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[libreriscv.git] / openpower / sv / bitmanip.mdwn

[[!tag standards]]

[[!toc levels=1]]

# Implementation Log

* ternlogi <https://bugs.libre-soc.org/show_bug.cgi?id=745>
* grev <https://bugs.libre-soc.org/show_bug.cgi?id=755>
* GF2^M <https://bugs.libre-soc.org/show_bug.cgi?id=782>
* binutils <https://bugs.libre-soc.org/show_bug.cgi?id=836>
* shift-and-add <https://bugs.libre-soc.org/show_bug.cgi?id=968>

# bitmanipulation

**DRAFT STATUS**

pseudocode: [[openpower/isa/bitmanip]]

this extension amalgamates bitmanipulation primitives from many sources,
including RISC-V bitmanip, Packed SIMD, AVX-512 and OpenPOWER VSX.
Also included are DSP/Multimedia operations suitable for Audio/Video.
Vectorisation and SIMD are removed: these are straight scalar (element)
operations making them suitable for embedded applications.  Vectorisation
Context is provided by [[openpower/sv]].

When combined with SV, scalar variants of bitmanip operations found in
VSX are added so that the Packed SIMD aspects of VSX may be retired as
"legacy" in the far future (10 to 20 years).  Also, VSX is hundreds of
opcodes, requires 128 bit pathways, and is wholly unsuited to low power
or embedded scenarios.

ternlogv is experimental and is the only operation that may be considered
a "Packed SIMD".  It is added as a variant of the already well-justified
ternlog operation (done in AVX512 as an immediate only) "because it
looks fun". As it is based on the LUT4 concept it will allow accelerated
emulation of FPGAs.  Other vendors of ISAs are buying FPGA companies to
achieve similar objectives.

general-purpose Galois Field 2^M operations are added so as to avoid
huge custom opcode proliferation across many areas of Computer Science.
however for convenience and also to avoid setup costs, some of the more
common operations (clmul, crc32) are also added.  The expectation is
that these operations would all be covered by the same pipeline.

note that there are brownfield spaces below that could incorporate
some of the set-before-first and other scalar operations listed in
[[sv/mv.swizzle]],
[[sv/vector_ops]], [[sv/int_fp_mv]] and the [[sv/av_opcodes]] as well as
[[sv/setvl]], [[sv/svstep]], [[sv/remap]]

Useful resource: 

* <https://en.wikiversity.org/wiki/Reed%E2%80%93Solomon_codes_for_coders>
* <https://maths-people.anu.edu.au/~brent/pd/rpb232tr.pdf>
* <https://gist.github.com/animetosho/d3ca95da2131b5813e16b5bb1b137ca0>
* <https://github.com/HJLebbink/asm-dude/wiki/GF2P8AFFINEINVQB>

[[!inline pages="openpower/sv/draft_opcode_tables" quick="yes" raw="yes" ]]

# binary and ternary bitops

Similar to FPGA LUTs: for two (binary) or three (ternary) inputs take
bits from each input, concatenate them and perform a lookup into a
table using an 8-8-bit immediate (for the ternary instructions), or in
another register (4-bit for the binary instructions).  The binary lookup
instructions have CR Field lookup variants due to CR Fields being 4 bit.

Like the x86 AVX512F
[vpternlogd/vpternlogq](https://www.felixcloutier.com/x86/vpternlogd:vpternlogq)
instructions.

## ternlogi

| 0.5|6.10|11.15|16.20| 21..28|29.30|31|
| -- | -- | --- | --- | ----- | --- |--|
| NN | RT | RA  | RB  | im0-7 |  00 |Rc|

    lut3(imm, a, b, c):
        idx = c << 2 | b << 1 | a
        return imm[idx] # idx by LSB0 order

    for i in range(64): 
        RT[i] = lut3(imm, RB[i], RA[i], RT[i]) 

## binlut

Binary lookup is a dynamic LUT2 version of ternlogi. Firstly, the
lookup table is 4 bits wide not 8 bits, and secondly the lookup
table comes from a register not an immediate.

| 0.5|6.10|11.15|16.20| 21..25|26..31  | Form    |
| -- | -- | --- | --- | ----- |--------|---------|
| NN | RT | RA  | RB  | RC    |nh 00001| VA-Form |
| NN | RT | RA  | RB  | /BFA/ |0  01001| VA-Form |

For binlut, the 4-bit LUT may be selected from either the high nibble
or the low nibble of the first byte of RC:

    lut2(imm, a, b):
        idx = b << 1 | a
        return imm[idx] # idx by LSB0 order

    imm = (RC>>(nh*4))&0b1111
    for i in range(64): 
        RT[i] = lut2(imm, RB[i], RA[i]) 

For bincrlut, `BFA` selects the 4-bit CR Field as the LUT2:

    for i in range(64): 
        RT[i] = lut2(CRs{BFA}, RB[i], RA[i]) 

When Vectorised with SVP64, as usual both source and destination may be
Vector or Scalar.

*Programmer's note: a dynamic ternary lookup may be synthesised from
a pair of `binlut` instructions followed by a `ternlogi` to select which
to merge. Use `nh` to select which nibble to use as the lookup table
from the RC source register (`nh=1` nibble high), i.e. keeping
an 8-bit LUT3 in RC, the first `binlut` instruction may set nh=0 and
the second nh=1.*

## crternlogi

another mode selection would be CRs not Ints.

CRB-Form:

| 0.5|6.8 |9.10|11.13|14.15|16.18|19.25|26.30| 31|
|----|----|----|-----|-----|-----|-----|-----|---|
| NN | BF | msk|BFA  | msk | BFB | TLI | XO  |TLI|

    for i in range(4):
        a,b,c = CRs[BF][i], CRs[BFA][i], CRs[BFB][i])
        if msk[i] CRs[BF][i] = lut3(imm, a, b, c)

This instruction is remarkably similar to the existing crops, `crand` etc.
which have been noted to be a 4-bit (binary) LUT.  In effect `crternlogi`
is the ternary LUT version of crops, having an 8-bit LUT.  However it
is an overwrite instruction in order to save on register file ports,
due to the mask requiring the contents of the BF to be both read and
written.

Programmer's note: This instruction is useful when combined with Matrix REMAP
in "Inner Product" Mode, creating Warshall Transitive Closure that has many
applications in Computer Science.

## crbinlog

With ternary (LUT3) dynamic instructions being very costly,
and CR Fields being only 4 bit, a binary (LUT2) variant is better

CRB-Form:

| 0.5|6.8 |9.10|11.13|14.15|16.18|19.25|26.30| 31|
|----|----|----|-----|-----|-----|-----|-----|---|
| NN | BF | msk|BFA  | msk | BFB | //  | XO  | //|

    for i in range(4):
        a,b = CRs[BF][i], CRs[BF][i])
        if msk[i] CRs[BF][i] = lut2(CRs[BFB], a, b)

When SVP64 Vectorised any of the 4 operands may be Scalar or
Vector, including `BFB` meaning that multiple different dynamic
lookups may be performed with a single instruction.  Note that
this instruction is deliberately an overwrite in order to reduce
the number of register file ports required: like `crternlogi`
the contents of `BF` **must** be read due to the mask only
writing back to non-masked-out bits of `BF`.

*Programmer's note: just as with binlut and ternlogi, a pair
 of crbinlog instructions followed by a merging crternlogi may
 be deployed to synthesise dynamic ternary (LUT3) CR Field
 manipulation*

# int ops

## min/m

required for the [[sv/av_opcodes]]

signed and unsigned min/max for integer.  this is sort-of partly
synthesiseable in [[sv/svp64]] with pred-result as long as the dest reg
is one of the sources, but not both signed and unsigned.  when the dest
is also one of the srces and the mv fails due to the CR bittest failing
this will only overwrite the dest where the src is greater (or less).

signed/unsigned min/max gives more flexibility.

\[un]signed min/max instructions are specifically needed for vector reduce min/max operations which are pretty common.

X-Form

* XO=0001001110, itype=0b00 min, unsigned
* XO=0101001110, itype=0b01 min, signed
* XO=0011001110, itype=0b10 max, unsigned
* XO=0111001110, itype=0b11 max, signed


```
uint_xlen_t mins(uint_xlen_t rs1, uint_xlen_t rs2)
{ return (int_xlen_t)rs1 < (int_xlen_t)rs2 ? rs1 : rs2;
}
uint_xlen_t maxs(uint_xlen_t rs1, uint_xlen_t rs2)
{ return (int_xlen_t)rs1 > (int_xlen_t)rs2 ? rs1 : rs2;
}
uint_xlen_t minu(uint_xlen_t rs1, uint_xlen_t rs2)
{ return rs1 < rs2 ? rs1 : rs2;
}
uint_xlen_t maxu(uint_xlen_t rs1, uint_xlen_t rs2)
{ return rs1 > rs2 ? rs1 : rs2;
}
```

## average

required for the [[sv/av_opcodes]], these exist in Packed SIMD (VSX)
but not scalar

```
uint_xlen_t intavg(uint_xlen_t rs1, uint_xlen_t rs2) {
     return (rs1 + rs2 + 1) >> 1:
}
```

## absdu

required for the [[sv/av_opcodes]], these exist in Packed SIMD (VSX)
but not scalar

```
uint_xlen_t absdu(uint_xlen_t rs1, uint_xlen_t rs2) {
     return (src1 > src2) ? (src1-src2) : (src2-src1)
}
```

## abs-accumulate

required for the [[sv/av_opcodes]], these are needed for motion estimation.
both are overwrite on RS.

```
uint_xlen_t uintabsacc(uint_xlen_t rs, uint_xlen_t ra, uint_xlen_t rb) {
     return rs + (src1 > src2) ? (src1-src2) : (src2-src1)
}
uint_xlen_t intabsacc(uint_xlen_t rs, int_xlen_t ra, int_xlen_t rb) {
     return rs + (src1 > src2) ? (src1-src2) : (src2-src1)
}
```

For SVP64, the twin Elwidths allows e.g. a 16 bit accumulator for 8 bit
differences.  Form is `RM-1P-3S1D` where RS-as-source has a separate
SVP64 designation from RS-as-dest. This gives a limited range of
non-overwrite capability.

# shift-and-add <a name="shift-add"> </a>

Power ISA is missing LD/ST with shift, which is present in both ARM and x86.
Too complex to add more LD/ST, a compromise is to add shift-and-add.
Replaces a pair of explicit instructions in hot-loops.

```
# 1.6.27 Z23-FORM
    |0     |6     |11    |15 |16     |21 |23    |31 |
    | PO   |  RT  |   RA     |   RB  |sm |   XO |Rc |
```

Pseudo-code (shadd):

    n <- (RB)
    m <- sm + 1
    RT <- (n[m:XLEN-1] || [0]*m) + (RA)

Pseudo-code (shaddw):

    shift <- sm + 1                # Shift is between 1-4
    n <- EXTS((RB)[XLEN/2:XLEN-1]) # Only use lower XLEN/2-bits of RB
    RT <- (n << shift) + (RA)      # Shift n, add RA

Pseudo-code (shadduw):

    n <- ([0]*(XLEN/2)) || (RB)[XLEN/2:XLEN-1]
    m <- sm + 1
    RT <- (n[m:XLEN-1] || [0]*m) + (RA)

```
uint_xlen_t shadd(uint_xlen_t RA, uint_xlen_t RB, uint8_t sm) {
    sm = sm & 0x3;
    return (RB << (sm+1)) + RA;
}

uint_xlen_t shaddw(uint_xlen_t RA, uint_xlen_t RB, uint8_t sm) {
    uint_xlen_t n = (int_xlen_t)(RB << XLEN / 2) >> XLEN / 2;
    sm = sm & 0x3;
    return (n << (sm+1)) + RA;
}

uint_xlen_t shadduw(uint_xlen_t RA, uint_xlen_t RB, uint8_t sm) {
    uint_xlen_t n = RB & 0xFFFFFFFF;
    sm = sm & 0x3;
    return (n << (sm+1)) + RA;
}
```

# bitmask set

based on RV bitmanip singlebit set, instruction format similar to shift
[[isa/fixedshift]].  bmext is actually covered already (shift-with-mask
rldicl but only immediate version).  however bitmask-invert is not,
and set/clr are not covered, although they can use the same Shift ALU.

bmext (RB) version is not the same as rldicl because bmext is a right
shift by RC, where rldicl is a left rotate.  for the immediate version
this does not matter, so a bmexti is not required.  bmrev however there
is no direct equivalent and consequently a bmrevi is required.

bmset (register for mask amount) is particularly useful for creating
predicate masks where the length is a dynamic runtime quantity.
bmset(RA=0, RB=0, RC=mask) will produce a run of ones of length "mask"
in a single instruction without needing to initialise or depend on any
other registers.

| 0.5|6.10|11.15|16.20|21.25| 26..30  |31| name  |
| -- | -- | --- | --- | --- | ------- |--| ----- |
| NN | RS | RA  | RB  | RC  | mode 010 |Rc| bm\*   |

Immediate-variant is an overwrite form:

| 0.5|6.10|11.15|16.20| 21 | 22.23 | 24....30 |31| name |
| -- | -- | --- | --- | -- | ----- | -------- |--| ---- |
| NN | RS | RB  | sh  | SH | itype | 1000 110 |Rc| bm\*i |

```
def MASK(x, y):
     if x < y:
         x = x+1
         mask_a = ((1 << x) - 1) & ((1 << 64) - 1)
         mask_b = ((1 << y) - 1) & ((1 << 64) - 1)
     elif x == y:
         return 1 << x
     else:
         x = x+1
         mask_a = ((1 << x) - 1) & ((1 << 64) - 1)
         mask_b = (~((1 << y) - 1)) & ((1 << 64) - 1)
     return mask_a ^ mask_b


uint_xlen_t bmset(RS, RB, sh)
{
    int shamt = RB & (XLEN - 1);
    mask = (2<<sh)-1;
    return RS | (mask << shamt);
}

uint_xlen_t bmclr(RS, RB, sh)
{
    int shamt = RB & (XLEN - 1);
    mask = (2<<sh)-1;
    return RS & ~(mask << shamt);
}

uint_xlen_t bminv(RS, RB, sh)
{
    int shamt = RB & (XLEN - 1);
    mask = (2<<sh)-1;
    return RS ^ (mask << shamt);
}

uint_xlen_t bmext(RS, RB, sh)
{
    int shamt = RB & (XLEN - 1);
    mask = (2<<sh)-1;
    return mask & (RS >> shamt);
}
```

bitmask extract with reverse.  can be done by bit-order-inverting all
of RB and getting bits of RB from the opposite end.

when RA is zero, no shift occurs. this makes bmextrev useful for
simply reversing all bits of a register.

```
msb = ra[5:0];
rev[0:msb] = rb[msb:0];
rt = ZE(rev[msb:0]);

uint_xlen_t bmrevi(RA, RB, sh)
{
    int shamt = XLEN-1;
    if (RA != 0) shamt = (GPR(RA) & (XLEN - 1));
    shamt = (XLEN-1)-shamt;  # shift other end
    brb = bitreverse(GPR(RB))     # swap LSB-MSB
    mask = (2<<sh)-1;
    return mask & (brb >> shamt);
}

uint_xlen_t bmrev(RA, RB, RC) {
    return bmrevi(RA, RB, GPR(RC) & 0b111111);
}
```

| 0.5|6.10|11.15|16.20|21.26| 27..30  |31| name   | Form     |
| -- | -- | --- | --- | --- | ------- |--| ------ | -------- |
| NN | RT | RA  | RB  | sh  | 1111    |Rc| bmrevi | MDS-Form |

| 0.5|6.10|11.15|16.20|21.25| 26..30  |31| name   | Form     |
| -- | -- | --- | --- | --- | ------- |--| ------ | -------- |
| NN | RT | RA  | RB  | RC  | 11110   |Rc| bmrev  | VA2-Form |

# grevlut <a name="grevlut"> </a>

generalised reverse combined with a pair of LUT2s and allowing
a constant `0b0101...0101` when RA=0, and an option to invert
(including when RA=0, giving a constant 0b1010...1010 as the
initial value) provides a wide range of instructions
and a means to set hundreds of regular 64 bit patterns with one
single 32 bit instruction.

the two LUT2s are applied left-half (when not swapping)
and right-half (when swapping) so as to allow a wider
range of options.

<img src="/openpower/sv/grevlut2x2.jpg" width=700 />

* A value of `0b11001010` for the immediate provides
the functionality of a standard "grev".  
* `0b11101110` provides gorc

grevlut should be arranged so as to produce the constants
needed to put into bext (bitextract) so as in turn to
be able to emulate x86 pmovmask instructions
<https://www.felixcloutier.com/x86/pmovmskb>.
This only requires 2 instructions (grevlut, bext).

Note that if the mask is required to be placed
directly into CR Fields (for use as CR Predicate
masks rather than a integer mask) then sv.cmpi or sv.ori
may be used instead, bearing in mind that sv.ori
is a 64-bit instruction, and `VL` must have been
set to the required length:

    sv.ori./elwid=8 r10.v, r10.v, 0

The following settings provide the required mask constants:

| RA=0     | RB      | imm        | iv | result        |
| -------  | ------- | ---------- | -- | ----------    |
| 0x555..  | 0b10    | 0b01101100 | 0  | 0x111111...   |
| 0x555..  | 0b110   | 0b01101100 | 0  | 0x010101...   |
| 0x555..  | 0b1110  | 0b01101100 | 0  | 0x00010001...   |
| 0x555..  | 0b10    | 0b11000110 | 1  | 0x88888...   |
| 0x555..  | 0b110   | 0b11000110 | 1  | 0x808080...   |
| 0x555..  | 0b1110  | 0b11000110 | 1  | 0x80008000...   |

Better diagram showing the correct ordering of shamt (RB).  A LUT2
is applied to all locations marked in red using the first 4
bits of the immediate, and a separate LUT2 applied to all
locations in green using the upper 4 bits of the immediate.

<img src="/openpower/sv/grevlut.png" width=700 />

demo code [[openpower/sv/grevlut.py]]

```
lut2(imm, a, b):
    idx = b << 1 | a
    return imm[idx] # idx by LSB0 order

dorow(imm8, step_i, chunksize, us32b):
    for j in 0 to 31 if is32b else 63:
        if (j&chunk_size) == 0
           imm = imm8[0..3]
        else
           imm = imm8[4..7]
        step_o[j] = lut2(imm, step_i[j], step_i[j ^ chunk_size])
    return step_o

uint64_t grevlut(uint64_t RA, uint64_t RB, uint8 imm, bool iv, bool is32b)
{
    uint64_t x = 0x5555_5555_5555_5555;
    if (RA != 0) x = GPR(RA);
    if (iv) x = ~x;
    int shamt = RB & 31 if is32b else 63
    for i in 0 to (6-is32b)
        step = 1<<i
        if (shamt & step) x = dorow(imm, x, step, is32b)
    return x;
}
```

A variant may specify different LUT-pairs per row,
using one byte of RB for each.  If it is desired that
a particular row-crossover shall not be applied it is
a simple matter to set the appropriate LUT-pair in RB
to effect an identity transform for that row (`0b11001010`).

```
uint64_t grevlutr(uint64_t RA, uint64_t RB, bool iv, bool is32b)
{
    uint64_t x = 0x5555_5555_5555_5555;
    if (RA != 0) x = GPR(RA);
    if (iv) x = ~x;
    for i in 0 to (6-is32b)
        step = 1<<i
        imm = (RB>>(i*8))&0xff
        x = dorow(imm, x, step, is32b)
    return x;
}

```

| 0.5|6.10|11.15|16.20 |21..28   | 29.30|31| name | Form |
| -- | -- | --- | ---  | -----   | -----|--| ------ | ----- |
| NN | RT | RA  | s0-4 | im0-7   | 1 iv |s5| grevlogi |      |
| NN | RT | RA  | RB   | im0-7   | 01   |0 | grevlog |           |

An equivalent to `grevlogw` may be synthesised by setting the
appropriate bits in RB to set the top half of RT to zero.
Thus an explicit grevlogw instruction is not necessary.

# xperm

based on RV bitmanip.

RA contains a vector of indices to select parts of RB to be
copied to RT.  The immediate-variant allows up to an 8 bit
pattern (repeated) to be targetted at different parts of RT.

xperm shares some similarity with one of the uses of bmator
in that xperm indices are binary addressing where bitmator
may be considered to be unary addressing.

```
uint_xlen_t xpermi(uint8_t imm8, uint_xlen_t RB, int sz_log2)
{
    uint_xlen_t r = 0;
    uint_xlen_t sz = 1LL << sz_log2;
    uint_xlen_t mask = (1LL << sz) - 1;
    uint_xlen_t RA = imm8 | imm8<<8 | ... | imm8<<56;
    for (int i = 0; i < XLEN; i += sz) {
        uint_xlen_t pos = ((RA >> i) & mask) << sz_log2;
        if (pos < XLEN)
            r |= ((RB >> pos) & mask) << i;
    }
    return r;
}
uint_xlen_t xperm(uint_xlen_t RA, uint_xlen_t RB, int sz_log2)
{
    uint_xlen_t r = 0;
    uint_xlen_t sz = 1LL << sz_log2;
    uint_xlen_t mask = (1LL << sz) - 1;
    for (int i = 0; i < XLEN; i += sz) {
        uint_xlen_t pos = ((RA >> i) & mask) << sz_log2;
        if (pos < XLEN)
            r |= ((RB >> pos) & mask) << i;
    }
    return r;
}
uint_xlen_t xperm_n (uint_xlen_t RA, uint_xlen_t RB)
{  return xperm(RA, RB, 2); }
uint_xlen_t xperm_b (uint_xlen_t RA, uint_xlen_t RB)
{  return xperm(RA, RB, 3); }
uint_xlen_t xperm_h (uint_xlen_t RA, uint_xlen_t RB)
{  return xperm(RA, RB, 4); }
uint_xlen_t xperm_w (uint_xlen_t RA, uint_xlen_t RB)
{  return xperm(RA, RB, 5); }
```

# bitmatrix

bmatflip and bmatxor is found in the Cray XMT, and in x86 is known
as GF2P8AFFINEQB. uses:

* <https://gist.github.com/animetosho/d3ca95da2131b5813e16b5bb1b137ca0>
* SM4, Reed Solomon, RAID6
  <https://stackoverflow.com/questions/59124720/what-are-the-avx-512-galois-field-related-instructions-for>
* Vector bit-reverse <https://reviews.llvm.org/D91515?id=305411>
* Affine Inverse <https://github.com/HJLebbink/asm-dude/wiki/GF2P8AFFINEINVQB>

| 0.5|6.10|11.15|16.20| 21 | 22.23 | 24....30 |31| name      |  Form   |
| -- | -- | --- | --- | -- | ----- | -------- |--| ----      | ------- |
| NN | RS | RA  |im04 | im5|  1 1  | im67 00 110 |Rc| bmatxori  | TODO    |


```
uint64_t bmatflip(uint64_t RA)
{
    uint64_t x = RA;
    x = shfl64(x, 31);
    x = shfl64(x, 31);
    x = shfl64(x, 31);
    return x;
}

uint64_t bmatxori(uint64_t RS, uint64_t RA, uint8_t imm) {
    // transpose of RA
    uint64_t RAt = bmatflip(RA);
    uint8_t u[8]; // rows of RS
    uint8_t v[8]; // cols of RA
    for (int i = 0; i < 8; i++) {
        u[i] = RS >> (i*8);
        v[i] = RAt >> (i*8);
    }
    uint64_t bit, x = 0;
    for (int i = 0; i < 64; i++) {
        bit = (imm >> (i%8)) & 1;
        bit ^= pcnt(u[i / 8] & v[i % 8]) & 1;
        x |= bit << i;
    }
    return x;
}

uint64_t bmatxor(uint64_t RA, uint64_t RB) {
    return bmatxori(RA, RB, 0xff)
}

uint64_t bmator(uint64_t RA, uint64_t RB) {
    // transpose of RB
    uint64_t RBt = bmatflip(RB);
    uint8_t u[8]; // rows of RA
    uint8_t v[8]; // cols of RB
    for (int i = 0; i < 8; i++) {
        u[i] = RA >> (i*8);
        v[i] = RBt >> (i*8);
    }
    uint64_t x = 0;
    for (int i = 0; i < 64; i++) {
        if ((u[i / 8] & v[i % 8]) != 0)
            x |= 1LL << i;
    }
    return x;
}

uint64_t bmatand(uint64_t RA, uint64_t RB) {
    // transpose of RB
    uint64_t RBt = bmatflip(RB);
    uint8_t u[8]; // rows of RA
    uint8_t v[8]; // cols of RB
    for (int i = 0; i < 8; i++) {
        u[i] = RA >> (i*8);
        v[i] = RBt >> (i*8);
    }
    uint64_t x = 0;
    for (int i = 0; i < 64; i++) {
        if ((u[i / 8] & v[i % 8]) == 0xff)
            x |= 1LL << i;
    }
    return x;
}
```

# Introduction to Carry-less and GF arithmetic

* obligatory xkcd <https://xkcd.com/2595/>

There are three completely separate types of Galois-Field-based arithmetic
that we implement which are not well explained even in introductory
literature.  A slightly oversimplified explanation is followed by more
accurate descriptions:

* `GF(2)` carry-less binary arithmetic. this is not actually a Galois Field,
  but is accidentally referred to as GF(2) - see below as to why.
* `GF(p)` modulo arithmetic with a Prime number, these are "proper"
   Galois Fields
* `GF(2^N)` carry-less binary arithmetic with two limits: modulo a power-of-2
  (2^N) and a second "reducing" polynomial (similar to a prime number), these
  are said to be GF(2^N) arithmetic.

further detailed and more precise explanations are provided below

* **Polynomials with coefficients in `GF(2)`**
  (aka. Carry-less arithmetic -- the `cl*` instructions).
  This isn't actually a Galois Field, but its coefficients are. This is
  basically binary integer addition, subtraction, and multiplication like
  usual, except that carries aren't propagated at all, effectively turning
  both addition and subtraction into the bitwise xor operation. Division and
  remainder are defined to match how addition and multiplication works.
* **Galois Fields with a prime size**
  (aka. `GF(p)` or Prime Galois Fields -- the `gfp*` instructions).
  This is basically just the integers mod `p`.
* **Galois Fields with a power-of-a-prime size**
  (aka. `GF(p^n)` or `GF(q)` where `q == p^n` for prime `p` and
  integer `n > 0`).
  We only implement these for `p == 2`, called Binary Galois Fields
  (`GF(2^n)` -- the `gfb*` instructions).
  For any prime `p`, `GF(p^n)` is implemented as polynomials with
  coefficients in `GF(p)` and degree `< n`, where the polynomials are the
  remainders of dividing by a specificly chosen polynomial in `GF(p)` called
  the Reducing Polynomial (we will denote that by `red_poly`). The Reducing
  Polynomial must be an irreducable polynomial (like primes, but for
  polynomials), as well as have degree `n`. All `GF(p^n)` for the same `p`
  and `n` are isomorphic to each other -- the choice of `red_poly` doesn't
  affect `GF(p^n)`'s mathematical shape, all that changes is the specific
  polynomials used to implement `GF(p^n)`.

Many implementations and much of the literature do not make a clear
distinction between these three categories, which makes it confusing
to understand what their purpose and value is.

* carry-less multiply is extremely common and is used for the ubiquitous
  CRC32 algorithm. [TODO add many others, helps justify to ISA WG]
* GF(2^N) forms the basis of Rijndael (the current AES standard) and
  has significant uses throughout cryptography
* GF(p) is the basis again of a significant quantity of algorithms
  (TODO, list them, jacob knows what they are), even though the
  modulo is limited to be below 64-bit (size of a scalar int)

# Instructions for Carry-less Operations

aka. Polynomials with coefficients in `GF(2)`

Carry-less addition/subtraction is simply XOR, so a `cladd`
instruction is not provided since the `xor[i]` instruction can be used instead.

These are operations on polynomials with coefficients in `GF(2)`, with the
polynomial's coefficients packed into integers with the following algorithm:

```python
[[!inline pagenames="gf_reference/pack_poly.py" raw="yes"]]
```

## Carry-less Multiply Instructions

based on RV bitmanip
see <https://en.wikipedia.org/wiki/CLMUL_instruction_set> and
<https://www.felixcloutier.com/x86/pclmulqdq> and
<https://en.m.wikipedia.org/wiki/Carry-less_product>

They are worth adding as their own non-overwrite operations
(in the same pipeline).

### `clmul` Carry-less Multiply

```python
[[!inline pagenames="gf_reference/clmul.py" raw="yes"]]
```

### `clmulh` Carry-less Multiply High

```python
[[!inline pagenames="gf_reference/clmulh.py" raw="yes"]]
```

### `clmulr` Carry-less Multiply (Reversed)

Useful for CRCs. Equivalent to bit-reversing the result of `clmul` on
bit-reversed inputs.

```python
[[!inline pagenames="gf_reference/clmulr.py" raw="yes"]]
```

## `clmadd` Carry-less Multiply-Add

```
clmadd RT, RA, RB, RC
```

```
(RT) = clmul((RA), (RB)) ^ (RC)
```

## `cltmadd` Twin Carry-less Multiply-Add (for FFTs)

Used in combination with SV FFT REMAP to perform a full Discrete Fourier
Transform of Polynomials over GF(2) in-place. Possible by having 3-in 2-out,
to avoid the need for a temp register. RS is written to as well as RT.

Note: Polynomials over GF(2) are a Ring rather than a Field, so, because the
definition of the Inverse Discrete Fourier Transform involves calculating a
multiplicative inverse, which may not exist in every Ring, therefore the
Inverse Discrete Fourier Transform may not exist. (AFAICT the number of inputs
to the IDFT must be odd for the IDFT to be defined for Polynomials over GF(2).
TODO: check with someone who knows for sure if that's correct.)

```
cltmadd RT, RA, RB, RC
```

TODO: add link to explanation for where `RS` comes from.

```
a = (RA)
c = (RC)
# read all inputs before writing to any outputs in case
# an input overlaps with an output register.
(RT) = clmul(a, (RB)) ^ c
(RS) = a ^ c
```

## `cldivrem` Carry-less Division and Remainder

`cldivrem` isn't an actual instruction, but is just used in the pseudo-code
for other instructions.

```python
[[!inline pagenames="gf_reference/cldivrem.py" raw="yes"]]
```

## `cldiv` Carry-less Division

```
cldiv RT, RA, RB
```

```
n = (RA)
d = (RB)
q, r = cldivrem(n, d, width=XLEN)
(RT) = q
```

## `clrem` Carry-less Remainder

```
clrem RT, RA, RB
```

```
n = (RA)
d = (RB)
q, r = cldivrem(n, d, width=XLEN)
(RT) = r
```

# Instructions for Binary Galois Fields `GF(2^m)`

see:

* <https://courses.csail.mit.edu/6.857/2016/files/ffield.py>
* <https://engineering.purdue.edu/kak/compsec/NewLectures/Lecture7.pdf>
* <https://foss.heptapod.net/math/libgf2/-/blob/branch/default/src/libgf2/gf2.py>

Binary Galois Field addition/subtraction is simply XOR, so a `gfbadd`
instruction is not provided since the `xor[i]` instruction can be used instead.

## `GFBREDPOLY` SPR -- Reducing Polynomial

In order to save registers and to make operations orthogonal with standard
arithmetic, the reducing polynomial is stored in a dedicated SPR `GFBREDPOLY`.
This also allows hardware to pre-compute useful parameters (such as the
degree, or look-up tables) based on the reducing polynomial, and store them
alongside the SPR in hidden registers, only recomputing them whenever the SPR
is written to, rather than having to recompute those values for every
instruction.

Because Galois Fields require the reducing polynomial to be an irreducible
polynomial, that guarantees that any polynomial of `degree > 1` must have
the LSB set, since otherwise it would be divisible by the polynomial `x`,
making it reducible, making whatever we're working on no longer a Field.
Therefore, we can reuse the LSB to indicate `degree == XLEN`.

```python
[[!inline pagenames="gf_reference/decode_reducing_polynomial.py" raw="yes"]]
```

## `gfbredpoly` -- Set the Reducing Polynomial SPR `GFBREDPOLY`

unless this is an immediate op, `mtspr` is completely sufficient.

```python
[[!inline pagenames="gf_reference/gfbredpoly.py" raw="yes"]]
```

## `gfbmul` -- Binary Galois Field `GF(2^m)` Multiplication

```
gfbmul RT, RA, RB
```

```python
[[!inline pagenames="gf_reference/gfbmul.py" raw="yes"]]
```

## `gfbmadd` -- Binary Galois Field `GF(2^m)` Multiply-Add

```
gfbmadd RT, RA, RB, RC
```

```python
[[!inline pagenames="gf_reference/gfbmadd.py" raw="yes"]]
```

## `gfbtmadd` -- Binary Galois Field `GF(2^m)` Twin Multiply-Add (for FFT)

Used in combination with SV FFT REMAP to perform a full `GF(2^m)` Discrete
Fourier Transform in-place. Possible by having 3-in 2-out, to avoid the need
for a temp register. RS is written to as well as RT.

```
gfbtmadd RT, RA, RB, RC
```

TODO: add link to explanation for where `RS` comes from.

```
a = (RA)
c = (RC)
# read all inputs before writing to any outputs in case
# an input overlaps with an output register.
(RT) = gfbmadd(a, (RB), c)
# use gfbmadd again since it reduces the result
(RS) = gfbmadd(a, 1, c) # "a * 1 + c"
```

## `gfbinv` -- Binary Galois Field `GF(2^m)` Inverse

```
gfbinv RT, RA
```

```python
[[!inline pagenames="gf_reference/gfbinv.py" raw="yes"]]
```

# Instructions for Prime Galois Fields `GF(p)`

## `GFPRIME` SPR -- Prime Modulus For `gfp*` Instructions

## `gfpadd` Prime Galois Field `GF(p)` Addition

```
gfpadd RT, RA, RB
```

```python
[[!inline pagenames="gf_reference/gfpadd.py" raw="yes"]]
```

the addition happens on infinite-precision integers

## `gfpsub` Prime Galois Field `GF(p)` Subtraction

```
gfpsub RT, RA, RB
```

```python
[[!inline pagenames="gf_reference/gfpsub.py" raw="yes"]]
```

the subtraction happens on infinite-precision integers

## `gfpmul` Prime Galois Field `GF(p)` Multiplication

```
gfpmul RT, RA, RB
```

```python
[[!inline pagenames="gf_reference/gfpmul.py" raw="yes"]]
```

the multiplication happens on infinite-precision integers

## `gfpinv` Prime Galois Field `GF(p)` Invert

```
gfpinv RT, RA
```

Some potential hardware implementations are found in:
<https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.5233&rep=rep1&type=pdf>

```python
[[!inline pagenames="gf_reference/gfpinv.py" raw="yes"]]
```

## `gfpmadd` Prime Galois Field `GF(p)` Multiply-Add

```
gfpmadd RT, RA, RB, RC
```

```python
[[!inline pagenames="gf_reference/gfpmadd.py" raw="yes"]]
```

the multiplication and addition happens on infinite-precision integers

## `gfpmsub` Prime Galois Field `GF(p)` Multiply-Subtract

```
gfpmsub RT, RA, RB, RC
```

```python
[[!inline pagenames="gf_reference/gfpmsub.py" raw="yes"]]
```

the multiplication and subtraction happens on infinite-precision integers

## `gfpmsubr` Prime Galois Field `GF(p)` Multiply-Subtract-Reversed

```
gfpmsubr RT, RA, RB, RC
```

```python
[[!inline pagenames="gf_reference/gfpmsubr.py" raw="yes"]]
```

the multiplication and subtraction happens on infinite-precision integers

## `gfpmaddsubr` Prime Galois Field `GF(p)` Multiply-Add and Multiply-Sub-Reversed (for FFT)

Used in combination with SV FFT REMAP to perform
a full Number-Theoretic-Transform in-place. Possible by having 3-in 2-out,
to avoid the need for a temp register. RS is written
to as well as RT.

```
gfpmaddsubr RT, RA, RB, RC
```

TODO: add link to explanation for where `RS` comes from.

```
factor1 = (RA)
factor2 = (RB)
term = (RC)
# read all inputs before writing to any outputs in case
# an input overlaps with an output register.
(RT) = gfpmadd(factor1, factor2, term)
(RS) = gfpmsubr(factor1, factor2, term)
```

# Already in POWER ISA or subsumed

Lists operations either included as part of
other bitmanip operations, or are already in
Power ISA.

## cmix

based on RV bitmanip, covered by ternlog bitops

```
uint_xlen_t cmix(uint_xlen_t RA, uint_xlen_t RB, uint_xlen_t RC) {
    return (RA & RB) | (RC & ~RB);
}
```

## count leading/trailing zeros with mask

in v3.1 p105

```
count = 0
do i = 0 to 63 if((RB)i=1) then do
if((RS)i=1) then break end end count ← count + 1
RA ← EXTZ64(count)
```

## bit deposit

pdepd VRT,VRA,VRB, identical to RV bitmamip bdep, found already in v3.1 p106

    do while(m < 64)
       if VSR[VRB+32].dword[i].bit[63-m]=1 then do
          result = VSR[VRA+32].dword[i].bit[63-k]
          VSR[VRT+32].dword[i].bit[63-m] = result
          k = k + 1
       m = m + 1

```

uint_xlen_t bdep(uint_xlen_t RA, uint_xlen_t RB)
{
    uint_xlen_t r = 0;
    for (int i = 0, j = 0; i < XLEN; i++)
        if ((RB >> i) & 1) {
            if ((RA >> j) & 1)
                r |= uint_xlen_t(1) << i;
            j++;
        }
    return r;
}

```

## bit extract

other way round: identical to RV bext: pextd, found in v3.1 p196

```
uint_xlen_t bext(uint_xlen_t RA, uint_xlen_t RB)
{
    uint_xlen_t r = 0;
    for (int i = 0, j = 0; i < XLEN; i++)
        if ((RB >> i) & 1) {
            if ((RA >> i) & 1)
                r |= uint_xlen_t(1) << j;
            j++;
        }
    return r;
}
```

## centrifuge

found in v3.1 p106 so not to be added here

```
ptr0 = 0
ptr1 = 0
do i = 0 to 63
    if((RB)i=0) then do
       resultptr0 = (RS)i
    end 
    ptr0 = ptr0 + 1
    if((RB)63-i==1) then do
        result63-ptr1 = (RS)63-i
    end
    ptr1 = ptr1 + 1
RA = result
```

## bit to byte permute

similar to matrix permute in RV bitmanip, which has XOR and OR variants,
these perform a transpose (bmatflip).
TODO this looks VSX is there a scalar variant
in v3.0/1 already

    do j = 0 to 7
      do k = 0 to 7
         b = VSR[VRB+32].dword[i].byte[k].bit[j]
         VSR[VRT+32].dword[i].byte[j].bit[k] = b

## grev

superceded by grevlut

based on RV bitmanip, this is also known as a butterfly network. however
where a butterfly network allows setting of every crossbar setting in
every row and every column, generalised-reverse (grev) only allows
a per-row decision: every entry in the same row must either switch or
not-switch.

<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Butterfly_Network.jpg/474px-Butterfly_Network.jpg" />

```
uint64_t grev64(uint64_t RA, uint64_t RB)
{
    uint64_t x = RA;
    int shamt = RB & 63;
    if (shamt & 1) x = ((x &  0x5555555555555555LL) <<  1) |
                        ((x & 0xAAAAAAAAAAAAAAAALL) >>  1);
    if (shamt & 2) x = ((x &  0x3333333333333333LL) <<  2) |
                        ((x & 0xCCCCCCCCCCCCCCCCLL) >>  2);
    if (shamt & 4) x = ((x &  0x0F0F0F0F0F0F0F0FLL) <<  4) |
                        ((x & 0xF0F0F0F0F0F0F0F0LL) >>  4);
    if (shamt & 8) x = ((x &  0x00FF00FF00FF00FFLL) <<  8) |
                        ((x & 0xFF00FF00FF00FF00LL) >>  8);
    if (shamt & 16) x = ((x & 0x0000FFFF0000FFFFLL) << 16) |
                        ((x & 0xFFFF0000FFFF0000LL) >> 16);
    if (shamt & 32) x = ((x & 0x00000000FFFFFFFFLL) << 32) |
                        ((x & 0xFFFFFFFF00000000LL) >> 32);
    return x;
}

```

## gorc

based on RV bitmanip, gorc is superceded by grevlut

```
uint32_t gorc32(uint32_t RA, uint32_t RB)
{
    uint32_t x = RA;
    int shamt = RB & 31;
    if (shamt & 1) x |= ((x & 0x55555555) << 1)   |  ((x &  0xAAAAAAAA) >> 1);
    if (shamt & 2) x |= ((x & 0x33333333) << 2)   |  ((x &  0xCCCCCCCC) >> 2);
    if (shamt & 4) x |= ((x & 0x0F0F0F0F) << 4)   |  ((x &  0xF0F0F0F0) >> 4);
    if (shamt & 8) x |= ((x & 0x00FF00FF) << 8)   |  ((x &  0xFF00FF00) >> 8);
    if (shamt & 16) x |= ((x & 0x0000FFFF) << 16) |  ((x &  0xFFFF0000) >> 16);
    return x;
}
uint64_t gorc64(uint64_t RA, uint64_t RB)
{
    uint64_t x = RA;
    int shamt = RB & 63;
    if (shamt & 1) x |= ((x & 0x5555555555555555LL)   <<   1) |
                         ((x & 0xAAAAAAAAAAAAAAAALL)  >>  1);
    if (shamt & 2) x |= ((x & 0x3333333333333333LL)   <<   2) |
                         ((x & 0xCCCCCCCCCCCCCCCCLL)  >>  2);
    if (shamt & 4) x |= ((x & 0x0F0F0F0F0F0F0F0FLL)   <<   4) |
                         ((x & 0xF0F0F0F0F0F0F0F0LL)  >>  4);
    if (shamt & 8) x |= ((x & 0x00FF00FF00FF00FFLL)   <<   8) |
                         ((x & 0xFF00FF00FF00FF00LL)  >>  8);
    if (shamt & 16) x |= ((x & 0x0000FFFF0000FFFFLL)  << 16) |
                         ((x & 0xFFFF0000FFFF0000LL)  >> 16);
    if (shamt & 32) x |= ((x & 0x00000000FFFFFFFFLL)  << 32) |
                         ((x & 0xFFFFFFFF00000000LL)  >> 32);
    return x;
}

```


# Appendix

see [[bitmanip/appendix]]