepoint(&self) -> EdwardsPoint
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Get the basepoint for this table as an EdwardsPoint
.
Trait Implementations
impl Clone for EdwardsBasepointTable
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fn clone(&self) -> EdwardsBasepointTable
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pub fn clone_from(&mut self, source: &Self)
1.0.0[src]
impl<'a, 'b> Mul<&'a EdwardsBasepointTable> for &'b Scalar
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type Output = EdwardsPoint
The resulting type after applying the *
operator.
fn mul(self, basepoint_table: &'a EdwardsBasepointTable) -> EdwardsPoint
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Construct an EdwardsPoint
from a Scalar
\(a\) by
computing the multiple \(aB\) of this basepoint \(B\).
impl<'a, 'b> Mul<&'b Scalar> for &'a EdwardsBasepointTable
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type Output = EdwardsPoint
The resulting type after applying the *
operator.
fn mul(self, scalar: &'b Scalar) -> EdwardsPoint
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Construct an EdwardsPoint
from a Scalar
\(a\) by
computing the multiple \(aB\) of this basepoint \(B\).
Auto Trait Implementations
impl RefUnwindSafe for EdwardsBasepointTable
impl Send for EdwardsBasepointTable
impl Sync for EdwardsBasepointTable
impl Unpin for EdwardsBasepointTable
impl UnwindSafe for EdwardsBasepointTable
Blanket Implementations
impl<T> Any for T where
T: 'static + ?Sized,
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T: 'static + ?Sized,
impl<T> Borrow<T> for T where
T: ?Sized,
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T: ?Sized,
impl<T> BorrowMut<T> for T where
T: ?Sized,
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T: ?Sized,
pub fn borrow_mut(&mut self) -> &mut T
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impl<T, U> Cast<U> for T where
U: FromCast<T>,
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U: FromCast<T>,
impl<T> From<T> for T
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impl<T> FromBits<T> for T
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impl<T> FromCast<T> for T
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impl<T, U> Into<U> for T where
U: From<T>,
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U: From<T>,
impl<T, U> IntoBits<U> for T where
U: FromBits<T>,
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U: FromBits<T>,
impl<T> Same<T> for T
type Output = T
Should always be Self
impl<T> ToOwned for T where
T: Clone,
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T: Clone,
type Owned = T
The resulting type after obtaining ownership.
pub fn to_owned(&self) -> T
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pub fn clone_into(&self, target: &mut T)
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impl<T, U> TryFrom<U> for T where
U: Into<T>,
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U: Into<T>,
type Error = Infallible
The type returned in the event of a conversion error.
pub fn try_from(value: U) -> Result<T, <T as TryFrom<U>>::Error>
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impl<T, U> TryInto<U> for T where
U: TryFrom<T>,
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U: TryFrom<T>,