curve25519-dalek-source/docs/avx2-notes.md

An implementation of group operations on the twisted Edwards form of
Curve25519, using AVX2 to implement the 4-way parallel formulas of
Hisil, Wong, Carter, and Dawson (HWCD).
Their 2008 paper [_Twisted Edwards Curves Revisited_][hwcd08], which
introduced the extended coordinates used in other parts of `-dalek`,
also describes 4-way parallel formulas for point addition and
doubling:

* a unified addition algorithm taking an effective \\(2\mathbf M +
1\mathbf D\\);

* a doubling algorithm taking an effective \\(1\mathbf M + 1\mathbf
S\\);

* a dedicated (i.e., for distinct points) addition algorithm taking
an effective \\(2 \mathbf M \\).

Here \\(\mathbf M\\) and \\(\mathbf S\\) represent the cost of
multiplication and squaring of generic field elements and \\(\mathbf
D\\) represents the cost of multiplication by a curve constant.

These formulas do not seem to have been implemented using SIMD before.
A 2015 paper by Hernández and López mentions using AVX2 for the X25519
Montgomery ladder, but neither the paper nor the code are publicly
available, and it apparently gives only a [slight speedup][avx2trac].
The 2008 HWCD paper also describes and analyzes a 2-wide variant of the
Montgomery ladder (for comparison with parallel Edwards formulas); this
strategy was used in 2015 by Tung Chou's `sandy2x` implementation, which
used a 2-wide field implementation in 128-bit vector registers.
Curiously, however, although the [`sandy2x` paper][sandy2x] also
implements Edwards arithmetic, and cites the HWCD paper, it doesn't
mention the parallel formulas from HWCD, suggesting that they have been
overlooked for software implementations.

The notes below describe a tweak to the \\( 2\mathbf M + 1\mathbf D \\)
unified addition formulas to give \\( 2\mathbf M \\) readdition with
\\(1\mathbf D\\) precomputation, and a tweak to the doubling formulas to
avoid an extra reduction.  These tweaked formulas are the ones used by
the `avx2` backend of `curve25519-dalek`.

# Parallel formulas in HWCD'08

The doubling formula is presented in the HWCD paper as follows:

| Cost             | Processor 1                    | Processor 2                    | Processor 3                    | Processor 4                    |
|------------------|--------------------------------|--------------------------------|--------------------------------|--------------------------------|
|                  | idle                           | idle                           | idle                           | \\( R\_1 \gets X\_1 + Y\_1 \\) |
| \\(1\mathbf S\\) | \\( R\_2 \gets X\_1\^2 \\)     | \\( R\_3 \gets Y\_1\^2   \\)   | \\( R\_4 \gets Z\_1\^2   \\)   | \\( R\_5 \gets R\_1\^2   \\)   |
|                  | \\( R\_6 \gets R\_2 + R\_3 \\) | \\( R\_7 \gets R\_2 - R\_3 \\) | \\( R\_4 \gets 2 R\_4 \\)      | idle                           |
|                  | idle                           | \\( R\_1 \gets R\_4 + R\_7 \\) | idle                           | \\( R\_2 \gets R\_6 - R\_5 \\) |
| \\(1\mathbf M\\) | \\( X\_3 \gets R\_1 R\_2 \\)   | \\( Y\_3 \gets R\_6 R\_7 \\)   | \\( T\_3 \gets R\_2 R\_6 \\)   | \\( Z\_3 \gets R\_1 R\_7 \\)   |

and the unified addition algorithm is presented as follows:

| Cost             | Processor 1                    | Processor 2                    | Processor 3                    | Processor 4                    |
|------------------|--------------------------------|--------------------------------|--------------------------------|--------------------------------|
|                  | \\( R\_1 \gets Y\_1 - X\_1 \\) | \\( R\_2 \gets Y\_2 - X\_2 \\) | \\( R\_3 \gets Y\_1 + X\_1 \\) | \\( R\_4 \gets Y\_2 + X\_2 \\) |
| \\(1\mathbf M\\) | \\( R\_5 \gets R\_1 R\_2 \\)   | \\( R\_6 \gets R\_3 R\_4 \\)   | \\( R\_7 \gets T\_1 T\_2 \\)   | \\( R\_8 \gets Z\_1 Z\_2 \\)   |
| \\(1\mathbf D\\) | idle                           | idle                           | \\( R\_7 \gets k R\_7 \\)      | \\( R\_8 \gets 2 R\_8 \\)      |
|                  | \\( R\_1 \gets R\_6 - R\_5 \\) | \\( R\_2 \gets R\_8 - R\_7 \\) | \\( R\_3 \gets R\_8 + R\_7 \\) | \\( R\_4 \gets R\_6 + R\_5 \\) |
| \\(1\mathbf M\\) | \\( X\_3 \gets R\_1 R\_2 \\)   | \\( Y\_3 \gets R\_3 R\_4 \\)   | \\( T\_3 \gets R\_1 R\_4 \\)   | \\( Z\_3 \gets R\_2 R\_3 \\)   |

Here \\( k = 2d \\) is a curve constant.

For a software implementation, each processor's operations are too
low-latency to parallelize across threads.  However, the main cost
is in the multiplication and squaring steps, which are uniform, while
the divergent steps involve inexpensive additions and subtractions.

This means we can use SIMD to implement the expensive portions in
parallel, and handle the instruction divergence on the inexpensive parts
using masking.

The remaining obstacle to parallelism is the multiplication by the curve
constant \\(k = 2d\\).  In the Curve25519 case, this is

$$ k \equiv 2 \frac{-121665}{121666} \\ \equiv 16295367250680780974490674513165176452449235426866156013048779062215315747161 \pmod p. $$

HWCD suggest parallelising this step by breaking \\(k\\) into four
parts as \\(k = k_0 + 2\^n k_1 + 2\^{2n} k_2 + 2\^{3n} k_3 \\) and
computing \\(k_i R_7 \\) in parallel.  However, this would be
somewhat awkward in our case, since we would normally represent
\\(k\\) as \\( 10 \\) 32-bit limbs, and \\(10 \\) is not divisible
by \\(4\\), so we would need a specialized routine to perform a
vectorized multiplication by 64-bit constants.

Instead, since we are working projectively, we can multiply
\\(R_7\\) by \\( -2\cdot 121665 \\) and multiply the other three
variables by \\(121666\\).  This trick was suggested by Mike
Hamburg.  Ignoring the sign for the moment, since
\\(2 \cdot 121666 < 2\^{18}\\), all these constants fit in 32 bits,
so (up to sign) this can be done in parallel as four multiplications
by small constants \\( (121666, 121666, 2\cdot 121665, 2\cdot 121666) \\).

How do we handle the sign?
Since we're primarily interested in Ristretto performance, not
Curve25519 performance, we could alternately work on the
\\(4\\)-isogenous "IsoEd25519" curve, which has \\(d = 121665\\).
However, this would only save the negation step, since multiplying
one field element by a 32-bit constant is not much easier than
multiplying four field elements by 32-bit constants, and it would
prevent accelerating Curve25519, so we don't make this choice.
Instead, we just negate one lane, and move the \\(1 \mathbf D\\)
into precomputation (see below).

# Tweaked formulas

After tweaking the formulas as described above, we obtain the
following.  To avoid confusion with the original HWCD formulas,
temporary variables are named \\(S\\) instead of \\(R\\) and are in
static single-assignment form.

## Addition

This implementation only implements readdition, but the tweaked addition
formulas are described first.  To add points \\(P_1 = (X_1 : Y_1 : Z_1 :
T_1) \\) and \\(P_2 = (X_2 : Y_2 : Z_2 : T_2 ) \\), we compute

$$
\begin{aligned}
S\_0 &\gets Y\_1 - X\_1 \\\\
S\_1 &\gets Y\_1 + X\_1 \\\\
S\_2 &\gets Y\_2 - X\_2 \\\\
S\_3 &\gets Y\_2 + X\_2
\end{aligned}
$$

$$
\begin{aligned}
S\_4 &\gets S\_0 S\_2 \\\\
S\_5 &\gets S\_1 S\_3 \\\\
S\_6 &\gets Z\_1 Z\_2 \\\\
S\_7 &\gets T\_1 T\_2
\end{aligned}
$$

$$
\begin{aligned}
S\_8    &\gets S\_4 \cdot 121666 \\\\
S\_9    &\gets S\_5 \cdot 121666 \\\\
S\_{10} &\gets S\_6 \cdot 2 \cdot 121666 \\\\
S\_{11} &\gets S\_7 \cdot -2 \cdot 121665
\end{aligned}
$$

$$
\begin{aligned}
S\_{12} &\gets S\_9 - S\_8 \\\\
S\_{13} &\gets S\_9 + S\_8 \\\\
S\_{14} &\gets S\_{10} - S\_{11} \\\\
S\_{15} &\gets S\_{10} + S\_{11}
\end{aligned}
$$

$$
\begin{aligned}
X\_3 &\gets S\_{12} S\_{14} \\\\
Y\_3 &\gets S\_{15} S\_{13} \\\\
Z\_3 &\gets S\_{15} S\_{14} \\\\
T\_3 &\gets S\_{12} S\_{13}
\end{aligned}
$$

to obtain \\( P\_3 = (X\_3 : Y\_3 : Z\_3 : T\_3) = P\_1 + P\_2 \\).

## Readdition

If the point \\( P_2 = (X\_2 : Y\_2 : Z\_2 : T\_2) \\) is fixed, we can precompute

$$
\begin{aligned}
S\_2 &\gets Y\_2 - X\_2 \\\\
S\_3 &\gets Y\_2 + X\_2
\end{aligned}
$$

$$
\begin{aligned}
S\_2'    &\gets S\_2 \cdot 121666 \\\\
S\_3'    &\gets S\_3 \cdot 121666 \\\\
Z\_2'    &\gets Z\_2 \cdot 2 \cdot 121666 \\\\
T\_2'    &\gets T\_2 \cdot -2 \cdot 121665 \\\\
\end{aligned}
$$

to obtain the `CachedPoint` \\( (S\_2', S\_3', Z\_2', T\_2') \\).
This precomputation is essentially the same as that suggested in
§3.1 of HWCD, with the difference that the multiplication by the curve
constant \\( -121665 / 121666 \\) is spread over all four
coordinates, to allow a vectorized computation of four
multiplications of small constants instead of a serial computation
of multiplication by a large constant.

To perform readdition of \\(P_1 = (X_1 : Y_1 : Z_1 : T_1) \\) and
\\(P_2 = (S\_2', S\_3', Z\_2', T\_2') \\), we compute

$$
\begin{aligned}
S\_0 &\gets Y\_1 - X\_1 \\\\
S\_1 &\gets Y\_1 + X\_1
\end{aligned}
$$

$$
\begin{aligned}
S\_8    &\gets S\_0 S\_2' \\\\
S\_9    &\gets S\_1 S\_3' \\\\
S\_{10} &\gets Z\_1 Z\_2' \\\\
S\_{11} &\gets T\_1 T\_2'
\end{aligned}
$$

$$
\begin{aligned}
S\_{12} &\gets S\_9 - S\_8 \\\\
S\_{13} &\gets S\_9 + S\_8 \\\\
S\_{14} &\gets S\_{10} - S\_{11} \\\\
S\_{15} &\gets S\_{10} + S\_{11}
\end{aligned}
$$

$$
\begin{aligned}
X\_3 &\gets S\_{12} S\_{14} \\\\
Y\_3 &\gets S\_{15} S\_{13} \\\\
Z\_3 &\gets S\_{15} S\_{14} \\\\
T\_3 &\gets S\_{12} S\_{13}
\end{aligned}
$$

to obtain \\( P\_3 = (X\_3 : Y\_3 : Z\_3 : T\_3) = P\_1 + P\_2 \\).

Compared to the addition formulas above, this saves \\( 1\mathbf D \\).

## Doubling

To double a point \\( P = (X\_1 : Y\_1 : Z\_1 : T\_1) \\), we compute

$$ S\_0 \gets X\_1 + Y\_1 $$

$$
\begin{aligned}
S\_1 &\gets X\_1\^2 \\\\
S\_2 &\gets Y\_1\^2 \\\\
S\_3 &\gets Z\_1\^2 \\\\
S\_4 &\gets S\_0\^2
\end{aligned}
$$

$$
\begin{aligned}
S\_5 &\gets S\_1 + S\_2 \\\\
S\_6 &\gets S\_1 - S\_2 \\\\
S\_7 &\gets 2S\_3 \\\\
S\_8 &\gets S\_7 + S\_6 = S\_1 + 2S\_3 - S\_2 \\\\
S\_9 &\gets S\_5 - S\_4 = S\_1 + S\_2 - S\_4
\end{aligned}
$$

$$
\begin{aligned}
X\_3 &\gets S\_8 S\_9 \\\\
Y\_3 &\gets S\_5 S\_6 \\\\
Z\_3 &\gets S\_8 S\_6 \\\\
T\_3 &\gets S\_5 S\_9
\end{aligned}
$$

to obtain \\( P\_3 = (X\_3 : Y\_3 : Z\_3 : T\_3) = [2]P\_1 \\).

Unlike the (re)addition formulas, the divergent parts of these formulas
are less nice.  However, with some careful bounds-juggling, it is
possible to implement them without inserting extra carry chains, as
described below.

# Field element representation

Our strategy is to implement 4-wide multiplication and squaring by
wordslicing, using one 64-bit AVX2 lane for each field element.  Field
elements are represented in the usual way as 10 `u32` limbs in radix
\\(25.5\\) (i.e., alternating between \\(2\^{26}\\) for even limbs and
\\(2\^{25}\\) for odd limbs).  This has the effect that passing between
the parallel 32-bit AVX2 representation and the serial 64-bit
representation (which uses radix \\(2^{51}\\)) amounts to regrouping
digits.

The field element representation is oriented around the AVX2
`vpmuluqdq` instruction, which multiplies the low 32 bits of each
64-bit lane of each operand to produce a 64-bit result.

```text,no_run
(a1 ?? b1 ?? c1 ?? d1 ??)
(a2 ?? b2 ?? c2 ?? d2 ??)

(a1*a2 b1*b2 c1*c2 d1*d2)
```

To unpack 32-bit values into 64-bit lanes for use in multiplication
it would be convenient to use the `vpunpck[lh]dq` instructions,
which unpack and interleave the low and high 32-bit lanes of two
source vectors.
However, the AVX2 versions of these instructions are designed to
operate only within 128-bit lanes of the 256-bit vectors, so that
interleaving the low lanes of `(a0 b0 c0 d0 a1 b1 c1 d1)` with zero
gives `(a0 00 b0 00 a1 00 b1 00)`.  Instead, we pre-shuffle the data
layout as `(a0 b0 a1 b1 c0 d0 c1 d1)` so that we can unpack the
"low" and "high" parts as

```text,no_run
(a0 00 b0 00 c0 00 d0 00)
(a1 00 b1 00 c1 00 d1 00)
```

The data layout for a vector of four field elements \\( (a,b,c,d)
\\) with limbs \\( a_0, a_1, \ldots, a_9 \\) is as `[u32x8; 5]` in
the form

```text,no_run
(a0 b0 a1 b1 c0 d0 c1 d1)
(a2 b2 a3 b3 c2 d2 c3 d3)
(a4 b4 a5 b5 c4 d4 c5 d5)
(a6 b6 a7 b7 c6 d6 c7 d7)
(a8 b8 a9 b9 c8 d8 c9 d9)
```

Since this breaks cleanly into two 128-bit lanes, it may be possible
to adapt it to 128-bit vector instructions such as NEON without too
much difficulty.  Going the other direction, to extend this to AVX512,
we could either run two point operations in parallel in lower and upper
halves of the registers, or use 2-way parallelism within a field operation.

# Handling the Doubling Formulas

The non-parallel portion of the doubling formulas is

$$
\begin{aligned}
S\_5 &\gets S\_1 + S\_2 \\\\
S\_6 &\gets S\_1 - S\_2 \\\\
S\_7 &\gets 2S\_3 \\\\
S\_8 &\gets S\_7 + S\_6 = S\_1 + 2S\_3 - S\_2 \\\\
S\_9 &\gets S\_5 - S\_4 = S\_1 + S\_2 - S\_4
\end{aligned}
$$

Performing too many intermediate additions and subtractions grows
the bounds beyond what is allowed as input to multiplication,
forcing an extra carry pass.  However, it is just possible to avoid
this by rearranging signs.

Assume that the bounds on the limbs of each field element are
parameterized by \\( b \in \mathbb R \\) representing the excess
bits, so that each limb is bounded by either 
\\( 2\^{25+b} \\) or \\( 2\^{26+b} \\).

The multiplication routine requires that its inputs are bounded by
\\( b < 1.75 \\), in order to fit a multiplication by \\( 19 \\)
into 32 bits.  Since \\( \lg 19 < 4.25 \\), \\( 19x < 2\^{32} \\)
when \\( x < 2\^{27.75} = 2\^{26 + 1.75} \\).  However, this is only
required for one of the inputs; the other can grow up to \\( b < 2.5
\\).

Computing \\( (S\_5, S\_6, S\_8, S\_9 ) \\) as

$$
\begin{matrix}
 & S\_1 & S\_1 & S\_1 & S\_1 \\\\
+& S\_2 &      &      & S\_2 \\\\
+&      &      & S\_3 &      \\\\
+&      &      & S\_3 &      \\\\
+&      & 2p   & 2p   & 2p   \\\\
-&      & S\_2 & S\_2 &      \\\\
-&      &      &      & S\_4 \\\\
=& S\_5 & S\_6 & S\_8 & S\_9
\end{matrix}
$$

results in bit-excesses \\( (1.00, 1.59, 2.33, 2.00)\\) for
\\( (S\_5, S\_6, S\_8, S\_9 ) \\).  The products we want to compute
are then

$$
\begin{aligned}
X\_3 &\gets S\_8 S\_9 \leftrightarrow (2.33, 2.00) \\\\
Y\_3 &\gets S\_5 S\_6 \leftrightarrow (1.00, 1.59) \\\\
Z\_3 &\gets S\_8 S\_6 \leftrightarrow (2.33, 1.59) \\\\
T\_3 &\gets S\_5 S\_9 \leftrightarrow (1.00, 2.00)
\end{aligned}
$$

which are too large.  However, if we flip the sign of \\( S\_4 =
S\_0\^2 \\) during squaring, so that we output \\(S\_4' = -S\_4
\pmod p\\), then we can compute

$$
\begin{matrix}
 & S\_1 & S\_1 & S\_1 & S\_1 \\\\
+& S\_2 &      &      & S\_2 \\\\
+&      &      & S\_3 &      \\\\
+&      &      & S\_3 &      \\\\
+&      &      &      & S\_4' \\\\
+&      & 2p   & 2p   &      \\\\
-&      & S\_2 & S\_2 &      \\\\
=& S\_5 & S\_6 & S\_8 & S\_9
\end{matrix}
$$

resulting in bit-excesses \\( (1.00, 1.59, 2.33, 1.59)\\) for
\\( (S\_5, S\_6, S\_8, S\_9 ) \\).  The products we want to compute
are then

$$
\begin{aligned}
X\_3 &\gets S\_8 S\_9 \leftrightarrow (2.33, 1.59) \\\\
Y\_3 &\gets S\_5 S\_6 \leftrightarrow (1.00, 1.59) \\\\
Z\_3 &\gets S\_8 S\_6 \leftrightarrow (2.33, 1.59) \\\\
T\_3 &\gets S\_5 S\_9 \leftrightarrow (1.00, 1.59)
\end{aligned}
$$

whose right-hand sides are all bounded with \\( b < 1.75 \\) and
whose left-hand sides are all bounded with \\( b < 2.5 \\).

# Comparison to non-vectorized formulas

HWCD also suggest using a mixed representation, passing between \\(
\mathbb P\^3 \\) "extended" coordinates and \\( \mathbb P\^2 \\)
"projective" coordinates, where doubling is slightly cheaper (saving
about \\(\mathbf 1M\\).  This approach is used for the
non-vectorized `u32` and `u64` backends, and more
details on the different coordinate systems can be found in the
`curve_models` module documentation.

This optimization is not compatible with the parallel formulas, which are
therefore slightly less efficient when counting the total number of
field multiplications and squarings.  In particular, vectorized doublings
are less efficient than serial doublings.

In addition, the parallel formulas can only use a \\( 32 \times 32
\rightarrow 64 \\)-bit integer multiplier, so the speedup from
vectorization must overcome the disadvantage of losing the \\( 64
\times 64 \rightarrow 128\\)-bit (serial) integer multiplier.

When compiling with AVX512VL, LLVM is able to use the extra
`ymm16..ymm31` registers to reduce register pressure, and avoid
spills during field multiplication.  This gives a small but
noticeable speedup.

Another concern with AVX2 is that currently-available Intel processors
(particularly Skylake and Skylake-X microarchitectures) perform thermal
throttling when using wide vector instructions.  For a mixed workload,
where point operations are interspersed with other tasks, this can
reduce overall performance.  This probably means that this
implementation is not suitable for basic applications, like signatures,
but could still be worthwhile for complex applications, like
zero-knowledge proofs, which do enough work to make it worthwhile.

On AMD's Zen microarchitecture, thermal throttling is not a concern,
since AVX2 is implemented at half rate, so there is no penalty for mixed
workloads (but also no speedup).

[sandy2x]: https://eprint.iacr.org/2015/943.pdf
[avx2trac]: https://trac.torproject.org/projects/tor/ticket/8897#comment:28
[hwcd08]: https://www.iacr.org/archive/asiacrypt2008/53500329/53500329.pdf