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pub struct EdwardsBasepointTableRadix16(_);
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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:

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.

Trait Implementations

Create a table of precomputed multiples of basepoint.

Get the basepoint for this table as an EdwardsPoint.

The computation uses Pippeneger’s algorithm, as described for the specific case of radix-16 on page 13 of the Ed25519 paper.

Piggenger’s Algorithm Generalised

Write the scalar \(a\) in radix-\(w\), where \(w\) is a power of 2, with coefficients in \([\frac{-w}{2},\frac{w}{2})\), i.e., $$ a = a_0 + a_1 w^1 + \cdots + a_{x} w^{x}, $$ with $$ \frac{-w}{2} \leq a_i < \frac{w}{2}, \cdots, \frac{-w}{2} \leq a_{x} \leq \frac{w}{2} $$ and the number of additions, \(x\), is given by \(x = \lceil \frac{256}{w} \rceil\). Then $$ a B = a_0 B + a_1 w^1 B + \cdots + a_{x-1} w^{x-1} B. $$ Grouping even and odd coefficients gives $$ \begin{aligned} a B = \quad a_0 w^0 B +& a_2 w^2 B + \cdots + a_{x-2} w^{x-2} B \\ + a_1 w^1 B +& a_3 w^3 B + \cdots + a_{x-1} w^{x-1} B \\ = \quad(a_0 w^0 B +& a_2 w^2 B + \cdots + a_{x-2} w^{x-2} B) \\ + w(a_1 w^0 B +& a_3 w^2 B + \cdots + a_{x-1} w^{x-2} B). \\ \end{aligned} $$ For each \(i = 0 \ldots 31\), we create a lookup table of $$ [w^{2i} B, \ldots, \frac{w}{2}\cdotw^{2i} B], $$ and use it to select \( y \cdot w^{2i} \cdot B \) in constant time.

The radix-\(w\) representation requires that the scalar is bounded by \(2^{255}\), which is always the case.

The above algorithm is trivially generalised to other powers-of-2 radices.

The type of point contained within this table.

Returns a copy of the value. Read more

Performs copy-assignment from source. Read more

Formats the value using the given formatter. Read more

Performs the conversion.

Performs the conversion.

Performs the conversion.

Performs the conversion.

Performs the conversion.

Performs the conversion.

Performs the conversion.

Performs the conversion.

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

The resulting type after applying the * operator.

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

The resulting type after applying the * operator.

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The type returned in the event of a conversion error.

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The type returned in the event of a conversion error.

Performs the conversion.