# XChaCha20 SVP64 Implementation Analysis

This document shows how xchacha20's core loop - all 20 rounds - was implemented in just 11 Vector instructions. There are an additional 9 instructions involved in establishing a REMAP Schedule (explained below), which if there are multiple blocks these 9 instructions do not need to be called again.

Firstly, we analyse the xchacha20 algorithm, showing what operations are performed and in what order. Secondly, two innovative features of SVP64 are described which are crucial to understanding of Simple-V Vectorization: Vertical-First Mode and Indexed REMAP. Then we show how Index REMAP eliminates the need entirely for inline-loop-unrolling, but note that in this particular algorithm REMAP is only useful for us in Vertical-First Mode.

## Description of XChacha20 Algorithm

We will first try to analyze the XChacha20 algorithm as found in:

https://github.com/spcnvdr/xchacha20/blob/master/src/xchacha20.c

The function under inspection is `xchacha_hchacha20`

. If we notice
we will that the main part of the computation, the main algorithm is
just a for loop -which is also the same in the `xchacha_encrypt_bytes`

function as well.

Main loop for `xchacha_hchacha20`

:

```
for (i = 0; i < 10; i++){
QUARTERROUND(x0, x4, x8, x12);
QUARTERROUND(x1, x5, x9, x13);
QUARTERROUND(x2, x6, x10, x14);
QUARTERROUND(x3, x7, x11, x15);
QUARTERROUND(x0, x5, x10, x15);
QUARTERROUND(x1, x6, x11, x12);
QUARTERROUND(x2, x7, x8, x13);
QUARTERROUND(x3, x4, x9, x14);
}
#define QUARTERROUND(a,b,c,d) \
a = PLUS(a,b); d = ROTATE(XOR(d,a),16); \
c = PLUS(c,d); b = ROTATE(XOR(b,c),12); \
a = PLUS(a,b); d = ROTATE(XOR(d,a), 8); \
c = PLUS(c,d); b = ROTATE(XOR(b,c), 7);
```

We see that the loop is split in two groups of `QUARTERROUND`

calls,
one with `step=4`

:

```
QUARTERROUND(x0, x4, x8, x12);
QUARTERROUND(x1, x5, x9, x13);
QUARTERROUND(x2, x6, x10, x14);
QUARTERROUND(x3, x7, x11, x15);
```

and another with `step=5`

:

```
QUARTERROUND(x0, x5, x10, x15);
QUARTERROUND(x1, x6, x11, x12);
QUARTERROUND(x2, x7, x8, x13);
QUARTERROUND(x3, x4, x9, x14);
```

Let's start with the first group of `QUARTERROUND`

s, by unrolling it,
essentially it results in the following instructions:

```
x0 = x0 + x4; x12 = ROTATE(x12 ^ x0, 16);
x8 = x8 + x12; x4 = ROTATE(x4 ^ x8, 12);
x0 = x0 + x4; x12 = ROTATE(x12 ^ x0, 8);
x8 = x8 + x12; x4 = ROTATE(x4 ^ x8, 7);
x1 = x1 + x5; x13 = ROTATE(x13 ^ x1, 16);
x9 = x9 + x13; x5 = ROTATE(x5 ^ x9, 12);
x1 = x1 + x5; x13 = ROTATE(x13 ^ x1, 8);
x9 = x9 + x13; x5 = ROTATE(x5 ^ x9, 7);
x2 = x2 + x6; x14 = ROTATE(x14 ^ x2, 16);
x10 = x10 + x14; x6 = ROTATE(x6 ^ x10, 12);
x2 = x2 + x6; x14 = ROTATE(x14 ^ x2, 8);
x10 = x10 + x14; x6 = ROTATE(x6 ^ x10, 7);
x3 = x3 + x7; x15 = ROTATE(x15 ^ x3, 16);
x11 = x11 + x15; x7 = ROTATE(x7 ^ x11, 12);
x3 = x3 + x7; x15 = ROTATE(x15 ^ x3, 8);
x11 = x11 + x15; x7 = ROTATE(x7 ^ x11, 7);
```

Second group of `QUARTERROUND`

s, unrolled:

```
x0 = x0 + x5; x15 = ROTATE(x15 ^ x0, 16);
x10 = x10 + x15; x5 = ROTATE(x5 ^ x10, 12);
x0 = x0 + x5; x12 = ROTATE(x15 ^ x0, 8);
x10 = x10 + x15; x5 = ROTATE(x5 ^ x10, 7);
x1 = x1 + x6; x12 = ROTATE(x12 ^ x1, 16);
x11 = x11 + x12; x6 = ROTATE(x6 ^ x11, 12);
x1 = x1 + x6; x12 = ROTATE(x12 ^ x1, 8);
x11 = x11 + x12; x6 = ROTATE(x6 ^ x11, 7);
x2 = x2 + x7; x13 = ROTATE(x13 ^ x2, 16);
x8 = x8 + x13; x7 = ROTATE(x7 ^ x8, 12);
x2 = x2 + x7; x13 = ROTATE(x13 ^ x2, 8);
x8 = x8 + x13; x7 = ROTATE(x7 ^ x8, 7);
x3 = x3 + x4; x14 = ROTATE(x14 ^ x3, 16);
x9 = x9 + x14; x4 = ROTATE(x4 ^ x9, 12);
x3 = x3 + x4; x14 = ROTATE(x14 ^ x3, 8);
x9 = x9 + x14; x4 = ROTATE(x4 ^ x9, 7);
```

Let's list the additions only:

```
x0 = x0 + x4
x8 = x8 + x12
x0 = x0 + x4
x8 = x8 + x12
x1 = x1 + x5
x9 = x9 + x13
x1 = x1 + x5
x9 = x9 + x13
x2 = x2 + x6
x10 = x10 + x14
x2 = x2 + x6
x10 = x10 + x14
x3 = x3 + x7
x11 = x11 + x15
x3 = x3 + x7
x11 = x11 + x15
x0 = x0 + x5
x10 = x10 + x15
x0 = x0 + x5
x10 = x10 + x15
x1 = x1 + x6
x11 = x11 + x12
x1 = x1 + x6
x11 = x11 + x12
x2 = x2 + x7
x8 = x8 + x13
x2 = x2 + x7
x8 = x8 + x13
x3 = x3 + x4
x9 = x9 + x14
x3 = x3 + x4
x9 = x9 + x14
```

## Introduction to REMAP Indexing

REMAP Indexing performs any arbitrary re-positioning of elements.
Where normally any other Vector Processor would only be able to do a
sequential element-level series of operations, and if re-ordering
of the elements is required use a special re-ordering instruction,
SVP64 can *in-place* reorder elements on *any* instruction, using
the REMAP subsystem.

Most of the REMAP systems are simple fixed-hardware Deterministic Schedules, but there is one that is general-purpose: Indexing. It requires specifying a group of GPRs (or indices packed into GPRs) that are to be used as the offsets.

This is a normal Simple-V operation:

```
for i in range(VL):
GPR[RT+i] = OPERATION(GPR[RA+i])
```

This is what happens when REMAP is enabled with Indexing:

```
def REMAP(SVSHAPE, i):
return GPR(SVSHAPE.GPR + i)
for i in range(VL):
idx_rt = REMAP(SVSHAPE0, i)
idx_ra = REMAP(SVSHAPE1, i)
GPR[RT+idx_rt] = OPERATION(GPR[RA+idx_ra])
```

In this way we can literally jump about, pretty much anywhere in
the register file, according to a Schedule that is determined by
the programmer. Therefore, if we can put all of the chacha20
round intermediary data into an array of registers, and can
analyse the *order* in which add-operations, xor-operations
and rotate-operations occur, it might just be possible to
eliminate **all** loop-unrolled inline assembler, replacing it
with three instructions and appropriate Indexed REMAP Schedules!
Turns out that this is indeed possible.

## Introduction to Vertical-First Mode

We're going to use Vertical-First mode (VF) to implement this, so we
will first do a short introduction to VF. In normal Horizontal Vector
mode, or even in traditional SIMD mode, the instruction is executed
across a (Horizontal) Vector. So, if we have, for example `VL=8`

, then
the instruction

```
sv.add *RT, *RA, *RB # RT = RA + RB
```

will be executed on all elements of the vector, **before** moving to
the next assembly instruction. This behaviour changes in Vertical-First
mode. In VF mode, the instruction is executed on a **single** element of
the vector and then moves to the next assembly instruction. It continues
to do so until execution reaches the `svstep`

instruction where processing
will be moved to the next element/register in the vector. Branching to
the beginning of the loop does not occur automatically though, a branch
instruction will have to be added manually.

The reason why Vertical-First is needed is because it should be clear
from xchacha20 that there are ordering dependencies between the three
operations `add, xor, rotate`

. It is not okay to perform the entire
suite of Vector-adds then move on to the Vector-xors then the Vector-rotates:
they have to be interleaved so as to respect the element-level ordering.
This is exactly what Vertical-First allows us to do:
element 0 add, element 0 xor, element 0 rotate, then element **1**
add, element **1** xor, element **1** rotate and so on. Vertical-First
*combined* with Index REMAP we can literally jump those operations around,
anywhere within the Vector.

## Application of VF mode in the Xchacha20 loop

Let's assume the values `x`

in the registers 24-36

GPR # | ||||
---|---|---|---|---|

24 | x0 | x1 | x2 | x3 |

28 | x4 | x5 | x6 | x7 |

32 | x8 | x9 | x10 | x11 |

36 | x12 | x13 | x14 | x15 |

So for the addition in Vertical-First mode, `RT`

(and `RA`

as they are the
same) indices are (in terms of x):

0 | 8 | 0 | 8 | 1 | 9 | 1 | 9 |

2 | 10 | 2 | 10 | 3 | 11 | 3 | 11 |

0 | 10 | 0 | 10 | 1 | 11 | 1 | 11 |

2 | 8 | 2 | 8 | 3 | 9 | 3 | 9 |

However, since the indices are small values, using a single 64-bit
register for a single index value is a waste so we will compress them,
8 indices in a 64-bit register:
So, `RT`

indices will fit inside these 4 registers (in Little Endian format):

SVSHAPE0: | 0x901090108000800 | 0xb030b030a020a02 | 0xb010b010a000a00 | 0x903090308020802 |

Similarly we find the RB indices:

4 | 12 | 4 | 12 | 5 | 13 | 5 | 13 |

6 | 14 | 6 | 14 | 7 | 15 | 7 | 15 |

5 | 15 | 5 | 15 | 6 | 12 | 6 | 12 |

7 | 13 | 7 | 13 | 4 | 14 | 7 | 14 |

Using a similar method, we find the final 4 registers with the `RB`

indices:

SVSHAPE1: | 0xd050d050c040c04 | 0xf070f070e060e06 | 0xc060c060f050f05 | 0xe040e040d070d07 |

Now, we can construct the Vertical First loop:

```
svindex 4, 0, 1, 3, 0, 1, 0 # SVSHAPE0, add RA/RT indices
svindex 6, 1, 1, 3, 0, 1, 0 # SVSHAPE1, add RB indices
setvl 0, 0, 32, 1, 1, 1 # MAXVL=VL=32, VF=1
svremap 31, 1, 0, 0, 0, 0, 0 # RA=1, RB=0, RT=0 (0b01011)
sv.add/w=32 *x, *x, *x # RT, RB: SHAPE0. RA: SHAPE1
svstep. 16, 1, 0 # step to next in-regs element
```

What this code snippet does is the following:

The first instruction

```
svindex 4, 0, 1, 3, 0, 1, 0
```

loads the add `RT`

indices in the `SVSHAPE0`

, in register 8. You will note
that 4 is listed, but that's because it only works on even registers,
so in order to save a bit, we have to double that number to get the
actual register. So, `SVSHAPE0`

will be listed in GPRs 8-12. The number
3 lists that the elements will be 8-bit long. 0=64-bit, 1=32-bit,
2=16-bit, 3=8-bit.

The next step instruction

```
svindex 6, 1, 1, 3, 0, 1, 0
```

loads the add `RB`

indices into `SVSHAPE1`

. Again, even though we list 6,
the actual registers will be loaded in GPR #12, again a use of 8-bit
elements is denoted.
Next, the `setvl`

instructions:

```
setvl 0, 0, 32, 1, 1, 1
```

We have to call `setvl`

to set `MAXVL`

and `VL`

to 32 and also configure
Vertical-First mode. Afterwards, we have to instruct the way we intend
to use the indices, and we do this using `svremap`

.

```
svremap 31, 1, 0, 0, 0, 0, 0
```

`svremap`

basically instructs the scheduler to use `SVSHAPE0`

for `RT`

and `RB`

,
`SVSHAPE1`

for `RA`

. The next instruction performs the **actual** addition:

```
sv.add/w=32 *x, *x, *x
```

Note the `/w=32`

suffix. This instructs the adder to perform the operation
in elements of `w=32`

bits. Since the Power CPU is a 64-bit CPU, this means
that we need to have 2 32-bit elements loaded in each register. Also,
note that in all parameters we use the `*x`

as argument. This instructs
the scheduler to act on the registers as a vector, or a sequence of
elements. But even though they are all the same, their indices will be
taken from the `SVSHAPE0`

/`SVSHAPE1`

indices as defined previously. Also
note that the indices are relative to the actual register used. So,
if `*x`

starts in GPR 24 for example, in essence this instruction will
issue the following sequence of instructions:

```
add/w=32 24 + 0, 24 + 4, 24 + 0
add/w=32 24 + 8, 24 + 12, 24 + 8
add/w=32 24 + 0, 24 + 4, 24 + 0
add/w=32 24 + 8, 24 + 12, 24 + 8
add/w=32 24 + 1, 24 + 5, 24 + 1
add/w=32 24 + 9, 24 + 13, 24 + 9
add/w=32 24 + 1, 24 + 5, 24 + 1
add/w=32 24 + 9, 24 + 13, 24 + 9
...
```

Finally, the `svstep.`

instruction steps to the next set of indices

We have shown how to do the additions in a Vertical-first mode. Now
let's add the rest of the instructions in the `QUARTERROUND`

s. For the
`XOR`

instructions of both `QUARTERROUND`

s groups only, assuming that ```
d =
XOR(d, a)
```

:

```
x12 = x12 ^ x0
x4 = x4 ^ x8
x12 = x12 ^ x0
x4 = x4 ^ x8
x13 = x13 ^ x1
x5 = x5 ^ x9
x13 = x13 ^ x1
x5 = x5 ^ x9
x14 = x14 ^ x2
x6 = x6 ^ x10
x14 = x14 ^ x2
x6 = x6 ^ x10
x15 = x15 ^ x3
x7 = x7 ^ x11
x15 = x15 ^ x3
x7 = x7 ^ x11
x15 = x15 ^ x0
x5 = x5 ^ x10
x12 = x15 ^ x0
x5 = x5 ^ x10
x12 = x12 ^ x1
x6 = x6 ^ x11
x12 = x12 ^ x1
x6 = x6 ^ x11
x13 = x13 ^ x2
x7 = x7 ^ x8
x13 = x13 ^ x1
x7 = x7 ^ x8
x14 = x14 ^ x3
x4 = x4 ^ x9
x14 = x14 ^ x3
x4 = x4 ^ x9
```

We will need to create another set of indices for the `XOR`

instructions. We
will only need one set as the other set of indices is the same as `RT`

for `sv.add`

(`SHAPE0`

). So, remembering that our

12 | 4 | 12 | 4 | 13 | 5 | 13 | 5 |

14 | 6 | 14 | 6 | 15 | 7 | 15 | 7 |

15 | 5 | 15 | 5 | 12 | 6 | 12 | 6 |

13 | 7 | 13 | 7 | 14 | 4 | 14 | 4 |

Again, we find

SVSHAPE2: | 0x50d050d040c040c | 0x70f070f060e060e | 0x60c060c050f050f | 0x40e040e070d070d |

The next operation is the `ROTATE`

which takes as operand the result of the
`XOR`

and a shift argument. You can easily see that the indices used in this
case are the same as the `XOR`

. However, the shift values cycle every 4:
16, 12, 8, 7. For the indices we can again use `svindex`

, like this:

```
svindex 8, 2, 1, 3, 0, 1, 0
```

Which again means `SVSHAPE2`

, operating on 8-bit elements, starting
from GPR #16 (`8*2`

). For the shift values cycling every 4 elements,
the `svshape2`

instruction will be used:

```
svshape2 0, 0, 3, 4, 0, 1
```

This will create an `SVSHAPE3`

, which will use a modulo 4 for all of its
elements. Now we can list both `XOR`

and `ROTATE`

instructions in assembly,
together with the respective svremap instructions:

```
svremap 31, 2, 0, 2, 2, 0, 0 # RA=2, RB=0, RS=2 (0b00111)
sv.xor/w=32 *x, *x, *x
svremap 31, 0, 3, 2, 2, 0, 0 # RA=2, RB=3, RS=2 (0b01110)
sv.rldcl/w=32 *x, *x, *SHIFTS, 0
```

So, in a similar fashion, we instruct `XOR`

(`sv.xor`

) to use `SVSHAPE2`

for
`RA`

and `RS`

and `SVSHAPE0`

for `RB`

, again for 32-bit elements, while `ROTATE`

(`sv.rldcl`

) will also use `SVSHAPE2`

for `RA`

and `RS`

, but `SVSHAPE3`

for `RB`

(the shift values, which cycle every 4 elements). Note that the actual
indices for `SVSHAPE3`

will have to be in 32-bit elements:

SHIFTS: | 0x0000000c00000010 | 0x0000000700000008 |

The complete algorithm for a loop with 10 iterations is as follows:

```
li 7, 10 # Load value 10 into GPR #7
mtctr 7 # Set up counter on GPR #7
# set up VL=32 vertical-first, and SVSHAPEs 0-2
setvl 0, 0, 32, 1, 1, 1
# SHAPE0, used by sv.add starts at GPR #8
svindex 8/2, 0, 1, 3, 0, 1, 0 # SVSHAPE0, a
# SHAPE1, used by sv.xor starts at GPR #12
svindex 12/2, 1, 1, 3, 0, 1, 0 # SVSHAPE1, b
# SHAPE2, used by sv.rldcl starts at GPR #16
svindex 16/2, 2, 1, 3, 0, 1, 0 # SVSHAPE2, c
# SHAPE3, used also by sv.rldcl to hold the shift values starts at GPR #20
# The inner loop will do 32 iterations, but there are only
# 4 shift values, so we mod by 4, and can cycle through them
svshape2 0, 0, 3, 4, 0, 1 # SVSHAPE3, shift amount, mod4
.outer:
# outer loop begins here (standard CTR loop)
setvl 0, 0, 32, 1, 1, 1 # MAXVL=VL=32, VF=1
# inner loop begins here. add-xor-rotl32 with remap, step, branch
.inner:
svremap 31, 1, 0, 0, 0, 0, 0 # RA=1, RB=0, RT=0 (0b01011)
sv.add/w=32 *x, *x, *x
svremap 31, 2, 0, 2, 2, 0, 0 # RA=2, RB=0, RS=2 (0b00111)
sv.xor/w=32 *x, *x, *x
svremap 31, 0, 3, 2, 2, 0, 0 # RA=2, RB=3, RS=2 (0b01110)
sv.rldcl/w=32 *x, *x, *SHIFTS, 0
# 16 is the destination containing the result of svstep.
# it overlaps with SHAPE2 which is also 16. the first 8 indices
# will get corrupted.
svstep. 7, 1, 0 # step to next in-regs element
bc 6, 3, .inner # svstep. Rc=1 loop-end-condition?
# inner-loop done: outer loop standard CTR-decrement to setvl again
bdnz .outer # Loop until CTR is zero
```