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: 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. Vectorization and SIMD are removed: these are straight scalar (element) operations making them suitable for embedded applications. Vectorization Context is provided by 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 mv.swizzle, vector ops, int fp mv and the av opcodes as well as setvl, svstep, 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

Draft Opcode tables

draft opcode tables

two major opcodes are needed

PO5 (temporary)

madd/sub:

PO22 - 2nd major opcode for other bitmanip: minor opcode allocation

minmax is allocated to PO 19 XO 000011

1-op and variants

2-op and variants

3 ops

• grevlog[w]
• GF mul-add
• bitmask-reverse

TODO: convert all instructions to use RT and not RS

[1] except bmrev

ops (note that av avg and abs as well as vec scalar mask are included here vector ops, and the av opcodes)

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 instructions.

ternlogi

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.

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 Vectorized 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:

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:

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 Vectorized 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 av opcodes

signed and unsigned min/max for integer.

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

• PO=19, XO=----000011 minmax RT, RA, RB, MMM
• PO=19, XO=----000011 minmax. RT, RA, RB, MMM

see ls013 for MMM definition and pseudo-code.

implements all of (and more):

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 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 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 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

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 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.

Immediate-variant is an overwrite form:

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);
}

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 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.

• 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:

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.

demo code grevlut.py

def lut2(imm, a, b):
    idx = b << 1 | a
    return (imm>>idx) & 1

def dorow(imm8, step_i, chunk_size):
    step_o = 0
    for j in range(64):
        if (j&chunk_size) == 0:
           imm = (imm8 & 0b1111)
        else:
           imm = (imm8>>4)
        a = (step_i>>j)&1
        b = (step_i>>(j ^ chunk_size))&1
        res = lut2(imm, a, b)
        #print(j, bin(imm), a, b, res)
        step_o |= (res<<j)
    #print ("  ", chunk_size, bin(step_o))
    return step_o

def grevlut64(RA, RB, imm, iv):
    x = 0
    if RA is None: # RA=0
        x = 0x5555555555555555
    else:
        x = RA
    if (iv): x = ~x;
    shamt = RB & 63;
    for i in range(6):
        step = 1<<i
        if (shamt & step):
            x = dorow(imm, x, step)
    return x & ((1<<64)-1)

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;
}

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

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:

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

clmulh Carry-less Multiply High

clmulr Carry-less Multiply (Reversed)

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

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.

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.

gfbredpoly -- Set the Reducing Polynomial SPR GFBREDPOLY

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

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

gfbmul RT, RA, RB

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

gfbmadd RT, RA, RB, RC

nmigen-gf.git/src/nmigen_gf/hdl/gfbmadd.py:419

tests and formal proofs

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

The main entry point in our hardware implementation can be found at:

nmigen-gf.git/src/nmigen_gf/hdl/gfbinv.py:386

tests and formal proofs

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

the addition happens on infinite-precision integers

gfpsub Prime Galois Field GF(p) Subtraction

gfpsub RT, RA, RB

the subtraction happens on infinite-precision integers

gfpmul Prime Galois Field GF(p) Multiplication

gfpmul RT, RA, RB

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

gfpmadd Prime Galois Field GF(p) Multiply-Add

gfpmadd RT, RA, RB, RC

the multiplication and addition happens on infinite-precision integers

gfpmsub Prime Galois Field GF(p) Multiply-Subtract

gfpmsub RT, RA, RB, RC

the multiplication and subtraction happens on infinite-precision integers

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

gfpmsubr RT, RA, RB, RC

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.

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 appendix