5 * ternlogi <https://bugs.libre-soc.org/show_bug.cgi?id=745>
6 * grev <https://bugs.libre-soc.org/show_bug.cgi?id=755>
7 * remove Rc=1 from ternlog due to conflicts in encoding as well
8 as saving space <https://bugs.libre-soc.org/show_bug.cgi?id=753#c5>
9 * GF2^M <https://bugs.libre-soc.org/show_bug.cgi?id=782>
15 pseudocode: <https://libre-soc.org/openpower/isa/bitmanip/>
17 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.
18 Vectorisation Context is provided by [[openpower/sv]].
20 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.
22 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.
24 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.
26 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
27 the [[sv/av_opcodes]] as well as [[sv/setvl]]
31 * <https://en.wikiversity.org/wiki/Reed%E2%80%93Solomon_codes_for_coders>
32 * <https://maths-people.anu.edu.au/~brent/pd/rpb232tr.pdf>
36 two major opcodes are needed
38 ternlog has its own major opcode
41 | ------ |--| --------- |
46 2nd major opcode for other bitmanip: minor opcode allocation
49 | ------ |--| --------- |
54 | 011 | | gf/cl madd* |
61 | dest | src1 | subop | op |
62 | ---- | ---- | ----- | -------- |
63 | RT | RA | .. | bmatflip |
67 | dest | src1 | src2 | subop | op |
68 | ---- | ---- | ---- | ----- | -------- |
69 | RT | RA | RB | or | bmatflip |
70 | RT | RA | RB | xor | bmatflip |
71 | RT | RA | RB | | grev |
72 | RT | RA | RB | | clmul* |
73 | RT | RA | RB | | gorc |
74 | RT | RA | RB | shuf | shuffle |
75 | RT | RA | RB | unshuf| shuffle |
76 | RT | RA | RB | width | xperm |
77 | RT | RA | RB | type | minmax |
78 | RT | RA | RB | | av abs avgadd |
79 | RT | RA | RB | type | vmask ops |
88 TODO: convert all instructions to use RT and not RS
90 | 0.5|6.8 | 9.11|12.14|15.17|18.20|21.28 | 29.30|31|name|
91 | -- | -- | --- | --- | --- |-----|----- | -----|--|----|
92 | NN | BT | BA | BB | BC |m0-2 | imm | 10 |m3|crternlog|
94 | 0.5|6.10|11.15|16.20 |21..25 | 26....30 |31| name |
95 | -- | -- | --- | --- | ----- | -------- |--| ------ |
96 | NN | RT | RA |itype/| im0-4 | im5-7 00 |0 | xpermi |
97 | NN | RT | RA | RB | im0-4 | im5-7 00 |1 | grevlog |
98 | NN | | | | | ..... 01 |0 | crternlog |
99 | NN | RT | RA | RB | RC | mode 010 |Rc| bitmask* |
100 | NN | RS | RA | RB | RC | 00 011 |0 | gfbmadd |
101 | NN | RS | RA | RB | RC | 00 011 |1 | gfbmaddsub |
102 | NN | RS | RA | RB | RC | 01 011 |0 | clmadd |
103 | NN | RS | RA | RB | RC | 01 011 |1 | clmaddsub |
104 | NN | RS | RA | RB | RC | 10 011 |0 | gfpmadd |
105 | NN | RS | RA | RB | RC | 10 011 |1 | gfpmaddsub |
106 | NN | RS | RA | RB | RC | 11 011 | | rsvd |
107 | NN | RT | RA | RB | sh0-4 | sh5 1 111 |Rc| bmrevi |
109 ops (note that av avg and abs as well as vec scalar mask
112 TODO: convert from RA, RB, and RC to correct field names of RT, RA, and RB, and
113 double check that instructions didn't need 3 inputs.
115 | 0.5|6.10|11.15|16.20| 21 | 22.23 | 24....30 |31| name |
116 | -- | -- | --- | --- | -- | ----- | -------- |--| ---- |
117 | NN | RT | RA | RB | 0 | | 0000 110 |Rc| rsvd |
118 | NN | RT | RA | RB | 1 | itype | 0000 110 |Rc| xperm |
119 | NN | RA | RB | RC | 0 | itype | 0100 110 |Rc| minmax |
120 | NN | RA | RB | RC | 1 | 00 | 0100 110 |Rc| av avgadd |
121 | NN | RA | RB | RC | 1 | 01 | 0100 110 |Rc| av abs |
122 | NN | RA | RB | | 1 | 10 | 0100 110 |Rc| rsvd |
123 | NN | RA | RB | | 1 | 11 | 0100 110 |Rc| rsvd |
124 | NN | RA | RB | sh | SH | itype | 1000 110 |Rc| bmopsi |
125 | NN | RT | RA | RB | | | 1100 110 |Rc| srsvd |
126 | NN | RT | RA | RB | 1 | 00 | 0001 110 |Rc| cldiv |
127 | NN | RT | RA | RB | 1 | 01 | 0001 110 |Rc| clmod |
128 | NN | RT | RA | RB | 1 | 10 | 0001 110 |Rc| |
129 | NN | RT | RB | RB | 1 | 11 | 0001 110 |Rc| clinv |
130 | NN | RA | RB | RC | 0 | 00 | 0001 110 |Rc| vec sbfm |
131 | NN | RA | RB | RC | 0 | 01 | 0001 110 |Rc| vec sofm |
132 | NN | RA | RB | RC | 0 | 10 | 0001 110 |Rc| vec sifm |
133 | NN | RA | RB | RC | 0 | 11 | 0001 110 |Rc| vec cprop |
134 | NN | RA | RB | | 0 | | 0101 110 |Rc| rsvd |
135 | NN | RA | RB | RC | 0 | 00 | 0010 110 |Rc| gorc |
136 | NN | RA | RB | sh | SH | 00 | 1010 110 |Rc| gorci |
137 | NN | RA | RB | RC | 0 | 00 | 0110 110 |Rc| gorcw |
138 | NN | RA | RB | sh | 0 | 00 | 1110 110 |Rc| gorcwi |
139 | NN | RA | RB | RC | 1 | 00 | 1110 110 |Rc| bmator |
140 | NN | RA | RB | RC | 0 | 01 | 0010 110 |Rc| grev |
141 | NN | RA | RB | RC | 1 | 01 | 0010 110 |Rc| clmul |
142 | NN | RA | RB | sh | SH | 01 | 1010 110 |Rc| grevi |
143 | NN | RA | RB | RC | 0 | 01 | 0110 110 |Rc| grevw |
144 | NN | RA | RB | sh | 0 | 01 | 1110 110 |Rc| grevwi |
145 | NN | RA | RB | RC | 1 | 01 | 1110 110 |Rc| bmatxor |
146 | NN | RA | RB | RC | | 10 | --10 110 |Rc| rsvd |
147 | NN | RA | RB | RC | 0 | 11 | 1110 110 |Rc| clmulr |
148 | NN | RA | RB | RC | 1 | 11 | 1110 110 |Rc| clmulh |
149 | NN | | | | | | --11 110 |Rc| setvl |
153 Similar to FPGA LUTs: for every bit perform a lookup into a table using an 8bit immediate, or in another register.
155 Like the x86 AVX512F [vpternlogd/vpternlogq](https://www.felixcloutier.com/x86/vpternlogd:vpternlogq) instructions.
159 | 0.5|6.10|11.15|16.20| 21..28|29.30|31|
160 | -- | -- | --- | --- | ----- | --- |--|
161 | NN | RT | RA | RB | im0-7 | 00 |Rc|
164 idx = c << 2 | b << 1 | a
165 return imm[idx] # idx by LSB0 order
168 RT[i] = lut3(imm, RB[i], RA[i], RT[i])
172 also, another possible variant involving swizzle-like selection
173 and masking, this only requires 3 64 bit registers (RA, RS, RB) and
176 Note however that unless XLEN matches sz, this instruction
177 is a Read-Modify-Write: RS must be read as a second operand
178 and all unmodified bits preserved. SVP64 may provide limited
179 alternative destination for RS from RS-as-source, but again
180 all unmodified bits must still be copied.
182 | 0.5|6.10|11.15|16.20|21.28 | 29.30 |31|
183 | -- | -- | --- | --- | ---- | ----- |--|
184 | NN | RS | RA | RB |idx0-3| 01 |sz|
186 SZ = (1+sz) * 8 # 8 or 16
187 raoff = MIN(XLEN, idx0 * SZ)
188 rboff = MIN(XLEN, idx1 * SZ)
189 rcoff = MIN(XLEN, idx2 * SZ)
190 rsoff = MIN(XLEN, idx3 * SZ)
192 for i in range(MIN(XLEN, SZ)):
196 res = lut3(imm, ra, rb, rc)
201 another mode selection would be CRs not Ints.
203 | 0.5|6.8 | 9.11|12.14|15.17|18.20|21.28 | 29.30|31|
204 | -- | -- | --- | --- | --- |-----|----- | -----|--|
205 | NN | BT | BA | BB | BC |m0-2 | imm | 10 |m3|
209 if not mask[i] continue
210 crregs[BT][i] = lut3(imm,
218 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).
220 signed/unsigned min/max gives more flexibility.
223 uint_xlen_t min(uint_xlen_t rs1, uint_xlen_t rs2)
224 { return (int_xlen_t)rs1 < (int_xlen_t)rs2 ? rs1 : rs2;
226 uint_xlen_t max(uint_xlen_t rs1, uint_xlen_t rs2)
227 { return (int_xlen_t)rs1 > (int_xlen_t)rs2 ? rs1 : rs2;
229 uint_xlen_t minu(uint_xlen_t rs1, uint_xlen_t rs2)
230 { return rs1 < rs2 ? rs1 : rs2;
232 uint_xlen_t maxu(uint_xlen_t rs1, uint_xlen_t rs2)
233 { return rs1 > rs2 ? rs1 : rs2;
240 based on RV bitmanip, covered by ternlog bitops
243 uint_xlen_t cmix(uint_xlen_t RA, uint_xlen_t RB, uint_xlen_t RC) {
244 return (RA & RB) | (RC & ~RB);
251 based on RV bitmanip singlebit set, instruction format similar to shift
252 [[isa/fixedshift]]. bmext is actually covered already (shift-with-mask rldicl but only immediate version).
253 however bitmask-invert is not, and set/clr are not covered, although they can use the same Shift ALU.
255 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.
256 bmrev however there is no direct equivalent and consequently a bmrevi is required.
258 bmset (register for mask amount) is particularly useful for creating
259 predicate masks where the length is a dynamic runtime quantity.
260 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.
262 | 0.5|6.10|11.15|16.20|21.25| 26..30 |31| name |
263 | -- | -- | --- | --- | --- | ------- |--| ----- |
264 | NN | RS | RA | RB | RC | mode 010 |Rc| bm* |
266 Immediate-variant is an overwrite form:
268 | 0.5|6.10|11.15|16.20| 21 | 22.23 | 24....30 |31| name |
269 | -- | -- | --- | --- | -- | ----- | -------- |--| ---- |
270 | NN | RS | RB | sh | SH | itype | 1000 110 |Rc| bm*i |
273 uint_xlen_t bmset(RS, RB, sh)
275 int shamt = RB & (XLEN - 1);
277 return RS | (mask << shamt);
280 uint_xlen_t bmclr(RS, RB, sh)
282 int shamt = RB & (XLEN - 1);
284 return RS & ~(mask << shamt);
287 uint_xlen_t bminv(RS, RB, sh)
289 int shamt = RB & (XLEN - 1);
291 return RS ^ (mask << shamt);
294 uint_xlen_t bmext(RS, RB, sh)
296 int shamt = RB & (XLEN - 1);
298 return mask & (RS >> shamt);
302 bitmask extract with reverse. can be done by bitinverting all of RB and getting bits of RB from the opposite end.
304 when RA is zero, no shift occurs. this makes bmextrev useful for
305 simply reversing all bits of a register.
309 rev[0:msb] = rb[msb:0];
312 uint_xlen_t bmextrev(RA, RB, sh)
315 if (RA != 0) (GPR(RA) & (XLEN - 1));
316 shamt = (XLEN-1)-shamt; # shift other end
317 bra = bitreverse(RB) # swap LSB-MSB
319 return mask & (bra >> shamt);
323 | 0.5|6.10|11.15|16.20|21.26| 27..30 |31| name |
324 | -- | -- | --- | --- | --- | ------- |--| ------ |
325 | NN | RT | RA | RB | sh | 1 011 |Rc| bmrevi |
330 generalised reverse combined with a pair of LUT2s and allowing
331 a constant `0b0101...0101` when RA=0, and an option to invert
332 (including when RA=0, giving a constant 0b1010...1010 as the
333 initial value) provides a wide range of instructions
334 and a means to set regular 64 bit patterns in one
337 the two LUT2s are applied left-half (when not swapping)
338 and right-half (when swapping) so as to allow a wider
341 <img src="/openpower/sv/grevlut2x2.jpg" width=700 />
343 * A value of `0b11001010` for the immediate provides
344 the functionality of a standard "grev".
345 * `0b11101110` provides gorc
347 grevlut should be arranged so as to produce the constants
348 needed to put into bext (bitextract) so as in turn to
349 be able to emulate x86 pmovmask instructions <https://www.felixcloutier.com/x86/pmovmskb>.
350 This only requires 2 instructions (grevlut, bext).
352 Note that if the mask is required to be placed
353 directly into CR Fields (for use as CR Predicate
354 masks rather than a integer mask) then sv.ori
355 may be used instead, bearing in mind that sv.ori
356 is a 64-bit instruction, and `VL` must have been
357 set to the required length:
359 sv.ori./elwid=8 r10.v, r10.v, 0
361 The following settings provide the required mask constants:
363 | RA | RB | imm | iv | result |
364 | ------- | ------- | ---------- | -- | ---------- |
365 | 0x555.. | 0b10 | 0b01101100 | 0 | 0x111111... |
366 | 0x555.. | 0b110 | 0b01101100 | 0 | 0x010101... |
367 | 0x555.. | 0b1110 | 0b01101100 | 0 | 0x00010001... |
368 | 0x555.. | 0b10 | 0b11000110 | 1 | 0x88888... |
369 | 0x555.. | 0b110 | 0b11000110 | 1 | 0x808080... |
370 | 0x555.. | 0b1110 | 0b11000110 | 1 | 0x80008000... |
372 Better diagram showing the correct ordering of shamt (RB). A LUT2
373 is applied to all locations marked in red using the first 4
374 bits of the immediate, and a separate LUT2 applied to all
375 locations in green using the upper 4 bits of the immediate.
377 <img src="/openpower/sv/grevlut.png" width=700 />
379 demo code [[openpower/sv/grevlut.py]]
384 return imm[idx] # idx by LSB0 order
386 dorow(imm8, step_i, chunksize):
388 if (j&chunk_size) == 0
392 step_o[j] = lut2(imm, step_i[j], step_i[j ^ chunk_size])
395 uint64_t grevlut64(uint64_t RA, uint64_t RB, uint8 imm, bool iv)
397 uint64_t x = 0x5555_5555_5555_5555;
398 if (RA != 0) x = GPR(RA);
403 if (shamt & step) x = dorow(imm, x, step)
409 | 0.5|6.10|11.15|16.20 |21..25 | 26....30 |31| name |
410 | -- | -- | --- | --- | ----- | -------- |--| ------ |
411 | NN | RT | RA | s0-4 | im0-4 | im5-7 1 iv |s5| grevlogi |
412 | NN | RT | RA | RB | im0-4 | im5-7 00 |1 | grevlog |
417 based on RV bitmanip, this is also known as a butterfly network. however
418 where a butterfly network allows setting of every crossbar setting in
419 every row and every column, generalised-reverse (grev) only allows
420 a per-row decision: every entry in the same row must either switch or
423 <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Butterfly_Network.jpg/474px-Butterfly_Network.jpg" />
426 uint64_t grev64(uint64_t RA, uint64_t RB)
430 if (shamt & 1) x = ((x & 0x5555555555555555LL) << 1) |
431 ((x & 0xAAAAAAAAAAAAAAAALL) >> 1);
432 if (shamt & 2) x = ((x & 0x3333333333333333LL) << 2) |
433 ((x & 0xCCCCCCCCCCCCCCCCLL) >> 2);
434 if (shamt & 4) x = ((x & 0x0F0F0F0F0F0F0F0FLL) << 4) |
435 ((x & 0xF0F0F0F0F0F0F0F0LL) >> 4);
436 if (shamt & 8) x = ((x & 0x00FF00FF00FF00FFLL) << 8) |
437 ((x & 0xFF00FF00FF00FF00LL) >> 8);
438 if (shamt & 16) x = ((x & 0x0000FFFF0000FFFFLL) << 16) |
439 ((x & 0xFFFF0000FFFF0000LL) >> 16);
440 if (shamt & 32) x = ((x & 0x00000000FFFFFFFFLL) << 32) |
441 ((x & 0xFFFFFFFF00000000LL) >> 32);
449 based on RV bitmanip.
451 RA contains a vector of indices to select parts of RB to be
452 copied to RT. The immediate-variant allows up to an 8 bit
453 pattern (repeated) to be targetted at different parts of RT
456 uint_xlen_t xpermi(uint8_t imm8, uint_xlen_t RB, int sz_log2)
459 uint_xlen_t sz = 1LL << sz_log2;
460 uint_xlen_t mask = (1LL << sz) - 1;
461 uint_xlen_t RA = imm8 | imm8<<8 | ... | imm8<<56;
462 for (int i = 0; i < XLEN; i += sz) {
463 uint_xlen_t pos = ((RA >> i) & mask) << sz_log2;
465 r |= ((RB >> pos) & mask) << i;
469 uint_xlen_t xperm(uint_xlen_t RA, uint_xlen_t RB, int sz_log2)
472 uint_xlen_t sz = 1LL << sz_log2;
473 uint_xlen_t mask = (1LL << sz) - 1;
474 for (int i = 0; i < XLEN; i += sz) {
475 uint_xlen_t pos = ((RA >> i) & mask) << sz_log2;
477 r |= ((RB >> pos) & mask) << i;
481 uint_xlen_t xperm_n (uint_xlen_t RA, uint_xlen_t RB)
482 { return xperm(RA, RB, 2); }
483 uint_xlen_t xperm_b (uint_xlen_t RA, uint_xlen_t RB)
484 { return xperm(RA, RB, 3); }
485 uint_xlen_t xperm_h (uint_xlen_t RA, uint_xlen_t RB)
486 { return xperm(RA, RB, 4); }
487 uint_xlen_t xperm_w (uint_xlen_t RA, uint_xlen_t RB)
488 { return xperm(RA, RB, 5); }
496 uint32_t gorc32(uint32_t RA, uint32_t RB)
500 if (shamt & 1) x |= ((x & 0x55555555) << 1) | ((x & 0xAAAAAAAA) >> 1);
501 if (shamt & 2) x |= ((x & 0x33333333) << 2) | ((x & 0xCCCCCCCC) >> 2);
502 if (shamt & 4) x |= ((x & 0x0F0F0F0F) << 4) | ((x & 0xF0F0F0F0) >> 4);
503 if (shamt & 8) x |= ((x & 0x00FF00FF) << 8) | ((x & 0xFF00FF00) >> 8);
504 if (shamt & 16) x |= ((x & 0x0000FFFF) << 16) | ((x & 0xFFFF0000) >> 16);
507 uint64_t gorc64(uint64_t RA, uint64_t RB)
511 if (shamt & 1) x |= ((x & 0x5555555555555555LL) << 1) |
512 ((x & 0xAAAAAAAAAAAAAAAALL) >> 1);
513 if (shamt & 2) x |= ((x & 0x3333333333333333LL) << 2) |
514 ((x & 0xCCCCCCCCCCCCCCCCLL) >> 2);
515 if (shamt & 4) x |= ((x & 0x0F0F0F0F0F0F0F0FLL) << 4) |
516 ((x & 0xF0F0F0F0F0F0F0F0LL) >> 4);
517 if (shamt & 8) x |= ((x & 0x00FF00FF00FF00FFLL) << 8) |
518 ((x & 0xFF00FF00FF00FF00LL) >> 8);
519 if (shamt & 16) x |= ((x & 0x0000FFFF0000FFFFLL) << 16) |
520 ((x & 0xFFFF0000FFFF0000LL) >> 16);
521 if (shamt & 32) x |= ((x & 0x00000000FFFFFFFFLL) << 32) |
522 ((x & 0xFFFFFFFF00000000LL) >> 32);
528 # Instructions for Carry-less Operations aka. Polynomials with coefficients in `GF(2)`
530 Carry-less addition/subtraction is simply XOR, so a `cladd`
531 instruction is not provided since the `xor[i]` instruction can be used instead.
533 These are operations on polynomials with coefficients in `GF(2)`, with the
534 polynomial's coefficients packed into integers with the following algorithm:
538 """`poly` is a list where `poly[i]` is the coefficient for `x ** i`"""
540 for i, v in enumerate(poly):
545 """returns a list `poly`, where `poly[i]` is the coefficient for `x ** i`.
554 ## Carry-less Multiply Instructions
557 see <https://en.wikipedia.org/wiki/CLMUL_instruction_set> and
558 <https://www.felixcloutier.com/x86/pclmulqdq> and
559 <https://en.m.wikipedia.org/wiki/Carry-less_product>
561 They are worth adding as their own non-overwrite operations
562 (in the same pipeline).
564 ### `clmul` Carry-less Multiply
567 uint_xlen_t clmul(uint_xlen_t RA, uint_xlen_t RB)
570 for (int i = 0; i < XLEN; i++)
577 ### `clmulh` Carry-less Multiply High
580 uint_xlen_t clmulh(uint_xlen_t RA, uint_xlen_t RB)
583 for (int i = 1; i < XLEN; i++)
590 ### `clmulr` Carry-less Multiply (Reversed)
592 Useful for CRCs. Equivalent to bit-reversing the result of `clmul` on
596 uint_xlen_t clmulr(uint_xlen_t RA, uint_xlen_t RB)
599 for (int i = 0; i < XLEN; i++)
601 x ^= RA >> (XLEN-i-1);
606 ## `clmadd` Carry-less Multiply-Add
609 clmadd RT, RA, RB, RC
613 (RT) = clmul((RA), (RB)) ^ (RC)
616 ## `cltmadd` Twin Carry-less Multiply-Add (for FFTs)
619 cltmadd RT, RA, RB, RC
622 TODO: add link to explanation for where `RS` comes from.
625 temp = clmul((RA), (RB)) ^ (RC)
630 ## `cldivrem` Carry-less Division and Remainder
632 `cldivrem` isn't an actual instruction, but is just used in the pseudo-code
633 for other instructions.
635 [[!inline pages="bitmanip_inlines/cldivrem.py" raw="true"]]
637 ## `cldiv` Carry-less Division
643 TODO: decide what happens on division by zero
646 (RT) = cldiv((RA), (RB))
649 ## `clrem` Carry-less Remainder
655 TODO: decide what happens on division by zero
658 (RT) = clrem((RA), (RB))
661 # Instructions for Binary Galois Fields `GF(2^m)`
665 * <https://courses.csail.mit.edu/6.857/2016/files/ffield.py>
666 * <https://engineering.purdue.edu/kak/compsec/NewLectures/Lecture7.pdf>
667 * <https://foss.heptapod.net/math/libgf2/-/blob/branch/default/src/libgf2/gf2.py>
669 Binary Galois Field addition/subtraction is simply XOR, so a `gfbadd`
670 instruction is not provided since the `xor[i]` instruction can be used instead.
672 ## `GFBREDPOLY` SPR -- Reducing Polynomial
674 In order to save registers and to make operations orthogonal with standard
675 arithmetic, the reducing polynomial is stored in a dedicated SPR `GFBREDPOLY`.
676 This also allows hardware to pre-compute useful parameters (such as the
677 degree, or look-up tables) based on the reducing polynomial, and store them
678 alongside the SPR in hidden registers, only recomputing them whenever the SPR
679 is written to, rather than having to recompute those values for every
682 Because Galois Fields require the reducing polynomial to be an irreducible
683 polynomial, that guarantees that any polynomial of `degree > 1` must have
684 the LSB set, since otherwise it would be divisible by the polynomial `x`,
685 making it reducible, making whatever we're working on no longer a Field.
686 Therefore, we can reuse the LSB to indicate `degree == XLEN`.
689 def decode_reducing_polynomial(GFBREDPOLY, XLEN):
690 """returns the decoded coefficient list in LSB to MSB order,
691 len(retval) == degree + 1"""
692 v = GFBREDPOLY & ((1 << XLEN) - 1) # mask to XLEN bits
693 if v == 0 or v == 2: # GF(2)
694 return [0, 1] # degree = 1, poly = x
696 degree = floor_log2(v)
698 # all reducing polynomials of degree > 1 must have the LSB set,
699 # because they must be irreducible polynomials (meaning they
700 # can't be factored), if the LSB was clear, then they would
701 # have `x` as a factor. Therefore, we can reuse the LSB clear
702 # to instead mean the polynomial has degree XLEN.
705 v |= 1 # LSB must be set
706 return [(v >> i) & 1 for i in range(1 + degree)]
709 ## `gfbredpoly` -- Set the Reducing Polynomial SPR `GFBREDPOLY`
711 unless this is an immediate op, `mtspr` is completely sufficient.
713 ## `gfbmul` -- Binary Galois Field `GF(2^m)` Multiplication
720 (RT) = gfbmul((RA), (RB))
723 ## `gfbmadd` -- Binary Galois Field `GF(2^m)` Multiply-Add
726 gfbmadd RT, RA, RB, RC
730 (RT) = gfbadd(gfbmul((RA), (RB)), (RC))
733 ## `gfbtmadd` -- Binary Galois Field `GF(2^m)` Twin Multiply-Add (for FFT)
736 gfbtmadd RT, RA, RB, RC
739 TODO: add link to explanation for where `RS` comes from.
742 temp = gfbadd(gfbmul((RA), (RB)), (RC))
747 ## `gfbinv` -- Binary Galois Field `GF(2^m)` Inverse
757 # Instructions for Prime Galois Fields `GF(p)`
762 def int_to_gfp(int_value, prime):
763 return int_value % prime # follows Python remainder semantics
766 ## `GFPRIME` SPR -- Prime Modulus For `gfp*` Instructions
768 ## `gfpadd` Prime Galois Field `GF(p)` Addition
775 (RT) = int_to_gfp((RA) + (RB), GFPRIME)
778 the addition happens on infinite-precision integers
780 ## `gfpsub` Prime Galois Field `GF(p)` Subtraction
787 (RT) = int_to_gfp((RA) - (RB), GFPRIME)
790 the subtraction happens on infinite-precision integers
792 ## `gfpmul` Prime Galois Field `GF(p)` Multiplication
799 (RT) = int_to_gfp((RA) * (RB), GFPRIME)
802 the multiplication happens on infinite-precision integers
804 ## `gfpinv` Prime Galois Field `GF(p)` Invert
810 Some potential hardware implementations are found in:
811 <https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.5233&rep=rep1&type=pdf>
814 (RT) = gfpinv((RA), GFPRIME)
817 the multiplication happens on infinite-precision integers
819 ## `gfpmadd` Prime Galois Field `GF(p)` Multiply-Add
822 gfpmadd RT, RA, RB, RC
826 (RT) = int_to_gfp((RA) * (RB) + (RC), GFPRIME)
829 the multiplication and addition happens on infinite-precision integers
831 ## `gfpmsub` Prime Galois Field `GF(p)` Multiply-Subtract
834 gfpmsub RT, RA, RB, RC
838 (RT) = int_to_gfp((RA) * (RB) - (RC), GFPRIME)
841 the multiplication and subtraction happens on infinite-precision integers
843 ## `gfpmsubr` Prime Galois Field `GF(p)` Multiply-Subtract-Reversed
846 gfpmsubr RT, RA, RB, RC
850 (RT) = int_to_gfp((RC) - (RA) * (RB), GFPRIME)
853 the multiplication and subtraction happens on infinite-precision integers
855 ## `gfpmaddsubr` Prime Galois Field `GF(p)` Multiply-Add and Multiply-Sub-Reversed (for FFT)
858 gfpmaddsubr RT, RA, RB, RC
861 TODO: add link to explanation for where `RS` comes from.
864 product = (RA) * (RB)
866 (RT) = int_to_gfp(product + term, GFPRIME)
867 (RS) = int_to_gfp(term - product, GFPRIME)
870 the multiplication, addition, and subtraction happens on infinite-precision integers
872 ## Twin Butterfly (Tukey-Cooley) Mul-add-sub
874 used in combination with SV FFT REMAP to perform
875 a full NTT in-place. possible by having 3-in 2-out,
876 to avoid the need for a temp register. RS is written
879 gffmadd RT,RA,RC,RB (Rc=0)
880 gffmadd. RT,RA,RC,RB (Rc=1)
884 RT <- GFADD(GFMUL(RA, RC), RB))
885 RS <- GFADD(GFMUL(RA, RC), RB))
890 with the modulo and degree being in an SPR, multiply can be identical
891 equivalent to standard integer add
895 | 0.5|6.10|11.15|16.20|21.25| 26..30 |31|
896 | -- | -- | --- | --- | --- | ------ |--|
897 | NN | RT | RA | RB |11000| 01110 |Rc|
902 from functools import reduce
912 # constants used in the multGF2 function
913 mask1 = mask2 = polyred = None
916 """Define parameters of binary finite field GF(2^m)/g(x)
917 - irPoly: coefficients of irreducible polynomial g(x)
919 # degree: extension degree of binary field
920 degree = gf_degree(irPoly)
923 """Convert an integer into a polynomial"""
924 return [(sInt >> i) & 1
925 for i in reversed(range(sInt.bit_length()))]
927 global mask1, mask2, polyred
928 mask1 = mask2 = 1 << degree
930 polyred = reduce(lambda x, y: (x << 1) + y, i2P(irPoly)[1:])
933 """Multiply two polynomials in GF(2^m)/g(x)"""
936 # standard long-multiplication: check LSB and add
940 # standard modulo: check MSB and add polynomial
946 if __name__ == "__main__":
948 # Define binary field GF(2^3)/x^3 + x + 1
949 setGF2(0b1011) # degree 3
951 # Evaluate the product (x^2 + x + 1)(x^2 + 1)
952 print("{:02x}".format(multGF2(0b111, 0b101)))
954 # Define binary field GF(2^8)/x^8 + x^4 + x^3 + x + 1
955 # (used in the Advanced Encryption Standard-AES)
956 setGF2(0b100011011) # degree 8
958 # Evaluate the product (x^7)(x^7 + x + 1)
959 print("{:02x}".format(multGF2(0b10000000, 0b10000011)))
965 # https://bugs.libre-soc.org/show_bug.cgi?id=782#c33
966 # https://ftp.libre-soc.org/ARITH18_Kobayashi.pdf
969 s = getGF2() # get the full polynomial (including the MSB)
975 for i in range(1, 2*degree+1):
976 # could use count-trailing-1s here to skip ahead
977 if r & mask1: # test MSB of r
978 if s & mask1: # test MSB of s
981 s <<= 1 # shift left 1
983 r, s = s, r # swap r,s
984 u, v = v<<1, u # shift v and swap
987 u >>= 1 # right shift left
990 r <<= 1 # shift left 1
991 u <<= 1 # shift left 1
999 ## GF2 (carryless) div and mod
1010 def FullDivision(self, f, v):
1012 Takes two arguments, f, v
1013 fDegree and vDegree are the degrees of the field elements
1014 f and v represented as a polynomials.
1015 This method returns the field elements a and b such that
1017 f(x) = a(x) * v(x) + b(x).
1019 That is, a is the divisor and b is the remainder, or in
1020 other words a is like floor(f/v) and b is like f modulo v.
1023 fDegree, vDegree = gf_degree(f), gf_degree(v)
1025 for i in reversed(range(vDegree, fDegree+1):
1026 if ((rem >> i) & 1): # check bit
1027 res ^= (1 << (i - vDegree))
1028 rem ^= ( v << (i - vDegree)))
1032 | 0.5|6.10|11.15|16.20| 21 | 22.23 | 24....30 |31| name |
1033 | -- | -- | --- | --- | -- | ----- | -------- |--| ---- |
1034 | NN | RT | RA | RB | 1 | 00 | 0001 110 |Rc| cldiv |
1035 | NN | RT | RA | RB | 1 | 01 | 0001 110 |Rc| clmod |
1037 ## GF2 carryless mul
1039 based on RV bitmanip
1040 see <https://en.wikipedia.org/wiki/CLMUL_instruction_set> and
1041 <https://www.felixcloutier.com/x86/pclmulqdq> and
1042 <https://en.m.wikipedia.org/wiki/Carry-less_product>
1044 these are GF2 operations with the modulo set to 2^degree.
1045 they are worth adding as their own non-overwrite operations
1046 (in the same pipeline).
1049 uint_xlen_t clmul(uint_xlen_t RA, uint_xlen_t RB)
1052 for (int i = 0; i < XLEN; i++)
1057 uint_xlen_t clmulh(uint_xlen_t RA, uint_xlen_t RB)
1060 for (int i = 1; i < XLEN; i++)
1062 x ^= RA >> (XLEN-i);
1065 uint_xlen_t clmulr(uint_xlen_t RA, uint_xlen_t RB)
1068 for (int i = 0; i < XLEN; i++)
1070 x ^= RA >> (XLEN-i-1);
1074 ## carryless Twin Butterfly (Tukey-Cooley) Mul-add-sub
1076 used in combination with SV FFT REMAP to perform
1077 a full NTT in-place. possible by having 3-in 2-out,
1078 to avoid the need for a temp register. RS is written
1081 clfmadd RT,RA,RC,RB (Rc=0)
1082 clfmadd. RT,RA,RC,RB (Rc=1)
1086 RT <- CLMUL(RA, RC) ^ RB
1087 RS <- CLMUL(RA, RC) ^ RB
1093 uint64_t bmatflip(uint64_t RA)
1101 uint64_t bmatxor(uint64_t RA, uint64_t RB)
1104 uint64_t RBt = bmatflip(RB);
1105 uint8_t u[8]; // rows of RA
1106 uint8_t v[8]; // cols of RB
1107 for (int i = 0; i < 8; i++) {
1109 v[i] = RBt >> (i*8);
1112 for (int i = 0; i < 64; i++) {
1113 if (pcnt(u[i / 8] & v[i % 8]) & 1)
1118 uint64_t bmator(uint64_t RA, uint64_t RB)
1121 uint64_t RBt = bmatflip(RB);
1122 uint8_t u[8]; // rows of RA
1123 uint8_t v[8]; // cols of RB
1124 for (int i = 0; i < 8; i++) {
1126 v[i] = RBt >> (i*8);
1129 for (int i = 0; i < 64; i++) {
1130 if ((u[i / 8] & v[i % 8]) != 0)
1138 # Already in POWER ISA
1140 ## count leading/trailing zeros with mask
1146 do i = 0 to 63 if((RB)i=1) then do
1147 if((RS)i=1) then break end end count ← count + 1
1153 vpdepd VRT,VRA,VRB, identical to RV bitmamip bdep, found already in v3.1 p106
1156 if VSR[VRB+32].dword[i].bit[63-m]=1 then do
1157 result = VSR[VRA+32].dword[i].bit[63-k]
1158 VSR[VRT+32].dword[i].bit[63-m] = result
1164 uint_xlen_t bdep(uint_xlen_t RA, uint_xlen_t RB)
1167 for (int i = 0, j = 0; i < XLEN; i++)
1168 if ((RB >> i) & 1) {
1170 r |= uint_xlen_t(1) << i;
1180 other way round: identical to RV bext, found in v3.1 p196
1183 uint_xlen_t bext(uint_xlen_t RA, uint_xlen_t RB)
1186 for (int i = 0, j = 0; i < XLEN; i++)
1187 if ((RB >> i) & 1) {
1189 r |= uint_xlen_t(1) << j;
1198 found in v3.1 p106 so not to be added here
1208 if((RB)63-i==1) then do
1209 result63-ptr1 = (RS)63-i
1215 # bit to byte permute
1217 similar to matrix permute in RV bitmanip, which has XOR and OR variants,
1218 these perform a transpose.
1222 b = VSR[VRB+32].dword[i].byte[k].bit[j]
1223 VSR[VRT+32].dword[i].byte[j].bit[k] = b