# [−][src]Struct curve25519_dalek::backend::serial::scalar_mul::straus::Straus

pub struct Straus {}

Perform multiscalar multiplication by the interleaved window method, also known as Straus' method (since it was apparently first published by Straus in 1964, as a solution to a problem posted in the American Mathematical Monthly in 1963).

It is easy enough to reinvent, and has been repeatedly. The basic idea is that when computing $Q = s_1 P_1 + \cdots + s_n P_n$ by means of additions and doublings, the doublings can be shared across the $$P_i$$.

We implement two versions, a constant-time algorithm using fixed windows and a variable-time algorithm using sliding windows. They are slight variations on the same idea, and are described in more detail in the respective implementations.

## Trait Implementations

### impl MultiscalarMul for Straus[src]

#### type Point = EdwardsPoint

The type of point being multiplied, e.g., RistrettoPoint.

#### fn multiscalar_mul<I, J>(scalars: I, points: J) -> EdwardsPoint where    I: IntoIterator,    I::Item: Borrow<Scalar>,    J: IntoIterator,    J::Item: Borrow<EdwardsPoint>, [src]

Constant-time Straus using a fixed window of size $$4$$.

Our goal is to compute $Q = s_1 P_1 + \cdots + s_n P_n.$

For each point $$P_i$$, precompute a lookup table of $P_i, 2P_i, 3P_i, 4P_i, 5P_i, 6P_i, 7P_i, 8P_i.$

For each scalar $$s_i$$, compute its radix-$$2^4$$ signed digits $$s_{i,j}$$, i.e., $s_i = s_{i,0} + s_{i,1} 16^1 + ... + s_{i,63} 16^{63},$ with $$-8 \leq s_{i,j} < 8$$. Since $$0 \leq |s_{i,j}| \leq 8$$, we can retrieve $$s_{i,j} P_i$$ from the lookup table with a conditional negation: using signed digits halves the required table size.

Then as in the single-base fixed window case, we have \begin{aligned} s_i P_i &= P_i (s_{i,0} + s_{i,1} 16^1 + \cdots + s_{i,63} 16^{63}) \\ s_i P_i &= P_i s_{i,0} + P_i s_{i,1} 16^1 + \cdots + P_i s_{i,63} 16^{63} \\ s_i P_i &= P_i s_{i,0} + 16(P_i s_{i,1} + 16( \cdots +16P_i s_{i,63})\cdots ) \end{aligned} so each $$s_i P_i$$ can be computed by alternately adding a precomputed multiple $$P_i s_{i,j}$$ of $$P_i$$ and repeatedly doubling.

Now consider the two-dimensional sum \begin{aligned} s_1 P_1 &=& P_1 s_{1,0} &+& 16 (P_1 s_{1,1} &+& 16 ( \cdots &+& 16 P_1 s_{1,63}&) \cdots ) \\ + & & + & & + & & & & + & \\ s_2 P_2 &=& P_2 s_{2,0} &+& 16 (P_2 s_{2,1} &+& 16 ( \cdots &+& 16 P_2 s_{2,63}&) \cdots ) \\ + & & + & & + & & & & + & \\ \vdots & & \vdots & & \vdots & & & & \vdots & \\ + & & + & & + & & & & + & \\ s_n P_n &=& P_n s_{n,0} &+& 16 (P_n s_{n,1} &+& 16 ( \cdots &+& 16 P_n s_{n,63}&) \cdots ) \end{aligned} The sum of the left-hand column is the result $$Q$$; by computing the two-dimensional sum on the right column-wise, top-to-bottom, then right-to-left, we need to multiply by $$16$$ only once per column, sharing the doublings across all of the input points.

### impl VartimeMultiscalarMul for Straus[src]

#### type Point = EdwardsPoint

The type of point being multiplied, e.g., RistrettoPoint.

#### fn optional_multiscalar_mul<I, J>(scalars: I, points: J) -> Option<EdwardsPoint> where    I: IntoIterator,    I::Item: Borrow<Scalar>,    J: IntoIterator<Item = Option<EdwardsPoint>>, [src]

Variable-time Straus using a non-adjacent form of width $$5$$.

This is completely similar to the constant-time code, but we use a non-adjacent form for the scalar, and do not do table lookups in constant time.

The non-adjacent form has signed, odd digits. Using only odd digits halves the table size (since we only need odd multiples), or gives fewer additions for the same table size.

## Blanket Implementations

### impl<T> Same<T> for T

#### type Output = T

Should always be Self

### impl<T, U> TryFrom<U> for T where    U: Into<T>, [src]

#### type Error = !

The type returned in the event of a conversion error.

### impl<T, U> TryInto<U> for T where    U: TryFrom<T>, [src]

#### type Error = <U as TryFrom<T>>::Error

The type returned in the event of a conversion error.