Struct curve25519_dalek::edwards::EdwardsBasepointTable

pub struct EdwardsBasepointTable(_);

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, 43A
• EdwardsBasepointTableRadix128: 240KB, 37A
• EdwardsBasepointTableRadix256: 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 i8s). 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

pub fn create(basepoint: &EdwardsPoint) -> EdwardsBasepointTable

Create a table of precomputed multiples of basepoint.

pub fn basepoint_mul(&self, scalar: &Scalar) -> EdwardsPoint

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

Get the basepoint for this table as an EdwardsPoint.

Trait Implementations

impl Clone for EdwardsBasepointTable

fn clone(&self) -> EdwardsBasepointTable

Returns a copy of the value.

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

Performs copy-assignment from source.

impl<'a, 'b> Mul<&'a EdwardsBasepointTable> for &'b Scalar

fn mul(self, basepoint_table: &'a EdwardsBasepointTable) -> EdwardsPoint

Construct an EdwardsPoint from a Scalar \(a\) by computing the multiple \(aB\) of this basepoint \(B\).

type Output = EdwardsPoint

The resulting type after applying the * operator.

impl<'a, 'b> Mul<&'b Scalar> for &'a EdwardsBasepointTable

fn mul(self, scalar: &'b Scalar) -> EdwardsPoint

Construct an EdwardsPoint from a Scalar \(a\) by computing the multiple \(aB\) of this basepoint \(B\).

type Output = EdwardsPoint

The resulting type after applying the * operator.