1263 lines
27 KiB
Go
1263 lines
27 KiB
Go
package reedsolomon
|
|
|
|
// This is a O(n*log n) implementation of Reed-Solomon
|
|
// codes, ported from the C++ library https://github.com/catid/leopard.
|
|
//
|
|
// The implementation is based on the paper
|
|
//
|
|
// S.-J. Lin, T. Y. Al-Naffouri, Y. S. Han, and W.-H. Chung,
|
|
// "Novel Polynomial Basis with Fast Fourier Transform
|
|
// and Its Application to Reed-Solomon Erasure Codes"
|
|
// IEEE Trans. on Information Theory, pp. 6284-6299, November, 2016.
|
|
|
|
import (
|
|
"bytes"
|
|
"io"
|
|
"math/bits"
|
|
"sync"
|
|
"unsafe"
|
|
|
|
"github.com/klauspost/cpuid/v2"
|
|
)
|
|
|
|
// leopardFF16 is like reedSolomon but for more than 256 total shards.
|
|
type leopardFF16 struct {
|
|
dataShards int // Number of data shards, should not be modified.
|
|
parityShards int // Number of parity shards, should not be modified.
|
|
totalShards int // Total number of shards. Calculated, and should not be modified.
|
|
|
|
workPool sync.Pool
|
|
|
|
o options
|
|
}
|
|
|
|
// newFF16 is like New, but for more than 256 total shards.
|
|
func newFF16(dataShards, parityShards int, opt options) (*leopardFF16, error) {
|
|
initConstants()
|
|
|
|
if dataShards <= 0 || parityShards <= 0 {
|
|
return nil, ErrInvShardNum
|
|
}
|
|
|
|
if dataShards+parityShards > 65536 {
|
|
return nil, ErrMaxShardNum
|
|
}
|
|
|
|
r := &leopardFF16{
|
|
dataShards: dataShards,
|
|
parityShards: parityShards,
|
|
totalShards: dataShards + parityShards,
|
|
o: opt,
|
|
}
|
|
return r, nil
|
|
}
|
|
|
|
var _ = Extensions(&leopardFF16{})
|
|
|
|
func (r *leopardFF16) ShardSizeMultiple() int {
|
|
return 64
|
|
}
|
|
|
|
func (r *leopardFF16) DataShards() int {
|
|
return r.dataShards
|
|
}
|
|
|
|
func (r *leopardFF16) ParityShards() int {
|
|
return r.parityShards
|
|
}
|
|
|
|
func (r *leopardFF16) TotalShards() int {
|
|
return r.totalShards
|
|
}
|
|
|
|
func (r *leopardFF16) AllocAligned(each int) [][]byte {
|
|
return AllocAligned(r.totalShards, each)
|
|
}
|
|
|
|
type ffe uint16
|
|
|
|
const (
|
|
bitwidth = 16
|
|
order = 1 << bitwidth
|
|
modulus = order - 1
|
|
polynomial = 0x1002D
|
|
)
|
|
|
|
var (
|
|
fftSkew *[modulus]ffe
|
|
logWalsh *[order]ffe
|
|
)
|
|
|
|
// Logarithm Tables
|
|
var (
|
|
logLUT *[order]ffe
|
|
expLUT *[order]ffe
|
|
)
|
|
|
|
// Stores the partial products of x * y at offset x + y * 65536
|
|
// Repeated accesses from the same y value are faster
|
|
var mul16LUTs *[order]mul16LUT
|
|
|
|
type mul16LUT struct {
|
|
// Contains Lo product as a single lookup.
|
|
// Should be XORed with Hi lookup for result.
|
|
Lo [256]ffe
|
|
Hi [256]ffe
|
|
}
|
|
|
|
// Stores lookup for avx2
|
|
var multiply256LUT *[order][8 * 16]byte
|
|
|
|
func (r *leopardFF16) Encode(shards [][]byte) error {
|
|
if len(shards) != r.totalShards {
|
|
return ErrTooFewShards
|
|
}
|
|
|
|
if err := checkShards(shards, false); err != nil {
|
|
return err
|
|
}
|
|
return r.encode(shards)
|
|
}
|
|
|
|
func (r *leopardFF16) encode(shards [][]byte) error {
|
|
shardSize := shardSize(shards)
|
|
if shardSize%64 != 0 {
|
|
return ErrInvalidShardSize
|
|
}
|
|
|
|
m := ceilPow2(r.parityShards)
|
|
var work [][]byte
|
|
if w, ok := r.workPool.Get().([][]byte); ok {
|
|
work = w
|
|
}
|
|
if cap(work) >= m*2 {
|
|
work = work[:m*2]
|
|
} else {
|
|
work = AllocAligned(m*2, shardSize)
|
|
}
|
|
for i := range work {
|
|
if cap(work[i]) < shardSize {
|
|
work[i] = AllocAligned(1, shardSize)[0]
|
|
} else {
|
|
work[i] = work[i][:shardSize]
|
|
}
|
|
}
|
|
defer r.workPool.Put(work)
|
|
|
|
mtrunc := m
|
|
if r.dataShards < mtrunc {
|
|
mtrunc = r.dataShards
|
|
}
|
|
|
|
skewLUT := fftSkew[m-1:]
|
|
|
|
sh := shards
|
|
ifftDITEncoder(
|
|
sh[:r.dataShards],
|
|
mtrunc,
|
|
work,
|
|
nil, // No xor output
|
|
m,
|
|
skewLUT,
|
|
&r.o,
|
|
)
|
|
|
|
lastCount := r.dataShards % m
|
|
if m >= r.dataShards {
|
|
goto skip_body
|
|
}
|
|
|
|
// For sets of m data pieces:
|
|
for i := m; i+m <= r.dataShards; i += m {
|
|
sh = sh[m:]
|
|
skewLUT = skewLUT[m:]
|
|
|
|
// work <- work xor IFFT(data + i, m, m + i)
|
|
|
|
ifftDITEncoder(
|
|
sh, // data source
|
|
m,
|
|
work[m:], // temporary workspace
|
|
work, // xor destination
|
|
m,
|
|
skewLUT,
|
|
&r.o,
|
|
)
|
|
}
|
|
|
|
// Handle final partial set of m pieces:
|
|
if lastCount != 0 {
|
|
sh = sh[m:]
|
|
skewLUT = skewLUT[m:]
|
|
|
|
// work <- work xor IFFT(data + i, m, m + i)
|
|
|
|
ifftDITEncoder(
|
|
sh, // data source
|
|
lastCount,
|
|
work[m:], // temporary workspace
|
|
work, // xor destination
|
|
m,
|
|
skewLUT,
|
|
&r.o,
|
|
)
|
|
}
|
|
|
|
skip_body:
|
|
// work <- FFT(work, m, 0)
|
|
fftDIT(work, r.parityShards, m, fftSkew[:], &r.o)
|
|
|
|
for i, w := range work[:r.parityShards] {
|
|
sh := shards[i+r.dataShards]
|
|
if cap(sh) >= shardSize {
|
|
sh = append(sh[:0], w...)
|
|
} else {
|
|
sh = w
|
|
}
|
|
shards[i+r.dataShards] = sh
|
|
}
|
|
|
|
return nil
|
|
}
|
|
|
|
func (r *leopardFF16) EncodeIdx(dataShard []byte, idx int, parity [][]byte) error {
|
|
return ErrNotSupported
|
|
}
|
|
|
|
func (r *leopardFF16) Join(dst io.Writer, shards [][]byte, outSize int) error {
|
|
// Do we have enough shards?
|
|
if len(shards) < r.dataShards {
|
|
return ErrTooFewShards
|
|
}
|
|
shards = shards[:r.dataShards]
|
|
|
|
// Do we have enough data?
|
|
size := 0
|
|
for _, shard := range shards {
|
|
if shard == nil {
|
|
return ErrReconstructRequired
|
|
}
|
|
size += len(shard)
|
|
|
|
// Do we have enough data already?
|
|
if size >= outSize {
|
|
break
|
|
}
|
|
}
|
|
if size < outSize {
|
|
return ErrShortData
|
|
}
|
|
|
|
// Copy data to dst
|
|
write := outSize
|
|
for _, shard := range shards {
|
|
if write < len(shard) {
|
|
_, err := dst.Write(shard[:write])
|
|
return err
|
|
}
|
|
n, err := dst.Write(shard)
|
|
if err != nil {
|
|
return err
|
|
}
|
|
write -= n
|
|
}
|
|
return nil
|
|
}
|
|
|
|
func (r *leopardFF16) Update(shards [][]byte, newDatashards [][]byte) error {
|
|
return ErrNotSupported
|
|
}
|
|
|
|
func (r *leopardFF16) Split(data []byte) ([][]byte, error) {
|
|
if len(data) == 0 {
|
|
return nil, ErrShortData
|
|
}
|
|
if r.totalShards == 1 && len(data)&63 == 0 {
|
|
return [][]byte{data}, nil
|
|
}
|
|
dataLen := len(data)
|
|
// Calculate number of bytes per data shard.
|
|
perShard := (len(data) + r.dataShards - 1) / r.dataShards
|
|
perShard = ((perShard + 63) / 64) * 64
|
|
needTotal := r.totalShards * perShard
|
|
|
|
if cap(data) > len(data) {
|
|
if cap(data) > needTotal {
|
|
data = data[:needTotal]
|
|
} else {
|
|
data = data[:cap(data)]
|
|
}
|
|
clear := data[dataLen:]
|
|
for i := range clear {
|
|
clear[i] = 0
|
|
}
|
|
}
|
|
|
|
// Only allocate memory if necessary
|
|
var padding [][]byte
|
|
if len(data) < needTotal {
|
|
// calculate maximum number of full shards in `data` slice
|
|
fullShards := len(data) / perShard
|
|
padding = AllocAligned(r.totalShards-fullShards, perShard)
|
|
if dataLen > perShard*fullShards {
|
|
// Copy partial shards
|
|
copyFrom := data[perShard*fullShards : dataLen]
|
|
for i := range padding {
|
|
if len(copyFrom) == 0 {
|
|
break
|
|
}
|
|
copyFrom = copyFrom[copy(padding[i], copyFrom):]
|
|
}
|
|
}
|
|
} else {
|
|
zero := data[dataLen : r.totalShards*perShard]
|
|
for i := range zero {
|
|
zero[i] = 0
|
|
}
|
|
}
|
|
|
|
// Split into equal-length shards.
|
|
dst := make([][]byte, r.totalShards)
|
|
i := 0
|
|
for ; i < len(dst) && len(data) >= perShard; i++ {
|
|
dst[i] = data[:perShard:perShard]
|
|
data = data[perShard:]
|
|
}
|
|
|
|
for j := 0; i+j < len(dst); j++ {
|
|
dst[i+j] = padding[0]
|
|
padding = padding[1:]
|
|
}
|
|
|
|
return dst, nil
|
|
}
|
|
|
|
func (r *leopardFF16) ReconstructSome(shards [][]byte, required []bool) error {
|
|
if len(required) == r.totalShards {
|
|
return r.reconstruct(shards, true)
|
|
}
|
|
return r.reconstruct(shards, false)
|
|
}
|
|
|
|
func (r *leopardFF16) Reconstruct(shards [][]byte) error {
|
|
return r.reconstruct(shards, true)
|
|
}
|
|
|
|
func (r *leopardFF16) ReconstructData(shards [][]byte) error {
|
|
return r.reconstruct(shards, false)
|
|
}
|
|
|
|
func (r *leopardFF16) Verify(shards [][]byte) (bool, error) {
|
|
if len(shards) != r.totalShards {
|
|
return false, ErrTooFewShards
|
|
}
|
|
if err := checkShards(shards, false); err != nil {
|
|
return false, err
|
|
}
|
|
|
|
// Re-encode parity shards to temporary storage.
|
|
shardSize := len(shards[0])
|
|
outputs := make([][]byte, r.totalShards)
|
|
copy(outputs, shards[:r.dataShards])
|
|
for i := r.dataShards; i < r.totalShards; i++ {
|
|
outputs[i] = make([]byte, shardSize)
|
|
}
|
|
if err := r.Encode(outputs); err != nil {
|
|
return false, err
|
|
}
|
|
|
|
// Compare.
|
|
for i := r.dataShards; i < r.totalShards; i++ {
|
|
if !bytes.Equal(outputs[i], shards[i]) {
|
|
return false, nil
|
|
}
|
|
}
|
|
return true, nil
|
|
}
|
|
|
|
func (r *leopardFF16) reconstruct(shards [][]byte, recoverAll bool) error {
|
|
if len(shards) != r.totalShards {
|
|
return ErrTooFewShards
|
|
}
|
|
|
|
if err := checkShards(shards, true); err != nil {
|
|
return err
|
|
}
|
|
|
|
// Quick check: are all of the shards present? If so, there's
|
|
// nothing to do.
|
|
numberPresent := 0
|
|
dataPresent := 0
|
|
for i := 0; i < r.totalShards; i++ {
|
|
if len(shards[i]) != 0 {
|
|
numberPresent++
|
|
if i < r.dataShards {
|
|
dataPresent++
|
|
}
|
|
}
|
|
}
|
|
if numberPresent == r.totalShards || !recoverAll && dataPresent == r.dataShards {
|
|
// Cool. All of the shards have data. We don't
|
|
// need to do anything.
|
|
return nil
|
|
}
|
|
|
|
// Use only if we are missing less than 1/4 parity.
|
|
useBits := r.totalShards-numberPresent <= r.parityShards/4
|
|
|
|
// Check if we have enough to reconstruct.
|
|
if numberPresent < r.dataShards {
|
|
return ErrTooFewShards
|
|
}
|
|
|
|
shardSize := shardSize(shards)
|
|
if shardSize%64 != 0 {
|
|
return ErrInvalidShardSize
|
|
}
|
|
|
|
m := ceilPow2(r.parityShards)
|
|
n := ceilPow2(m + r.dataShards)
|
|
|
|
const LEO_ERROR_BITFIELD_OPT = true
|
|
|
|
// Fill in error locations.
|
|
var errorBits errorBitfield
|
|
var errLocs [order]ffe
|
|
for i := 0; i < r.parityShards; i++ {
|
|
if len(shards[i+r.dataShards]) == 0 {
|
|
errLocs[i] = 1
|
|
if LEO_ERROR_BITFIELD_OPT && recoverAll {
|
|
errorBits.set(i)
|
|
}
|
|
}
|
|
}
|
|
for i := r.parityShards; i < m; i++ {
|
|
errLocs[i] = 1
|
|
if LEO_ERROR_BITFIELD_OPT && recoverAll {
|
|
errorBits.set(i)
|
|
}
|
|
}
|
|
for i := 0; i < r.dataShards; i++ {
|
|
if len(shards[i]) == 0 {
|
|
errLocs[i+m] = 1
|
|
if LEO_ERROR_BITFIELD_OPT {
|
|
errorBits.set(i + m)
|
|
}
|
|
}
|
|
}
|
|
|
|
if LEO_ERROR_BITFIELD_OPT && useBits {
|
|
errorBits.prepare()
|
|
}
|
|
|
|
// Evaluate error locator polynomial
|
|
fwht(&errLocs, order, m+r.dataShards)
|
|
|
|
for i := 0; i < order; i++ {
|
|
errLocs[i] = ffe((uint(errLocs[i]) * uint(logWalsh[i])) % modulus)
|
|
}
|
|
|
|
fwht(&errLocs, order, order)
|
|
|
|
var work [][]byte
|
|
if w, ok := r.workPool.Get().([][]byte); ok {
|
|
work = w
|
|
}
|
|
if cap(work) >= n {
|
|
work = work[:n]
|
|
} else {
|
|
work = make([][]byte, n)
|
|
}
|
|
for i := range work {
|
|
if cap(work[i]) < shardSize {
|
|
work[i] = make([]byte, shardSize)
|
|
} else {
|
|
work[i] = work[i][:shardSize]
|
|
}
|
|
}
|
|
defer r.workPool.Put(work)
|
|
|
|
// work <- recovery data
|
|
|
|
for i := 0; i < r.parityShards; i++ {
|
|
if len(shards[i+r.dataShards]) != 0 {
|
|
mulgf16(work[i], shards[i+r.dataShards], errLocs[i], &r.o)
|
|
} else {
|
|
memclr(work[i])
|
|
}
|
|
}
|
|
for i := r.parityShards; i < m; i++ {
|
|
memclr(work[i])
|
|
}
|
|
|
|
// work <- original data
|
|
|
|
for i := 0; i < r.dataShards; i++ {
|
|
if len(shards[i]) != 0 {
|
|
mulgf16(work[m+i], shards[i], errLocs[m+i], &r.o)
|
|
} else {
|
|
memclr(work[m+i])
|
|
}
|
|
}
|
|
for i := m + r.dataShards; i < n; i++ {
|
|
memclr(work[i])
|
|
}
|
|
|
|
// work <- IFFT(work, n, 0)
|
|
|
|
ifftDITDecoder(
|
|
m+r.dataShards,
|
|
work,
|
|
n,
|
|
fftSkew[:],
|
|
&r.o,
|
|
)
|
|
|
|
// work <- FormalDerivative(work, n)
|
|
|
|
for i := 1; i < n; i++ {
|
|
width := ((i ^ (i - 1)) + 1) >> 1
|
|
slicesXor(work[i-width:i], work[i:i+width], &r.o)
|
|
}
|
|
|
|
// work <- FFT(work, n, 0) truncated to m + dataShards
|
|
|
|
outputCount := m + r.dataShards
|
|
|
|
if LEO_ERROR_BITFIELD_OPT && useBits {
|
|
errorBits.fftDIT(work, outputCount, n, fftSkew[:], &r.o)
|
|
} else {
|
|
fftDIT(work, outputCount, n, fftSkew[:], &r.o)
|
|
}
|
|
|
|
// Reveal erasures
|
|
//
|
|
// Original = -ErrLocator * FFT( Derivative( IFFT( ErrLocator * ReceivedData ) ) )
|
|
// mul_mem(x, y, log_m, ) equals x[] = y[] * log_m
|
|
//
|
|
// mem layout: [Recovery Data (Power of Two = M)] [Original Data (K)] [Zero Padding out to N]
|
|
end := r.dataShards
|
|
if recoverAll {
|
|
end = r.totalShards
|
|
}
|
|
for i := 0; i < end; i++ {
|
|
if len(shards[i]) != 0 {
|
|
continue
|
|
}
|
|
if cap(shards[i]) >= shardSize {
|
|
shards[i] = shards[i][:shardSize]
|
|
} else {
|
|
shards[i] = make([]byte, shardSize)
|
|
}
|
|
if i >= r.dataShards {
|
|
// Parity shard.
|
|
mulgf16(shards[i], work[i-r.dataShards], modulus-errLocs[i-r.dataShards], &r.o)
|
|
} else {
|
|
// Data shard.
|
|
mulgf16(shards[i], work[i+m], modulus-errLocs[i+m], &r.o)
|
|
}
|
|
}
|
|
return nil
|
|
}
|
|
|
|
// Basic no-frills version for decoder
|
|
func ifftDITDecoder(mtrunc int, work [][]byte, m int, skewLUT []ffe, o *options) {
|
|
// Decimation in time: Unroll 2 layers at a time
|
|
dist := 1
|
|
dist4 := 4
|
|
for dist4 <= m {
|
|
// For each set of dist*4 elements:
|
|
for r := 0; r < mtrunc; r += dist4 {
|
|
iend := r + dist
|
|
log_m01 := skewLUT[iend-1]
|
|
log_m02 := skewLUT[iend+dist-1]
|
|
log_m23 := skewLUT[iend+dist*2-1]
|
|
|
|
// For each set of dist elements:
|
|
for i := r; i < iend; i++ {
|
|
ifftDIT4(work[i:], dist, log_m01, log_m23, log_m02, o)
|
|
}
|
|
}
|
|
dist = dist4
|
|
dist4 <<= 2
|
|
}
|
|
|
|
// If there is one layer left:
|
|
if dist < m {
|
|
// Assuming that dist = m / 2
|
|
if dist*2 != m {
|
|
panic("internal error")
|
|
}
|
|
|
|
log_m := skewLUT[dist-1]
|
|
|
|
if log_m == modulus {
|
|
slicesXor(work[dist:2*dist], work[:dist], o)
|
|
} else {
|
|
for i := 0; i < dist; i++ {
|
|
ifftDIT2(
|
|
work[i],
|
|
work[i+dist],
|
|
log_m,
|
|
o,
|
|
)
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// In-place FFT for encoder and decoder
|
|
func fftDIT(work [][]byte, mtrunc, m int, skewLUT []ffe, o *options) {
|
|
// Decimation in time: Unroll 2 layers at a time
|
|
dist4 := m
|
|
dist := m >> 2
|
|
for dist != 0 {
|
|
// For each set of dist*4 elements:
|
|
for r := 0; r < mtrunc; r += dist4 {
|
|
iend := r + dist
|
|
log_m01 := skewLUT[iend-1]
|
|
log_m02 := skewLUT[iend+dist-1]
|
|
log_m23 := skewLUT[iend+dist*2-1]
|
|
|
|
// For each set of dist elements:
|
|
for i := r; i < iend; i++ {
|
|
fftDIT4(
|
|
work[i:],
|
|
dist,
|
|
log_m01,
|
|
log_m23,
|
|
log_m02,
|
|
o,
|
|
)
|
|
}
|
|
}
|
|
dist4 = dist
|
|
dist >>= 2
|
|
}
|
|
|
|
// If there is one layer left:
|
|
if dist4 == 2 {
|
|
for r := 0; r < mtrunc; r += 2 {
|
|
log_m := skewLUT[r+1-1]
|
|
|
|
if log_m == modulus {
|
|
sliceXor(work[r], work[r+1], o)
|
|
} else {
|
|
fftDIT2(work[r], work[r+1], log_m, o)
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// 4-way butterfly
|
|
func fftDIT4Ref(work [][]byte, dist int, log_m01, log_m23, log_m02 ffe, o *options) {
|
|
// First layer:
|
|
if log_m02 == modulus {
|
|
sliceXor(work[0], work[dist*2], o)
|
|
sliceXor(work[dist], work[dist*3], o)
|
|
} else {
|
|
fftDIT2(work[0], work[dist*2], log_m02, o)
|
|
fftDIT2(work[dist], work[dist*3], log_m02, o)
|
|
}
|
|
|
|
// Second layer:
|
|
if log_m01 == modulus {
|
|
sliceXor(work[0], work[dist], o)
|
|
} else {
|
|
fftDIT2(work[0], work[dist], log_m01, o)
|
|
}
|
|
|
|
if log_m23 == modulus {
|
|
sliceXor(work[dist*2], work[dist*3], o)
|
|
} else {
|
|
fftDIT2(work[dist*2], work[dist*3], log_m23, o)
|
|
}
|
|
}
|
|
|
|
// Unrolled IFFT for encoder
|
|
func ifftDITEncoder(data [][]byte, mtrunc int, work [][]byte, xorRes [][]byte, m int, skewLUT []ffe, o *options) {
|
|
// I tried rolling the memcpy/memset into the first layer of the FFT and
|
|
// found that it only yields a 4% performance improvement, which is not
|
|
// worth the extra complexity.
|
|
for i := 0; i < mtrunc; i++ {
|
|
copy(work[i], data[i])
|
|
}
|
|
for i := mtrunc; i < m; i++ {
|
|
memclr(work[i])
|
|
}
|
|
|
|
// I tried splitting up the first few layers into L3-cache sized blocks but
|
|
// found that it only provides about 5% performance boost, which is not
|
|
// worth the extra complexity.
|
|
|
|
// Decimation in time: Unroll 2 layers at a time
|
|
dist := 1
|
|
dist4 := 4
|
|
for dist4 <= m {
|
|
// For each set of dist*4 elements:
|
|
for r := 0; r < mtrunc; r += dist4 {
|
|
iend := r + dist
|
|
log_m01 := skewLUT[iend]
|
|
log_m02 := skewLUT[iend+dist]
|
|
log_m23 := skewLUT[iend+dist*2]
|
|
|
|
// For each set of dist elements:
|
|
for i := r; i < iend; i++ {
|
|
ifftDIT4(
|
|
work[i:],
|
|
dist,
|
|
log_m01,
|
|
log_m23,
|
|
log_m02,
|
|
o,
|
|
)
|
|
}
|
|
}
|
|
|
|
dist = dist4
|
|
dist4 <<= 2
|
|
// I tried alternating sweeps left->right and right->left to reduce cache misses.
|
|
// It provides about 1% performance boost when done for both FFT and IFFT, so it
|
|
// does not seem to be worth the extra complexity.
|
|
}
|
|
|
|
// If there is one layer left:
|
|
if dist < m {
|
|
// Assuming that dist = m / 2
|
|
if dist*2 != m {
|
|
panic("internal error")
|
|
}
|
|
|
|
logm := skewLUT[dist]
|
|
|
|
if logm == modulus {
|
|
slicesXor(work[dist:dist*2], work[:dist], o)
|
|
} else {
|
|
for i := 0; i < dist; i++ {
|
|
ifftDIT2(work[i], work[i+dist], logm, o)
|
|
}
|
|
}
|
|
}
|
|
|
|
// I tried unrolling this but it does not provide more than 5% performance
|
|
// improvement for 16-bit finite fields, so it's not worth the complexity.
|
|
if xorRes != nil {
|
|
slicesXor(xorRes[:m], work[:m], o)
|
|
}
|
|
}
|
|
|
|
func ifftDIT4Ref(work [][]byte, dist int, log_m01, log_m23, log_m02 ffe, o *options) {
|
|
// First layer:
|
|
if log_m01 == modulus {
|
|
sliceXor(work[0], work[dist], o)
|
|
} else {
|
|
ifftDIT2(work[0], work[dist], log_m01, o)
|
|
}
|
|
|
|
if log_m23 == modulus {
|
|
sliceXor(work[dist*2], work[dist*3], o)
|
|
} else {
|
|
ifftDIT2(work[dist*2], work[dist*3], log_m23, o)
|
|
}
|
|
|
|
// Second layer:
|
|
if log_m02 == modulus {
|
|
sliceXor(work[0], work[dist*2], o)
|
|
sliceXor(work[dist], work[dist*3], o)
|
|
} else {
|
|
ifftDIT2(work[0], work[dist*2], log_m02, o)
|
|
ifftDIT2(work[dist], work[dist*3], log_m02, o)
|
|
}
|
|
}
|
|
|
|
// Reference version of muladd: x[] ^= y[] * log_m
|
|
func refMulAdd(x, y []byte, log_m ffe) {
|
|
lut := &mul16LUTs[log_m]
|
|
|
|
for len(x) >= 64 {
|
|
// Assert sizes for no bounds checks in loop
|
|
hiA := y[32:64]
|
|
loA := y[:32]
|
|
dst := x[:64] // Needed, but not checked...
|
|
for i, lo := range loA {
|
|
hi := hiA[i]
|
|
prod := lut.Lo[lo] ^ lut.Hi[hi]
|
|
|
|
dst[i] ^= byte(prod)
|
|
dst[i+32] ^= byte(prod >> 8)
|
|
}
|
|
x = x[64:]
|
|
y = y[64:]
|
|
}
|
|
}
|
|
|
|
func memclr(s []byte) {
|
|
for i := range s {
|
|
s[i] = 0
|
|
}
|
|
}
|
|
|
|
// slicesXor calls xor for every slice pair in v1, v2.
|
|
func slicesXor(v1, v2 [][]byte, o *options) {
|
|
for i, v := range v1 {
|
|
sliceXor(v2[i], v, o)
|
|
}
|
|
}
|
|
|
|
// Reference version of mul: x[] = y[] * log_m
|
|
func refMul(x, y []byte, log_m ffe) {
|
|
lut := &mul16LUTs[log_m]
|
|
|
|
for off := 0; off < len(x); off += 64 {
|
|
loA := y[off : off+32]
|
|
hiA := y[off+32:]
|
|
hiA = hiA[:len(loA)]
|
|
for i, lo := range loA {
|
|
hi := hiA[i]
|
|
prod := lut.Lo[lo] ^ lut.Hi[hi]
|
|
|
|
x[off+i] = byte(prod)
|
|
x[off+i+32] = byte(prod >> 8)
|
|
}
|
|
}
|
|
}
|
|
|
|
// Returns a * Log(b)
|
|
func mulLog(a, log_b ffe) ffe {
|
|
/*
|
|
Note that this operation is not a normal multiplication in a finite
|
|
field because the right operand is already a logarithm. This is done
|
|
because it moves K table lookups from the Decode() method into the
|
|
initialization step that is less performance critical. The LogWalsh[]
|
|
table below contains precalculated logarithms so it is easier to do
|
|
all the other multiplies in that form as well.
|
|
*/
|
|
if a == 0 {
|
|
return 0
|
|
}
|
|
return expLUT[addMod(logLUT[a], log_b)]
|
|
}
|
|
|
|
// z = x + y (mod kModulus)
|
|
func addMod(a, b ffe) ffe {
|
|
sum := uint(a) + uint(b)
|
|
|
|
// Partial reduction step, allowing for kModulus to be returned
|
|
return ffe(sum + sum>>bitwidth)
|
|
}
|
|
|
|
// z = x - y (mod kModulus)
|
|
func subMod(a, b ffe) ffe {
|
|
dif := uint(a) - uint(b)
|
|
|
|
// Partial reduction step, allowing for kModulus to be returned
|
|
return ffe(dif + dif>>bitwidth)
|
|
}
|
|
|
|
// ceilPow2 returns power of two at or above n.
|
|
func ceilPow2(n int) int {
|
|
const w = int(unsafe.Sizeof(n) * 8)
|
|
return 1 << (w - bits.LeadingZeros(uint(n-1)))
|
|
}
|
|
|
|
// Decimation in time (DIT) Fast Walsh-Hadamard Transform
|
|
// Unrolls pairs of layers to perform cross-layer operations in registers
|
|
// mtrunc: Number of elements that are non-zero at the front of data
|
|
func fwht(data *[order]ffe, m, mtrunc int) {
|
|
// Decimation in time: Unroll 2 layers at a time
|
|
dist := 1
|
|
dist4 := 4
|
|
for dist4 <= m {
|
|
// For each set of dist*4 elements:
|
|
for r := 0; r < mtrunc; r += dist4 {
|
|
// For each set of dist elements:
|
|
// Use 16 bit indices to avoid bounds check on [65536]ffe.
|
|
dist := uint16(dist)
|
|
off := uint16(r)
|
|
for i := uint16(0); i < dist; i++ {
|
|
// fwht4(data[i:], dist) inlined...
|
|
// Reading values appear faster than updating pointers.
|
|
// Casting to uint is not faster.
|
|
t0 := data[off]
|
|
t1 := data[off+dist]
|
|
t2 := data[off+dist*2]
|
|
t3 := data[off+dist*3]
|
|
|
|
t0, t1 = fwht2alt(t0, t1)
|
|
t2, t3 = fwht2alt(t2, t3)
|
|
t0, t2 = fwht2alt(t0, t2)
|
|
t1, t3 = fwht2alt(t1, t3)
|
|
|
|
data[off] = t0
|
|
data[off+dist] = t1
|
|
data[off+dist*2] = t2
|
|
data[off+dist*3] = t3
|
|
off++
|
|
}
|
|
}
|
|
dist = dist4
|
|
dist4 <<= 2
|
|
}
|
|
|
|
// If there is one layer left:
|
|
if dist < m {
|
|
dist := uint16(dist)
|
|
for i := uint16(0); i < dist; i++ {
|
|
fwht2(&data[i], &data[i+dist])
|
|
}
|
|
}
|
|
}
|
|
|
|
func fwht4(data []ffe, s int) {
|
|
s2 := s << 1
|
|
|
|
t0 := &data[0]
|
|
t1 := &data[s]
|
|
t2 := &data[s2]
|
|
t3 := &data[s2+s]
|
|
|
|
fwht2(t0, t1)
|
|
fwht2(t2, t3)
|
|
fwht2(t0, t2)
|
|
fwht2(t1, t3)
|
|
}
|
|
|
|
// {a, b} = {a + b, a - b} (Mod Q)
|
|
func fwht2(a, b *ffe) {
|
|
sum := addMod(*a, *b)
|
|
dif := subMod(*a, *b)
|
|
*a = sum
|
|
*b = dif
|
|
}
|
|
|
|
// fwht2alt is as fwht2, but returns result.
|
|
func fwht2alt(a, b ffe) (ffe, ffe) {
|
|
return addMod(a, b), subMod(a, b)
|
|
}
|
|
|
|
var initOnce sync.Once
|
|
|
|
func initConstants() {
|
|
initOnce.Do(func() {
|
|
initLUTs()
|
|
initFFTSkew()
|
|
initMul16LUT()
|
|
})
|
|
}
|
|
|
|
// Initialize logLUT, expLUT.
|
|
func initLUTs() {
|
|
cantorBasis := [bitwidth]ffe{
|
|
0x0001, 0xACCA, 0x3C0E, 0x163E,
|
|
0xC582, 0xED2E, 0x914C, 0x4012,
|
|
0x6C98, 0x10D8, 0x6A72, 0xB900,
|
|
0xFDB8, 0xFB34, 0xFF38, 0x991E,
|
|
}
|
|
|
|
expLUT = &[order]ffe{}
|
|
logLUT = &[order]ffe{}
|
|
|
|
// LFSR table generation:
|
|
state := 1
|
|
for i := ffe(0); i < modulus; i++ {
|
|
expLUT[state] = i
|
|
state <<= 1
|
|
if state >= order {
|
|
state ^= polynomial
|
|
}
|
|
}
|
|
expLUT[0] = modulus
|
|
|
|
// Conversion to Cantor basis:
|
|
|
|
logLUT[0] = 0
|
|
for i := 0; i < bitwidth; i++ {
|
|
basis := cantorBasis[i]
|
|
width := 1 << i
|
|
|
|
for j := 0; j < width; j++ {
|
|
logLUT[j+width] = logLUT[j] ^ basis
|
|
}
|
|
}
|
|
|
|
for i := 0; i < order; i++ {
|
|
logLUT[i] = expLUT[logLUT[i]]
|
|
}
|
|
|
|
for i := 0; i < order; i++ {
|
|
expLUT[logLUT[i]] = ffe(i)
|
|
}
|
|
|
|
expLUT[modulus] = expLUT[0]
|
|
}
|
|
|
|
// Initialize fftSkew.
|
|
func initFFTSkew() {
|
|
var temp [bitwidth - 1]ffe
|
|
|
|
// Generate FFT skew vector {1}:
|
|
|
|
for i := 1; i < bitwidth; i++ {
|
|
temp[i-1] = ffe(1 << i)
|
|
}
|
|
|
|
fftSkew = &[modulus]ffe{}
|
|
logWalsh = &[order]ffe{}
|
|
|
|
for m := 0; m < bitwidth-1; m++ {
|
|
step := 1 << (m + 1)
|
|
|
|
fftSkew[1<<m-1] = 0
|
|
|
|
for i := m; i < bitwidth-1; i++ {
|
|
s := 1 << (i + 1)
|
|
|
|
for j := 1<<m - 1; j < s; j += step {
|
|
fftSkew[j+s] = fftSkew[j] ^ temp[i]
|
|
}
|
|
}
|
|
|
|
temp[m] = modulus - logLUT[mulLog(temp[m], logLUT[temp[m]^1])]
|
|
|
|
for i := m + 1; i < bitwidth-1; i++ {
|
|
sum := addMod(logLUT[temp[i]^1], temp[m])
|
|
temp[i] = mulLog(temp[i], sum)
|
|
}
|
|
}
|
|
|
|
for i := 0; i < modulus; i++ {
|
|
fftSkew[i] = logLUT[fftSkew[i]]
|
|
}
|
|
|
|
// Precalculate FWHT(Log[i]):
|
|
|
|
for i := 0; i < order; i++ {
|
|
logWalsh[i] = logLUT[i]
|
|
}
|
|
logWalsh[0] = 0
|
|
|
|
fwht(logWalsh, order, order)
|
|
}
|
|
|
|
func initMul16LUT() {
|
|
mul16LUTs = &[order]mul16LUT{}
|
|
|
|
// For each log_m multiplicand:
|
|
for log_m := 0; log_m < order; log_m++ {
|
|
var tmp [64]ffe
|
|
for nibble, shift := 0, 0; nibble < 4; {
|
|
nibble_lut := tmp[nibble*16:]
|
|
|
|
for xnibble := 0; xnibble < 16; xnibble++ {
|
|
prod := mulLog(ffe(xnibble<<shift), ffe(log_m))
|
|
nibble_lut[xnibble] = prod
|
|
}
|
|
nibble++
|
|
shift += 4
|
|
}
|
|
lut := &mul16LUTs[log_m]
|
|
for i := range lut.Lo[:] {
|
|
lut.Lo[i] = tmp[i&15] ^ tmp[((i>>4)+16)]
|
|
lut.Hi[i] = tmp[((i&15)+32)] ^ tmp[((i>>4)+48)]
|
|
}
|
|
}
|
|
if cpuid.CPU.Has(cpuid.SSSE3) || cpuid.CPU.Has(cpuid.AVX2) || cpuid.CPU.Has(cpuid.AVX512F) {
|
|
multiply256LUT = &[order][16 * 8]byte{}
|
|
|
|
for logM := range multiply256LUT[:] {
|
|
// For each 4 bits of the finite field width in bits:
|
|
shift := 0
|
|
for i := 0; i < 4; i++ {
|
|
// Construct 16 entry LUT for PSHUFB
|
|
prodLo := multiply256LUT[logM][i*16 : i*16+16]
|
|
prodHi := multiply256LUT[logM][4*16+i*16 : 4*16+i*16+16]
|
|
for x := range prodLo[:] {
|
|
prod := mulLog(ffe(x<<shift), ffe(logM))
|
|
prodLo[x] = byte(prod)
|
|
prodHi[x] = byte(prod >> 8)
|
|
}
|
|
shift += 4
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
const kWordMips = 5
|
|
const kWords = order / 64
|
|
const kBigMips = 6
|
|
const kBigWords = (kWords + 63) / 64
|
|
const kBiggestMips = 4
|
|
|
|
// errorBitfield contains progressive errors to help indicate which
|
|
// shards need reconstruction.
|
|
type errorBitfield struct {
|
|
Words [kWordMips][kWords]uint64
|
|
BigWords [kBigMips][kBigWords]uint64
|
|
BiggestWords [kBiggestMips]uint64
|
|
}
|
|
|
|
func (e *errorBitfield) set(i int) {
|
|
e.Words[0][i/64] |= uint64(1) << (i & 63)
|
|
}
|
|
|
|
func (e *errorBitfield) isNeededFn(mipLevel int) func(bit int) bool {
|
|
if mipLevel >= 16 {
|
|
return func(bit int) bool {
|
|
return true
|
|
}
|
|
}
|
|
if mipLevel >= 12 {
|
|
w := e.BiggestWords[mipLevel-12]
|
|
return func(bit int) bool {
|
|
bit /= 4096
|
|
return 0 != (w & (uint64(1) << bit))
|
|
}
|
|
}
|
|
if mipLevel >= 6 {
|
|
w := e.BigWords[mipLevel-6][:]
|
|
return func(bit int) bool {
|
|
bit /= 64
|
|
return 0 != (w[bit/64] & (uint64(1) << (bit & 63)))
|
|
}
|
|
}
|
|
if mipLevel > 0 {
|
|
w := e.Words[mipLevel-1][:]
|
|
return func(bit int) bool {
|
|
return 0 != (w[bit/64] & (uint64(1) << (bit & 63)))
|
|
}
|
|
}
|
|
return nil
|
|
}
|
|
|
|
func (e *errorBitfield) isNeeded(mipLevel int, bit uint) bool {
|
|
if mipLevel >= 16 {
|
|
return true
|
|
}
|
|
if mipLevel >= 12 {
|
|
bit /= 4096
|
|
return 0 != (e.BiggestWords[mipLevel-12] & (uint64(1) << bit))
|
|
}
|
|
if mipLevel >= 6 {
|
|
bit /= 64
|
|
return 0 != (e.BigWords[mipLevel-6][bit/64] & (uint64(1) << (bit % 64)))
|
|
}
|
|
return 0 != (e.Words[mipLevel-1][bit/64] & (uint64(1) << (bit % 64)))
|
|
}
|
|
|
|
var kHiMasks = [5]uint64{
|
|
0xAAAAAAAAAAAAAAAA,
|
|
0xCCCCCCCCCCCCCCCC,
|
|
0xF0F0F0F0F0F0F0F0,
|
|
0xFF00FF00FF00FF00,
|
|
0xFFFF0000FFFF0000,
|
|
}
|
|
|
|
func (e *errorBitfield) prepare() {
|
|
// First mip level is for final layer of FFT: pairs of data
|
|
for i := 0; i < kWords; i++ {
|
|
w_i := e.Words[0][i]
|
|
hi2lo0 := w_i | ((w_i & kHiMasks[0]) >> 1)
|
|
lo2hi0 := (w_i & (kHiMasks[0] >> 1)) << 1
|
|
w_i = hi2lo0 | lo2hi0
|
|
e.Words[0][i] = w_i
|
|
|
|
bits := 2
|
|
for j := 1; j < kWordMips; j++ {
|
|
hi2lo_j := w_i | ((w_i & kHiMasks[j]) >> bits)
|
|
lo2hi_j := (w_i & (kHiMasks[j] >> bits)) << bits
|
|
w_i = hi2lo_j | lo2hi_j
|
|
e.Words[j][i] = w_i
|
|
bits <<= 1
|
|
}
|
|
}
|
|
|
|
for i := 0; i < kBigWords; i++ {
|
|
w_i := uint64(0)
|
|
bit := uint64(1)
|
|
src := e.Words[kWordMips-1][i*64 : i*64+64]
|
|
for _, w := range src {
|
|
w_i |= (w | (w >> 32) | (w << 32)) & bit
|
|
bit <<= 1
|
|
}
|
|
e.BigWords[0][i] = w_i
|
|
|
|
bits := 1
|
|
for j := 1; j < kBigMips; j++ {
|
|
hi2lo_j := w_i | ((w_i & kHiMasks[j-1]) >> bits)
|
|
lo2hi_j := (w_i & (kHiMasks[j-1] >> bits)) << bits
|
|
w_i = hi2lo_j | lo2hi_j
|
|
e.BigWords[j][i] = w_i
|
|
bits <<= 1
|
|
}
|
|
}
|
|
|
|
w_i := uint64(0)
|
|
bit := uint64(1)
|
|
for _, w := range e.BigWords[kBigMips-1][:kBigWords] {
|
|
w_i |= (w | (w >> 32) | (w << 32)) & bit
|
|
bit <<= 1
|
|
}
|
|
e.BiggestWords[0] = w_i
|
|
|
|
bits := uint64(1)
|
|
for j := 1; j < kBiggestMips; j++ {
|
|
hi2lo_j := w_i | ((w_i & kHiMasks[j-1]) >> bits)
|
|
lo2hi_j := (w_i & (kHiMasks[j-1] >> bits)) << bits
|
|
w_i = hi2lo_j | lo2hi_j
|
|
e.BiggestWords[j] = w_i
|
|
bits <<= 1
|
|
}
|
|
}
|
|
|
|
func (e *errorBitfield) fftDIT(work [][]byte, mtrunc, m int, skewLUT []ffe, o *options) {
|
|
// Decimation in time: Unroll 2 layers at a time
|
|
mipLevel := bits.Len32(uint32(m)) - 1
|
|
|
|
dist4 := m
|
|
dist := m >> 2
|
|
needed := e.isNeededFn(mipLevel)
|
|
for dist != 0 {
|
|
// For each set of dist*4 elements:
|
|
for r := 0; r < mtrunc; r += dist4 {
|
|
if !needed(r) {
|
|
continue
|
|
}
|
|
iEnd := r + dist
|
|
logM01 := skewLUT[iEnd-1]
|
|
logM02 := skewLUT[iEnd+dist-1]
|
|
logM23 := skewLUT[iEnd+dist*2-1]
|
|
|
|
// For each set of dist elements:
|
|
for i := r; i < iEnd; i++ {
|
|
fftDIT4(
|
|
work[i:],
|
|
dist,
|
|
logM01,
|
|
logM23,
|
|
logM02,
|
|
o)
|
|
}
|
|
}
|
|
dist4 = dist
|
|
dist >>= 2
|
|
mipLevel -= 2
|
|
needed = e.isNeededFn(mipLevel)
|
|
}
|
|
|
|
// If there is one layer left:
|
|
if dist4 == 2 {
|
|
for r := 0; r < mtrunc; r += 2 {
|
|
if !needed(r) {
|
|
continue
|
|
}
|
|
logM := skewLUT[r+1-1]
|
|
|
|
if logM == modulus {
|
|
sliceXor(work[r], work[r+1], o)
|
|
} else {
|
|
fftDIT2(work[r], work[r+1], logM, o)
|
|
}
|
|
}
|
|
}
|
|
}
|