[[!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>
-* remove Rc=1 from ternlog due to conflicts in encoding as well
- as saving space <https://bugs.libre-soc.org/show_bug.cgi?id=753#c5>
* GF2^M <https://bugs.libre-soc.org/show_bug.cgi?id=782>
+
# bitmanipulation
**DRAFT STATUS**
-pseudocode: <https://libre-soc.org/openpower/isa/bitmanip/>
+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]].
| RT | RA | RB | shuf | shuffle |
| RT | RA | RB | unshuf| shuffle |
| RT | RA | RB | width | xperm |
-| RT | RA | RB | type | minmax |
+| RT | RA | RB | type | av minmax |
| RT | RA | RB | | av abs avgadd |
| RT | RA | RB | type | vmask ops |
| RT | RA | RB | | |
| 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 | | | | | 00 011 | | rsvd |
+| NN | | | | | 01 011 | | rsvd |
+| NN | | | | | 10 011 | | rsvd |
+| NN | | | | | 11 011 |Rc| setvl |
| 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)
+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 | nn00 110 |Rc| bmopsi |
-| NN | RT | RA | RB | | | 1100 110 |Rc| srsvd |
+| 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| |
| 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 | 0 | | 0101 110 |Rc| rsvd |
| NN | RT | RA | RB | 1 | itype | 0101 110 |Rc| xperm |
-| NN | RA | RB | RC | 0 | itype | 0101 110 |Rc| minmax |
-| NN | RA | RB | RC | 1 | 00 | 0101 110 |Rc| av avgadd |
-| NN | RA | RB | RC | 1 | 01 | 0101 110 |Rc| av abs |
-| NN | RA | RB | | 1 | 10 | 0101 110 |Rc| rsvd |
-| NN | RA | RB | | 1 | 11 | 0101 110 |Rc| rsvd |
+| NN | RA | RB | RC | 0 | itype | 1001 110 |Rc| av minmax |
+| NN | RA | RB | RC | 1 | 00 | 1001 110 |Rc| av abss |
+| NN | RA | RB | RC | 1 | 01 | 1001 110 |Rc| av absu|
+| NN | RA | RB | | 1 | 10 | 1001 110 |Rc| av avgadd |
+| NN | RA | RB | | 1 | 11 | 1001 110 |Rc| rsvd |
+| NN | RA | RB | | | | 1101 110 |Rc| rsvd |
| NN | RA | RB | RC | 0 | 00 | 0010 110 |Rc| gorc |
| NN | RA | RB | sh | SH | 00 | 1010 110 |Rc| gorci |
| NN | RA | RB | RC | 0 | 00 | 0110 110 |Rc| gorcw |
| NN | RA | RB | RC | | 10 | --10 110 |Rc| rsvd |
| NN | RA | RB | RC | 0 | 11 | 1110 110 |Rc| clmulr |
| NN | RA | RB | RC | 1 | 11 | 1110 110 |Rc| clmulh |
-| NN | | | | | | --11 110 |Rc| setvl |
+| NN | | | | | | --11 110 |Rc| rsvd |
# ternlog bitops
# 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 bmset(RS, RB, sh)
{
int shamt = RB & (XLEN - 1);
- return RS | MASK(shamt, sh)
+ mask = (2<<sh)-1;
+ return RS | (mask << shamt);
}
uint_xlen_t bmclr(RS, RB, sh)
{
int shamt = RB & (XLEN - 1);
- return RS & ~MASK(shamt, sh)
+ mask = (2<<sh)-1;
+ return RS & ~(mask << shamt);
}
uint_xlen_t bminv(RS, RB, sh)
{
int shamt = RB & (XLEN - 1);
- return RS ^ MASK(shamt, sh)
+ mask = (2<<sh)-1;
+ return RS ^ (mask << shamt);
}
uint_xlen_t bmext(RS, RB, sh)
}
```
-
-# Instructions for Carry-less Operations aka. Polynomials with coefficients in `GF(2)`
+# 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:
-[[!inline pagenames="openpower/sv/bitmanip/pack_poly.py" raw="true" feeds="no" actions="yes"]]
+```python
+[[!inline pagenames="gf_reference/pack_poly.py" raw="yes"]]
+```
## Carry-less Multiply Instructions
### `clmul` Carry-less Multiply
-[[!inline pagenames="openpower/sv/bitmanip/clmul.py" raw="true" feeds="no" actions="yes"]]
+```python
+[[!inline pagenames="gf_reference/clmul.py" raw="yes"]]
+```
### `clmulh` Carry-less Multiply High
-[[!inline pagenames="openpower/sv/bitmanip/clmulh.py" raw="true" feeds="no" actions="yes"]]
+```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.
-[[!inline pagenames="openpower/sv/bitmanip/clmulr.py" raw="true" feeds="no" actions="yes"]]
+```python
+[[!inline pagenames="gf_reference/clmulr.py" raw="yes"]]
+```
## `clmadd` Carry-less Multiply-Add
## `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.
-```
-(RT) = RC ^ clmul((RA), (RB))
-(RS) = RA ^ RC
+```
+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.
-[[!inline pagenames="openpower/sv/bitmanip/cldivrem.py" raw="true" feeds="no" actions="yes"]]
+```python
+[[!inline pagenames="gf_reference/cldivrem.py" raw="yes"]]
+```
## `cldiv` Carry-less Division
making it reducible, making whatever we're working on no longer a Field.
Therefore, we can reuse the LSB to indicate `degree == XLEN`.
-[[!inline pagenames="openpower/sv/bitmanip/decode_reducing_polynomial.py" raw="true" feeds="no" actions="yes"]]
+```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.
-[[!inline pagenames="openpower/sv/bitmanip/gfbredpoly.py" raw="true" feeds="no" actions="yes"]]
+```python
+[[!inline pagenames="gf_reference/gfbredpoly.py" raw="yes"]]
+```
## `gfbmul` -- Binary Galois Field `GF(2^m)` Multiplication
gfbmul RT, RA, RB
```
-[[!inline pagenames="openpower/sv/bitmanip/gfbmul.py" raw="true" feeds="no" actions="yes"]]
+```python
+[[!inline pagenames="gf_reference/gfbmul.py" raw="yes"]]
+```
## `gfbmadd` -- Binary Galois Field `GF(2^m)` Multiply-Add
gfbmadd RT, RA, RB, RC
```
-[[!inline pagenames="openpower/sv/bitmanip/gfbmadd.py" raw="true" feeds="no" actions="yes"]]
+```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.
-```
-(RT) = gfbmadd((RA), (RB), (RC))
-(RS) = RA ^ RC
+```
+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
```
-[[!inline pagenames="openpower/sv/bitmanip/gfbinv.py" raw="true" feeds="no" actions="yes"]]
-
-# Instructions for Prime Galois Fields `GF(p)`
-
-## Helper algorithms
-
```python
-def int_to_gfp(int_value, prime):
- return int_value % prime # follows Python remainder semantics
+[[!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
```
-```
-(RT) = int_to_gfp((RA) + (RB), GFPRIME)
+```python
+[[!inline pagenames="gf_reference/gfpadd.py" raw="yes"]]
```
the addition happens on infinite-precision integers
gfpsub RT, RA, RB
```
-```
-(RT) = int_to_gfp((RA) - (RB), GFPRIME)
+```python
+[[!inline pagenames="gf_reference/gfpsub.py" raw="yes"]]
```
the subtraction happens on infinite-precision integers
gfpmul RT, RA, RB
```
-```
-(RT) = int_to_gfp((RA) * (RB), GFPRIME)
+```python
+[[!inline pagenames="gf_reference/gfpmul.py" raw="yes"]]
```
the multiplication happens on infinite-precision integers
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"]]
```
-(RT) = gfpinv((RA), GFPRIME)
-```
-
-the multiplication happens on infinite-precision integers
## `gfpmadd` Prime Galois Field `GF(p)` Multiply-Add
gfpmadd RT, RA, RB, RC
```
-```
-(RT) = int_to_gfp((RA) * (RB) + (RC), GFPRIME)
+```python
+[[!inline pagenames="gf_reference/gfpmadd.py" raw="yes"]]
```
the multiplication and addition happens on infinite-precision integers
gfpmsub RT, RA, RB, RC
```
-```
-(RT) = int_to_gfp((RA) * (RB) - (RC), GFPRIME)
+```python
+[[!inline pagenames="gf_reference/gfpmsub.py" raw="yes"]]
```
the multiplication and subtraction happens on infinite-precision integers
gfpmsubr RT, RA, RB, RC
```
-```
-(RT) = int_to_gfp((RC) - (RA) * (RB), GFPRIME)
+```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.
```
-product = (RA) * (RB)
+factor1 = (RA)
+factor2 = (RB)
term = (RC)
-(RT) = int_to_gfp(product + term, GFPRIME)
-(RS) = int_to_gfp(term - product, GFPRIME)
+# 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)
```
-the multiplication, addition, and subtraction happens on infinite-precision integers
-
-## Twin Butterfly (Tukey-Cooley) Mul-add-sub
-
-used in combination with SV FFT REMAP to perform
-a full NTT 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.
-
- gffmadd RT,RA,RC,RB (Rc=0)
- gffmadd. RT,RA,RC,RB (Rc=1)
-
-Pseudo-code:
-
- RT <- GFADD(GFMUL(RA, RC), RB))
- RS <- GFADD(GFMUL(RA, RC), RB))
-
-
-## Multiply
-
-with the modulo and degree being in an SPR, multiply can be identical
-equivalent to standard integer add
-
- RS = GFMUL(RA, RB)
-
-| 0.5|6.10|11.15|16.20|21.25| 26..30 |31|
-| -- | -- | --- | --- | --- | ------ |--|
-| NN | RT | RA | RB |11000| 01110 |Rc|
-
-
-
-```
-from functools import reduce
-
-def gf_degree(a) :
- res = 0
- a >>= 1
- while (a != 0) :
- a >>= 1;
- res += 1;
- return res
-
-# constants used in the multGF2 function
-mask1 = mask2 = polyred = None
-
-def setGF2(irPoly):
- """Define parameters of binary finite field GF(2^m)/g(x)
- - irPoly: coefficients of irreducible polynomial g(x)
- """
- # degree: extension degree of binary field
- degree = gf_degree(irPoly)
-
- def i2P(sInt):
- """Convert an integer into a polynomial"""
- return [(sInt >> i) & 1
- for i in reversed(range(sInt.bit_length()))]
-
- global mask1, mask2, polyred
- mask1 = mask2 = 1 << degree
- mask2 -= 1
- polyred = reduce(lambda x, y: (x << 1) + y, i2P(irPoly)[1:])
-
-def multGF2(p1, p2):
- """Multiply two polynomials in GF(2^m)/g(x)"""
- p = 0
- while p2:
- # standard long-multiplication: check LSB and add
- if p2 & 1:
- p ^= p1
- p1 <<= 1
- # standard modulo: check MSB and add polynomial
- if p1 & mask1:
- p1 ^= polyred
- p2 >>= 1
- return p & mask2
-
-if __name__ == "__main__":
-
- # Define binary field GF(2^3)/x^3 + x + 1
- setGF2(0b1011) # degree 3
-
- # Evaluate the product (x^2 + x + 1)(x^2 + 1)
- print("{:02x}".format(multGF2(0b111, 0b101)))
-
- # Define binary field GF(2^8)/x^8 + x^4 + x^3 + x + 1
- # (used in the Advanced Encryption Standard-AES)
- setGF2(0b100011011) # degree 8
-
- # Evaluate the product (x^7)(x^7 + x + 1)
- print("{:02x}".format(multGF2(0b10000000, 0b10000011)))
-```
-
-## carryless Twin Butterfly (Tukey-Cooley) Mul-add-sub
-
-used in combination with SV FFT REMAP to perform
-a full NTT 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.
-
- clfmadd RT,RA,RC,RB (Rc=0)
- clfmadd. RT,RA,RC,RB (Rc=1)
-
-Pseudo-code:
-
- RT <- CLMUL(RA, RC) ^ RB
- RS <- CLMUL(RA, RC) ^ RB
-
-
# bitmatrix
```
## bit deposit
-vpdepd VRT,VRA,VRB, identical to RV bitmamip bdep, found already in v3.1 p106
+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
```
-# bit extract
+## bit extract
-other way round: identical to RV bext, found in v3.1 p196
+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)
}
```
-# centrifuge
+## centrifuge
found in v3.1 p106 so not to be added here
RA = result
```
-# bit to byte permute
+## bit to byte permute
similar to matrix permute in RV bitmanip, which has XOR and OR variants,
-these perform a transpose.
+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]]
+