[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>


# 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.  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 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/vector_ops]], and
the [[sv/av_opcodes]] as well as [[sv/setvl]]

Useful resource: 

* <https://en.wikiversity.org/wiki/Reed%E2%80%93Solomon_codes_for_coders>
* <https://maths-people.anu.edu.au/~brent/pd/rpb232tr.pdf>

# summary

two major opcodes are needed

ternlog has its own major opcode

|  29.30 |31| name      |
| ------ |--| --------- |
|   0  0   |Rc| ternlogi  |
|   0  1   |sz| ternlogv  |
|   1 iv   |  | grevlogi |

2nd major opcode for other bitmanip: minor opcode allocation

|  28.30 |31| name      |
| ------ |--| --------- |
|  -00   |0 | xpermi    |
|  -00   |1 | grevlog   |
|  -01   |  | crternlog  |
|  010   |Rc| bitmask   |
|  011   |  | gf/cl madd*  |
|  110   |Rc| 1/2-op    |
|  111   |  | bmrevi   |


1-op and variants

| dest | src1 | subop | op       |
| ---- | ---- | ----- | -------- |
| RT   | RA   | ..    | bmatflip | 

2-op and variants

| dest | src1 | src2 | subop | op       |
| ---- | ---- | ---- | ----- | -------- |
| RT   | RA   | RB   | or    | bmatflip | 
| RT   | RA   | RB   | xor   | bmatflip | 
| RT   | RA   | RB   |       | grev  |
| RT   | RA   | RB   |       | clmul*  |
| RT   | RA   | RB   |       | gorc |  
| RT   | RA   | RB   | shuf  | shuffle | 
| RT   | RA   | RB   | unshuf| shuffle | 
| RT   | RA   | RB   | width | xperm  | 
| RT   | RA   | RB   | type | av minmax | 
| RT   | RA   | RB   |      | av abs avgadd  | 
| RT   | RA   | RB   | type | vmask ops | 
| RT   | RA   | RB   |      |       | 

3 ops 

* grevlog
* GF mul-add
* bitmask-reverse

TODO: convert all instructions to use RT and not RS

| 0.5|6.8 | 9.11|12.14|15.17|18.20|21.28 | 29.30|31|name|
| -- | -- | --- | --- | --- |-----|----- | -----|--|----|
| NN | BT | BA  | BB  | BC  |m0-2 | imm  |  10  |m3|crternlog|

| 0.5|6.10|11.15|16.20 |21..25   | 26....30  |31| name |
| -- | -- | --- | ---  | -----   | --------  |--| ------ |
| NN | RT | RA  |itype/| im0-4   | im5-7  00 |0 | xpermi  |
| NN | RT | RA  | RB   | im0-4   | im5-7  00 |1 | grevlog |
| NN |    |     |      |         | .....  01 |0 | crternlog |
| NN | RT | RA  | RB   | RC      | mode  010 |Rc| bitmask* |
| NN | RS | RA  | RB   | RC      | 00    011 |0 | gfbmadd |
| NN | RS | RA  | RB   | RC      | 00    011 |1 | gfbmaddsub |
| NN | RS | RA  | RB   | RC      | 01    011 |0 | clmadd |
| NN | RS | RA  | RB   | RC      | 01    011 |1 | clmaddsub |
| NN | RS | RA  | RB   | RC      | 10    011 |0 | gfpmadd |
| NN | RS | RA  | RB   | RC      | 10    011 |1 | gfpmaddsub |
| NN | RS | RA  | RB   | RC      | 11    011 |  | rsvd |
| NN | RT | RA  | RB   | sh0-4   | sh5 1 111 |Rc| bmrevi |

ops (note that av avg and abs as well as vec scalar mask
are included here [[sv/vector_ops]], and
the [[sv/av_opcodes]])

TODO: convert from RA, RB, and RC to correct field names of RT, RA, and RB, and
double check that instructions didn't need 3 inputs.

| 0.5|6.10|11.15|16.20| 21 | 22.23 | 24....30 |31| name |
| -- | -- | --- | --- | -- | ----- | -------- |--| ---- |
| NN | RS | me  | sh  | SH | ME 0  | nn00 110 |Rc| bmopsi |
| NN | RS | RB  | sh  | SH | 0   1 | nn00 110 |Rc| bmopsi |
| NN | RT | RA  | RB  | 1  |  00   | 0001 110 |Rc| cldiv |
| NN | RT | RA  | RB  | 1  |  01   | 0001 110 |Rc| clmod |
| NN | RT | RA  | RB  | 1  |  10   | 0001 110 |Rc| clmul |
| NN | RT | RB  | RB  | 1  |  11   | 0001 110 |Rc| clinv |
| NN | RA | RB  | RC  | 0  |   00  | 0001 110 |Rc| vec sbfm |
| NN | RA | RB  | RC  | 0  |   01  | 0001 110 |Rc| vec sofm |
| NN | RA | RB  | RC  | 0  |   10  | 0001 110 |Rc| vec sifm |
| NN | RA | RB  | RC  | 0  |   11  | 0001 110 |Rc| vec cprop |
| NN | RT | RA  | RB  | 1  | itype | 0101 110 |Rc| xperm |
| NN | RA | RB  | RC  | 0  | itype | 0101 110 |Rc| av minmax |
| NN | RA | RB  | RC  | 1  |   00  | 0101 110 |Rc| av abss |
| NN | RA | RB  | RC  | 1  |   01  | 0101 110 |Rc| av absu|
| NN | RA | RB  |     | 1  |   10  | 0101 110 |Rc| av avgadd |
| NN | RA | RB  | RC  | 1  |   11  | 0101 110 |Rc| grevw |
| NN | RA | RB  |     |    |       | 1001 110 |Rc| rsvd |
| NN | RA | RB  | RC  | 0  | 00    | 1101 110 |Rc| bmator  |
| NN | RA | RB  | RC  | 1  | 00    | 1101 110 |Rc| bmatxor   |
| NN | RA | RB  | sh  | 0  | 01    | 1101 110 |Rc| grevwi |
| NN | RA | RB  | RC  | 1  | 01    | 1101 110 |Rc| grev |
| NN | RA | RB  | sh  | SH | 10    | 1101 110 |Rc| grevi |
| NN | RA | RB  | RC  | 0  | 11    | 1101 110 |Rc| clmulr  |
| NN | RA | RB  | RC  | 1  | 11    | 1101 110 |Rc| clmulh  |
| NN | RA | RB  | RC  |    |       | --10 110 |Rc| rsvd  |
| NN |    |     |     |    |       | --11 110 |Rc| setvl  |

# ternlog bitops

Similar to FPGA LUTs: for every bit perform a lookup into a table using an 8bit immediate, or in another register.

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]) 

## ternlogv

also, another possible variant involving swizzle-like selection
and masking, this only requires 3 64 bit registers (RA, RS, RB) and
only 16 LUT3s.

Note however that unless XLEN matches sz, this instruction
is a Read-Modify-Write: RS must be read as a second operand
and all unmodified bits preserved.  SVP64 may provide limited
alternative destination for RS from RS-as-source, but again
all unmodified bits must still be copied.

| 0.5|6.10|11.15|16.20|21.28 | 29.30 |31|
| -- | -- | --- | --- | ---- | ----- |--|
| NN | RS | RA  | RB  |idx0-3|  01   |sz|

    SZ = (1+sz) * 8 # 8 or 16
    raoff = MIN(XLEN, idx0 * SZ)
    rboff = MIN(XLEN, idx1 * SZ)
    rcoff = MIN(XLEN, idx2 * SZ)
    rsoff = MIN(XLEN, idx3 * SZ)
    imm = RB[0:8]
    for i in range(MIN(XLEN, SZ)):
        ra = RA[raoff:+i]
        rb = RA[rboff+i]
        rc = RA[rcoff+i]
        res = lut3(imm, ra, rb, rc)
        RS[rsoff+i] = res

## ternlogcr

another mode selection would be CRs not Ints. 

| 0.5|6.8 | 9.11|12.14|15.17|18.20|21.28 | 29.30|31|
| -- | -- | --- | --- | --- |-----|----- | -----|--|
| NN | BT | BA  | BB  | BC  |m0-2 | imm  |  10  |m3|

    mask = m0-3,m4
    for i in range(4):
        if not mask[i] continue
        crregs[BT][i] = lut3(imm,
                             crregs[BA][i],
                             crregs[BB][i],
                             crregs[BC][i])


# int min/max

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.

```
uint_xlen_t min(uint_xlen_t rs1, uint_xlen_t rs2)
{ return (int_xlen_t)rs1 < (int_xlen_t)rs2 ? rs1 : rs2;
}
uint_xlen_t max(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;
}
```


## 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);
}
```


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

| 0.5|6.10|11.15|16.20|21.26| 27..30  |31| name   |
| -- | -- | --- | --- | --- | ------- |--| ------ |
| NN | RT | RA  | RB  | sh  | 1   011 |Rc| bmrevi |


# grevlut

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 regular 64 bit patterns in one
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.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       | 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):
    for j in 0 to 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 grevlut64(uint64_t RA, uint64_t RB, uint8 imm, bool iv)
{
    uint64_t x = 0x5555_5555_5555_5555;
    if (RA != 0) x = GPR(RA);
    if (iv) x = ~x;
    int shamt = RB & 63;
    for i in 0 to 6
        step = 1<<i
        if (shamt & step) x = dorow(imm, x, step)
    return x;
}

```

| 0.5|6.10|11.15|16.20 |21..25   | 26....30    |31| name |
| -- | -- | --- | ---  | -----   | --------    |--| ------ |
| NN | RT | RA  | s0-4 | im0-4   | im5-7  1 iv |s5| grevlogi |
| NN | RT | RA  | RB   | im0-4   | im5-7  00   |1 | grevlog |


# grev

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;
}

```

# 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

```
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); }
```

# gorc

based on RV bitmanip

```
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;
}

```
# 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)
```

# bitmatrix

```
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 bmatxor(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 (pcnt(u[i / 8] & v[i % 8]) & 1)
            x |= 1LL << i;
    }
    return x;
}
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;
}

```

# Already in POWER ISA

## 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. 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

# Appendix

see [[bitmanip/appendix]]