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@ -1,42 +1,48 @@
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 vectorized implementation of group operations on the twisted Edwards
form of Curve25519, using a modification of the 4-way parallel
formulas of Hisil, Wong, Carter, and Dawson.
* a unified addition algorithm taking an effective \\(2\mathbf M +
1\mathbf D\\);
# Overview
* a doubling algorithm taking an effective \\(1\mathbf M + 1\mathbf
S\\);
The 2008 paper [_Twisted Edwards Curves Revisited_][hwcd08] by Hisil,
Wong, Carter, and Dawson (HWCD) introduced the “extended coordinates”
and mixed-model representations which are used by most Edwards curve
implementations.
* a dedicated (i.e., for distinct points) addition algorithm taking
an effective \\(2 \mathbf M \\).
However, they also describe 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\\), and a dedicated (i.e., for
distinct points) addition algorithm taking an effective \\(2 \mathbf M
\\). They compare these formulas with a 2-way parallel variant of the
Montgomery ladder.
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.
Unlike their serial formulas, which are used widely, their parallel
formulas do not seem to have been implemented in software before. The
2-way parallel Montgomery ladder was used in 2015 by Tung Chou's
`sandy2x` implementation. Curiously, however, although the [`sandy2x`
paper][sandy2x] also implements Edwards arithmetic, and cites HWCD08,
it doesn't mention their parallel Edwards formulas.
A 2015 paper by Hernández and López describes an AVX2 implementation
of X25519. Neither the paper nor the code are publicly available, but
it apparently gives only a [slight speedup][avx2trac], suggesting that
it uses a 4-way parallel Montgomery ladder rather than parallel
Edwards formulas.
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 reason may be that HWCD08 describe their formulas as operating on
four independent processors, which would make a software
implementation impractical: all of the operations are too low-latency
to effectively synchronize. But a closer inspection reveals that the
(more expensive) multiplication and squaring steps are uniform, while
the instruction divergence occurs in the (much cheaper) addition and
subtraction steps. This means that a SIMD implementation can perform
the expensive steps uniformly, and handle divergence in the
inexpensive steps using masking.
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`.
These notes describe modifications to the original parallel formulas
to allow a SIMD implementation, and this module contains an
implementation of the modified formulas using 256-bit AVX2 vector
operations.
# Parallel formulas in HWCD'08
@ -60,218 +66,153 @@ and the unified addition algorithm is presented as follows:
| | \\( 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.
Here \\(\mathbf M\\) and \\(\mathbf S\\) represent the cost of
multiplication and squaring of generic field elements, \\(\mathbf D\\)
represents the cost of multiplication by a curve constant (in this
case \\( k = 2d \\)).
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.
Notice that the \\(1\mathbf M\\) and \\(1\mathbf S\\) steps are
uniform. The non-uniform steps are all inexpensive additions or
subtractions, with the exception of the multiplication by the curve
constant \\(k = 2d\\):
$$
R\_7 \gets 2 d R\_7.
$$
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
HWCD suggest parallelising this step by breaking \\(k = 2d\\) 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.
computing \\(k_i R_7 \\) in parallel. This is quite awkward, but if
the curve constant is a ratio \\( d = d\_1/d\_2 \\), then projective
coordinates allow us to instead compute
$$
(R\_5, R\_6, R\_7, R\_8) \gets (d\_2 R\_5, d\_2 R\_6, 2d\_1 R\_7, d\_2 R\_8).
$$
This can be performed as a uniform multiplication by a vector of
constants, and if \\(d\_1, d\_2\\) are small, it is relatively
inexpensive. (This trick was suggested by Mike Hamburg).
In the Curve25519 case, we have
$$
d = \frac{d\_1}{d\_2} = \frac{-121665}{121666};
$$
Since \\(2 \cdot 121666 < 2\^{18}\\), all the constants above fit (up
to sign) in 32 bits, so this can be done in parallel as four
multiplications by small constants \\( (121666, 121666, 2\cdot 121665,
2\cdot 121666) \\), followed by a negation to compute \\( - 2\cdot 121665\\).
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) \\).
# Modified parallel formulas
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.
Using the modifications sketched above, we can write SIMD-friendly
versions of the parallel formulas as follows. To avoid confusion with
the original 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
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
$$
\begin{aligned}
S\_0 &\gets Y\_1 - X\_1 \\\\
S\_1 &\gets Y\_1 + X\_1 \\\\
S\_2 &\gets Y\_2 - X\_2 \\\\
S\_3 &\gets Y\_2 + X\_2
(S\_0 &&,&& S\_1 &&,&& S\_2 &&,&& S\_3 )
&\gets
(Y\_1 - X\_1&&,&& Y\_1 + X\_1&&,&& Y\_2 - X\_2&&,&& Y\_2 + X\_2)
\\\\
(S\_4 &&,&& S\_5 &&,&& S\_6 &&,&& S\_7 )
&\gets
(S\_0 \cdot S\_2&&,&& S\_1 \cdot S\_3&&,&& Z\_1 \cdot Z\_2&&,&& T\_1 \cdot T\_2)
\\\\
(S\_8 &&,&& S\_9 &&,&& S\_{10} &&,&& S\_{11} )
&\gets
(d\_2 \cdot S\_4 &&,&& d\_2 \cdot S\_5 &&,&& 2 d\_2 \cdot S\_6 &&,&& 2 d\_1 \cdot S\_7 )
\\\\
(S\_{12} &&,&& S\_{13} &&,&& S\_{14} &&,&& S\_{15})
&\gets
(S\_9 - S\_8&&,&& S\_9 + S\_8&&,&& S\_{10} - S\_{11}&&,&& S\_{10} + S\_{11})
\\\\
(X\_3&&,&& Y\_3&&,&& Z\_3&&,&& T\_3)
&\gets
(S\_{12} \cdot S\_{14}&&,&& S\_{15} \cdot S\_{13}&&,&& S\_{15} \cdot S\_{14}&&,&& S\_{12} \cdot S\_{13})
\end{aligned}
$$
$$
\begin{aligned}
S\_4 &\gets S\_0 S\_2 \\\\
S\_5 &\gets S\_1 S\_3 \\\\
S\_6 &\gets Z\_1 Z\_2 \\\\
S\_7 &\gets T\_1 T\_2
\end{aligned}
$$
$$
\begin{aligned}
S\_8 &\gets S\_4 \cdot 121666 \\\\
S\_9 &\gets S\_5 \cdot 121666 \\\\
S\_{10} &\gets S\_6 \cdot 2 \cdot 121666 \\\\
S\_{11} &\gets S\_7 \cdot -2 \cdot 121665
\end{aligned}
$$
$$
\begin{aligned}
S\_{12} &\gets S\_9 - S\_8 \\\\
S\_{13} &\gets S\_9 + S\_8 \\\\
S\_{14} &\gets S\_{10} - S\_{11} \\\\
S\_{15} &\gets S\_{10} + S\_{11}
\end{aligned}
$$
$$
\begin{aligned}
X\_3 &\gets S\_{12} S\_{14} \\\\
Y\_3 &\gets S\_{15} S\_{13} \\\\
Z\_3 &\gets S\_{15} S\_{14} \\\\
T\_3 &\gets S\_{12} S\_{13}
\end{aligned}
$$
to obtain \\( P\_3 = (X\_3 : Y\_3 : Z\_3 : T\_3) = P\_1 + P\_2 \\).
This costs \\( 2\mathbf M + 1 \mathbf D\\).
## Readdition
If the point \\( P_2 = (X\_2 : Y\_2 : Z\_2 : T\_2) \\) is fixed, we can precompute
If the point \\( P_2 = (X\_2 : Y\_2 : Z\_2 : T\_2) \\) is fixed, we
can cache the multiplication of the curve constants by computing
$$
\begin{aligned}
S\_2 &\gets Y\_2 - X\_2 \\\\
S\_3 &\gets Y\_2 + X\_2
(S\_2' &&,&& S\_3' &&,&& Z\_2' &&,&& T\_2' )
&\gets
(d\_2 \cdot (Y\_2 - X\_2)&&,&& d\_2 \cdot (Y\_1 + X\_1)&&,&& 2d\_2 \cdot Z\_2 &&,&& 2d\_1 \cdot T\_2).
\end{aligned}
$$
This costs \\( 1\mathbf D\\); with \\( (S\_2', S\_3', Z\_2', T\_2')\\)
in hand, the addition formulas above become
$$
\begin{aligned}
S\_2' &\gets S\_2 \cdot 121666 \\\\
S\_3' &\gets S\_3 \cdot 121666 \\\\
Z\_2' &\gets Z\_2 \cdot 2 \cdot 121666 \\\\
T\_2' &\gets T\_2 \cdot -2 \cdot 121665 \\\\
(S\_0 &&,&& S\_1 &&,&& Z\_1 &&,&& T\_1 )
&\gets
(Y\_1 - X\_1&&,&& Y\_1 + X\_1&&,&& Z\_1 &&,&& T\_1)
\\\\
(S\_8 &&,&& S\_9 &&,&& S\_{10} &&,&& S\_{11} )
&\gets
(S\_0 \cdot S\_2' &&,&& S\_1 \cdot S\_3'&&,&& Z\_1 \cdot Z\_2' &&,&& T\_1 \cdot T\_2')
\\\\
(S\_{12} &&,&& S\_{13} &&,&& S\_{14} &&,&& S\_{15})
&\gets
(S\_9 - S\_8&&,&& S\_9 + S\_8&&,&& S\_{10} - S\_{11}&&,&& S\_{10} + S\_{11})
\\\\
(X\_3&&,&& Y\_3&&,&& Z\_3&&,&& T\_3)
&\gets
(S\_{12} \cdot S\_{14}&&,&& S\_{15} \cdot S\_{13}&&,&& S\_{15} \cdot S\_{14}&&,&& S\_{12} \cdot S\_{13})
\end{aligned}
$$
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 \\).
which costs only \\( 2\mathbf M \\). This precomputation is
essentially similar to the precomputation that HWCD suggest for their
serial formulas. Because the cost of precomputation and then
readdition is the same as addition, it's sufficient to only
implement caching and readdition.
## Doubling
The non-uniform portions of the (re)addition formulas have a fairly
regular structure. Unfortunately, this is not the case for the
doubling formulas, which are much less nice.
To double a point \\( P = (X\_1 : Y\_1 : Z\_1 : T\_1) \\), we compute
$$ S\_0 \gets X\_1 + Y\_1 $$
$$
\begin{aligned}
S\_1 &\gets X\_1\^2 \\\\
S\_2 &\gets Y\_1\^2 \\\\
S\_3 &\gets Z\_1\^2 \\\\
S\_4 &\gets S\_0\^2
(X\_1 &&,&& Y\_1 &&,&& Z\_1 &&,&& S\_0)
&\gets
(X\_1 &&,&& Y\_1 &&,&& Z\_1 &&,&& X\_1 + Y\_1)
\\\\
(S\_1 &&,&& S\_2 &&,&& S\_3 &&,&& S\_4 )
&\gets
(X\_1\^2 &&,&& Y\_1\^2&&,&& Z\_1\^2 &&,&& S\_0\^2)
\\\\
(S\_5 &&,&& S\_6 &&,&& S\_8 &&,&& S\_9 )
&\gets
(S\_1 + S\_2 &&,&& S\_1 - S\_2 &&,&& S\_1 + 2S\_3 - S\_2 &&,&& S\_1 + S\_2 - S\_4)
\\\\
(X\_3 &&,&& Y\_3 &&,&& Z\_3 &&,&& T\_3 )
&\gets
(S\_8 \cdot S\_9 &&,&& S\_5 \cdot S\_6 &&,&& S\_8 \cdot S\_6 &&,&& S\_5 \cdot S\_9)
\end{aligned}
$$
$$
\begin{aligned}
S\_5 &\gets S\_1 + S\_2 \\\\
S\_6 &\gets S\_1 - S\_2 \\\\
S\_7 &\gets 2S\_3 \\\\
S\_8 &\gets S\_7 + S\_6 = S\_1 + 2S\_3 - S\_2 \\\\
S\_9 &\gets S\_5 - S\_4 = S\_1 + S\_2 - S\_4
\end{aligned}
$$
$$
\begin{aligned}
X\_3 &\gets S\_8 S\_9 \\\\
Y\_3 &\gets S\_5 S\_6 \\\\
Z\_3 &\gets S\_8 S\_6 \\\\
T\_3 &\gets S\_5 S\_9
\end{aligned}
$$
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.
The intermediate step between the squaring and multiplication requires
a long chain of additions, but with some care and finesse,
described below, it is possible (in our case) to arrange this
computation without requiring an intermediate reduction.
However, it does mean that the doubling formulas have proportionately
more vectorization overhead than the (re)addition formulas. The
effects of this are discussed in the comparison section below.
# Field element representation
@ -329,39 +270,36 @@ 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
# Avoiding Overflow in Doubling
The non-parallel portion of the doubling formulas is
To analyze the size of the field element coefficients during the
computations, we can parameterize the bounds on the limbs of each
field element by \\( b \in \mathbb R \\) representing the excess bits
above that limb's radix, so that each limb is bounded by either
\\(2\^{25+b} \\) or \\( 2\^{26+b} \\), as appropriate.
$$
\begin{aligned}
S\_5 &\gets S\_1 + S\_2 \\\\
S\_6 &\gets S\_1 - S\_2 \\\\
S\_7 &\gets 2S\_3 \\\\
S\_8 &\gets S\_7 + S\_6 = S\_1 + 2S\_3 - S\_2 \\\\
S\_9 &\gets S\_5 - S\_4 = S\_1 + S\_2 - S\_4
\end{aligned}
$$
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
The multiplication routine requires that its inputs are bounded with
\\( 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
In addition, the multiplication and squaring routines do not
canonically reduce their outputs, but can leave some small uncarried
excesses, so that their reduced outputs are bounded with
\\( b < 0.007 \\).
The non-parallel portion of the doubling formulas is
$$
\begin{aligned}
(S\_5 &&,&& S\_6 &&,&& S\_8 &&,&& S\_9 )
&\gets
(S\_1 + S\_2 &&,&& S\_1 - S\_2 &&,&& S\_1 + 2S\_3 - S\_2 &&,&& S\_1 + S\_2 - S\_4)
\end{aligned}
$$
Computing \\( (S\_5, S\_6, S\_8, S\_9 ) \\) as
$$
\begin{matrix}
& S\_1 & S\_1 & S\_1 & S\_1 \\\\
@ -374,24 +312,22 @@ $$
=& S\_5 & S\_6 & S\_8 & S\_9
\end{matrix}
$$
results in bit-excesses \\( (1.00, 1.59, 2.33, 2.00)\\) for
results in bit-excesses \\( < (1.01, 1.60, 2.33, 2.01)\\) for
\\( (S\_5, S\_6, S\_8, S\_9 ) \\). The products we want to compute
are then
$$
\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)
X\_3 &\gets S\_8 S\_9 \leftrightarrow (2.33, 2.01) \\\\
Y\_3 &\gets S\_5 S\_6 \leftrightarrow (1.01, 1.60) \\\\
Z\_3 &\gets S\_8 S\_6 \leftrightarrow (2.33, 1.60) \\\\
T\_3 &\gets S\_5 S\_9 \leftrightarrow (1.01, 2.01)
\end{aligned}
$$
which are too large. However, if we flip the sign of \\( S\_4 =
S\_0\^2 \\) during squaring, so that we output \\(S\_4' = -S\_4
\pmod p\\), then we can compute
which are too large: it's not possible to arrange the multiplicands so
that one vector has \\(b < 2.5\\) and the other has \\( b < 1.75 \\).
However, if we flip the sign of \\( S\_4 = S\_0\^2 \\) during
squaring, so that we output \\(S\_4' = -S\_4 \pmod p\\), then we can
compute
$$
\begin{matrix}
& S\_1 & S\_1 & S\_1 & S\_1 \\\\
@ -404,61 +340,120 @@ $$
=& S\_5 & S\_6 & S\_8 & S\_9
\end{matrix}
$$
resulting in bit-excesses \\( (1.00, 1.59, 2.33, 1.59)\\) for
resulting in bit-excesses \\( < (1.01, 1.60, 2.33, 1.60)\\) for
\\( (S\_5, S\_6, S\_8, S\_9 ) \\). The products we want to compute
are then
$$
\begin{aligned}
X\_3 &\gets S\_8 S\_9 \leftrightarrow (2.33, 1.59) \\\\
Y\_3 &\gets S\_5 S\_6 \leftrightarrow (1.00, 1.59) \\\\
Z\_3 &\gets S\_8 S\_6 \leftrightarrow (2.33, 1.59) \\\\
T\_3 &\gets S\_5 S\_9 \leftrightarrow (1.00, 1.59)
X\_3 &\gets S\_8 S\_9 \leftrightarrow (2.33, 1.60) \\\\
Y\_3 &\gets S\_5 S\_6 \leftrightarrow (1.01, 1.60) \\\\
Z\_3 &\gets S\_8 S\_6 \leftrightarrow (2.33, 1.60) \\\\
T\_3 &\gets S\_5 S\_9 \leftrightarrow (1.01, 1.60)
\end{aligned}
$$
whose right-hand sides are all bounded with \\( b < 1.75 \\) and
whose left-hand sides are all bounded with \\( b < 2.5 \\).
whose left-hand sides are all bounded with \\( b < 2.5 \\),
so that we can avoid any intermediate reductions.
# 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.
In theory, the parallel Edwards formulas seem to allow a \\(4\\)-way
speedup from parallelism. However, an actual vectorized
implementation has several slowdowns that cut into this speedup.
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
First, 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.
\times 64 \rightarrow 128\\)-bit (serial) integer multiplier. The
effect of this slowdown is microarchitecture-dependent, since it
requires accounting for the total number of multiplications and
additions and their relative costs. In the future, it will probably
be possible to avoid this slowdown by using the `IFMA52` instructions,
whose parallelism is perfectly suited to these formulas.
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.
Second, the parallel doubling formulas incur both a theoretical and
practical slowdown. The parallel formulas described above work on the
\\( \mathbb P\^3 \\) “extended” coordinates. The \\( \mathbb P\^2 \\)
model introduced earlier by [Bernstein, Birkner, Joye, Lange, and
Peters][bbjlp08] allows slightly faster doublings, so HWCD suggest
mixing coordinate systems while performing scalar multiplication
(attributing the idea to [a 1998 paper][cmo98] by Cohen, Miyagi, and
Ono). The \\( T \\) coordinate is not required for doublings, so when
doublings are followed by doublings, its computation can be skipped.
More details on this approach and the different coordinate systems can
be found in the [`curve_models` module documentation][curve_models].
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,
Unfortunately, this optimization is not compatible with the parallel
formulas, which cannot save time by skipping a single variable, so the
parallel doubling formulas do slightly more work when counting the
total number of field multiplications and squarings.
In addition, the parallel doubling formulas have a less regular
pattern of additions and subtractions than the parallel addition
formulas, so the vectorization overhead is proportionately greater.
Both the parallel addition and parallel doubling formulas also require
some shuffling to rearrange data within the vectors, which places more
pressure on the shuffle unit than is desirable.
This means that the speedup from using a vectorized implementation of
parallel Edwards formulas is likely to be greatest in applications
that do fewer doublings and more additions (like a large multiscalar
multiplication) rather than applications that do fewer additions and
more doublings (like a double-base scalar multiplication).
Third, current Intel CPUs perform thermal throttling when using wide
vector instructions. A detailed description can be found in §15.26 of
[the Intel Optimization Manual][intel], but using wide vector
instructions prevents the core from operating at higher frequencies.
The core can return to the higher-frequency state after 2
milliseconds, but this timer is reset every time high-power
instructions are used.
Any speedup from vectorization therefore has to be weighed against a
slowdown for the next few million 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.
reduce overall performance. This implementation is therefore probably
not suitable for basic applications, like signatures, but is
worthwhile for complex applications, like zero-knowledge proofs, which
do sustained work.
For this reason, the AVX2 backend is not enabled by default, but can
be selected using the `avx2_backend` feature.
# Future work
There are several directions for future improvement:
* Using the vectorized field arithmetic code to parallelize across
point operations rather than within a single point operation. This
is less flexible, but would give a speedup both from allowing use of
the faster mixed-model arithmetic and from reducing shuffle
pressure. One approach in this direction would be to implement
batched scalar-point operations using vectors of points (AoSoA
layout). This less generally useful but would give a speedup for
Bulletproofs.
* Extending the implementation to use the full width of AVX512, either
handling the extra parallelism internally to a single point
operation (by using a 2-way parallel implementation of field
arithmetic instead of a wordsliced one), or externally,
parallelizing across point operations. Internal parallelism would
be preferable but might require too much shuffle pressure.
* Generalizing the implementation to non-AVX2 instructions,
particularly NEON. The current point arithmetic code is written in
terms of field element vectors, which are in turn implemented using
platform SIMD vectors. It should be possible to write an alternate
implementation of the `FieldElement32x4` using NEON without changing
the point arithmetic. NEON has 128-bit vectors rather than 256-bit
vectors, but this may still be worthwhile compared to a serial
implementation.
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
[curve_models]: https://doc-internal.dalek.rs/curve25519_dalek/curve_models/index.html
[bbjlp08]: https://eprint.iacr.org/2008/013
[cmo98]: https://link.springer.com/content/pdf/10.1007%2F3-540-49649-1_6.pdf
[intel]: https://software.intel.com/sites/default/files/managed/9e/bc/64-ia-32-architectures-optimization-manual.pdf

48
src/backend/avx2/edwards.rs
View file

@ -8,7 +8,29 @@
// - Isis Agora Lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>
//! Extended Twisted Edwards for Curve25519, using AVX2.
//! Parallel Edwards Arithmetic for Curve25519.
//!
//! This module currently has two point types:
//!
//! * `ExtendedPoint`: a point stored in vector-friendly format, with
//! vectorized doubling and addition;
//!
//! * `CachedPoint`: used for readdition.
//!
//! Details on the formulas can be found in the documentation for the
//! parent `avx2` module.
//!
//! This API is designed to be safe: vectorized points can only be
//! created from serial points (which do validation on decompression),
//! and operations on valid points return valid points, so invalid
//! point states should be unrepresentable.
//!
//! This design goal is met, with one exception: the `Neg`
//! implementation for the `CachedPoint` performs a lazy negation, so
//! that subtraction can be efficiently implemented as a negation and
//! an addition. Repeatedly negating a `CachedPoint` will cause its
//! coefficients to grow and eventually overflow. Repeatedly negating
//! a point should not be necessary anyways.
#![allow(non_snake_case)]
@ -26,7 +48,13 @@ use traits::Identity;
use backend::avx2::field::{FieldElement32x4, Lanes, Shuffle};
use backend::avx2::constants;
/// A point on Curve25519, represented in an AVX2-friendly format.
/// A point on Curve25519, using parallel Edwards formulas for curve
/// operations.
///
/// # Invariant
///
/// The coefficients of an `ExtendedPoint` are bounded with
/// \\( b < 0.007 \\).
#[derive(Copy, Clone, Debug)]
pub struct ExtendedPoint(pub(super) FieldElement32x4);
@ -67,6 +95,7 @@ impl Identity for ExtendedPoint {
}
impl ExtendedPoint {
/// Compute the double of this point.
pub fn double(&self) -> ExtendedPoint {
// Want to compute (X1 Y1 Z1 X1+Y1).
// Not sure how to do this less expensively than computing
@ -138,6 +167,15 @@ impl ExtendedPoint {
}
/// A cached point with some precomputed variables used for readdition.
///
/// # Warning
///
/// It is not safe to negate this point more than once.
///
/// # Invariant
///
/// As long as the `CachedPoint` is not repeatedly negated, its
/// coefficients will be bounded with \\( b < 1.0 \\).
#[derive(Copy, Clone, Debug)]
pub struct CachedPoint(pub(super) FieldElement32x4);
@ -196,6 +234,12 @@ impl<'a, 'b> Add<&'b CachedPoint> for &'a ExtendedPoint {
/// Add an `ExtendedPoint` and a `CachedPoint`.
fn add(self, other: &'b CachedPoint) -> ExtendedPoint {
// The coefficients of an `ExtendedPoint` are reduced after
// every operation. If the `CachedPoint` was negated, its
// coefficients grow by one bit. So on input, `self` is
// bounded with `b < 0.007` and `other` is bounded with
// `b < 1.0`.
let mut tmp = self.0;
tmp = tmp.blend(tmp.diff_sum(), Lanes::AB);

55
src/backend/avx2/field.rs
View file

@ -8,7 +8,20 @@
// - Isis Agora Lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>
//! 4-way vectorized 32bit field arithmetic using AVX2.
//! An implementation of 4-way vectorized 32bit field arithmetic using
//! AVX2.
//!
//! The `FieldElement32x4` struct provides a vector of four field
//! elements, implemented using AVX2 operations. Its API is designed
//! to abstract away the platform-dependent details, so that point
//! arithmetic can be implemented only in terms of a vector of field
//! elements.
//!
//! At this level, the API is optimized for speed and not safety. The
//! `FieldElement32x4` does not always perform reductions. The pre-
//! and post-conditions on the bounds of the coefficients are
//! documented for each method, but it is the caller's responsibility
//! to ensure that there are no overflows.
#![allow(non_snake_case)]
@ -32,7 +45,7 @@ use core::simd::{i32x8, u32x8, u64x4, IntoBits};
use backend::avx2::constants::{P_TIMES_16_HI, P_TIMES_16_LO, P_TIMES_2_HI, P_TIMES_2_LO};
use backend::u64::field::FieldElement64;
/// Unpack
/// Unpack 32-bit lanes into 64-bit lanes:
/// ```
/// (a0, b0, a1, b1, c0, d0, c1, d1)
/// ```
@ -55,7 +68,7 @@ fn unpack_pair(src: u32x8) -> (u32x8, u32x8) {
(a, b)
}
/// Repack
/// Repack 64-bit lanes into 32-bit lanes:
/// ```
/// (a0, 0, b0, 0, c0, 0, d0, 0)
/// (a1, 0, b1, 0, c1, 0, d1, 0)
@ -102,6 +115,10 @@ pub enum Lanes {
}
/// The `Shuffle` enum represents a shuffle of a `FieldElement32x4`.
///
/// The enum variants are named by what they do to a vector \\(
/// (A,B,C,D) \\); for instance, `Shuffle::BADC` turns \\( (A, B, C,
/// D) \\) into \\( (B, A, D, C) \\).
#[derive(Copy, Clone, Debug)]
pub enum Shuffle {
AAAA,
@ -116,7 +133,13 @@ pub enum Shuffle {
ABDC,
}
/// A vector of four field elements, in an AVX2-friendly format.
/// A vector of four field elements.
///
/// Each operation on a `FieldElement32x4` has documented effects on
/// the bounds of the coefficients. This API is designed for speed
/// and not safety; it is the caller's responsibility to ensure that
/// the post-conditions of one operation are compatible with the
/// pre-conditions of the next.
#[derive(Clone, Copy, Debug)]
pub struct FieldElement32x4(pub(crate) [u32x8; 5]);
@ -134,6 +157,8 @@ impl ConditionallyAssignable for FieldElement32x4 {
}
impl FieldElement32x4 {
/// Split this vector into an array of four (serial) field
/// elements.
pub fn split(&self) -> [FieldElement64; 4] {
let mut out = [FieldElement64::zero(); 4];
for i in 0..5 {
@ -155,6 +180,11 @@ impl FieldElement32x4 {
out
}
/// Rearrange the elements of this vector according to `control`.
///
/// The `control` parameter should be a compile-time constant, so
/// that when this function is inlined, LLVM is able to lower the
/// shuffle using an immediate.
#[inline]
pub fn shuffle(&self, control: Shuffle) -> FieldElement32x4 {
#[inline(always)]
@ -190,8 +220,13 @@ impl FieldElement32x4 {
])
}
/// Blend `self` with `other`, taking lanes specified in `control` from `other`.
///
/// The `control` parameter should be a compile-time constant, so
/// that this function can be inlined and LLVM can lower it to a
/// blend instruction using an immediate.
#[inline]
pub fn blend(&self, b: FieldElement32x4, control: Lanes) -> FieldElement32x4 {
pub fn blend(&self, other: FieldElement32x4, control: Lanes) -> FieldElement32x4 {
#[inline(always)]
fn blend_lanes(x: u32x8, y: u32x8, control: Lanes) -> u32x8 {
unsafe {
@ -254,11 +289,11 @@ impl FieldElement32x4 {
}
FieldElement32x4([
blend_lanes(self.0[0], b.0[0], control),
blend_lanes(self.0[1], b.0[1], control),
blend_lanes(self.0[2], b.0[2], control),
blend_lanes(self.0[3], b.0[3], control),
blend_lanes(self.0[4], b.0[4], control),
blend_lanes(self.0[0], other.0[0], control),
blend_lanes(self.0[1], other.0[1], control),
blend_lanes(self.0[2], other.0[2], control),
blend_lanes(self.0[3], other.0[3], control),
blend_lanes(self.0[4], other.0[4], control),
])
}