algorithm.go

// Copyright 2017 The go-ethereum Authors
// This file is part of the go-ethereum library.
//
// The go-ethereum library is free software: you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// The go-ethereum library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU Lesser General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public License
// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.

package ethash

import (
	"encoding/binary"
	"hash"
	"math/big"
	"reflect"
	"runtime"
	"sync"
	"sync/atomic"
	"time"
	"unsafe"

	"github.com/ethereum/go-ethereum/common"
	"github.com/ethereum/go-ethereum/common/bitutil"
	"github.com/ethereum/go-ethereum/crypto"
	"github.com/ethereum/go-ethereum/log"
	"golang.org/x/crypto/sha3"
)

const (
	datasetInitBytes   = 1 << 30 // Bytes in dataset at genesis
	datasetGrowthBytes = 1 << 23 // Dataset growth per epoch
	cacheInitBytes     = 1 << 24 // Bytes in cache at genesis
	cacheGrowthBytes   = 1 << 17 // Cache growth per epoch
	epochLength        = 30000   // Blocks per epoch
	mixBytes           = 128     // Width of mix
	hashBytes          = 64      // Hash length in bytes
	hashWords          = 16      // Number of 32 bit ints in a hash
	datasetParents     = 256     // Number of parents of each dataset element
	cacheRounds        = 3       // Number of rounds in cache production
	loopAccesses       = 64      // Number of accesses in hashimoto loop
)

// cacheSize returns the size of the ethash verification cache that belongs to a certain
// block number.
func cacheSize(block uint64) uint64 {
	epoch := int(block / epochLength)
	if epoch < maxEpoch {
		return cacheSizes[epoch]
	}
	return calcCacheSize(epoch)
}

// calcCacheSize calculates the cache size for epoch. The cache size grows linearly,
// however, we always take the highest prime below the linearly growing threshold in order
// to reduce the risk of accidental regularities leading to cyclic behavior.
func calcCacheSize(epoch int) uint64 {
	size := cacheInitBytes + cacheGrowthBytes*uint64(epoch) - hashBytes
	for !new(big.Int).SetUint64(size / hashBytes).ProbablyPrime(1) { // Always accurate for n < 2^64
		size -= 2 * hashBytes
	}
	return size
}

// datasetSize returns the size of the ethash mining dataset that belongs to a certain
// block number.
func datasetSize(block uint64) uint64 {
	epoch := int(block / epochLength)
	if epoch < maxEpoch {
		return datasetSizes[epoch]
	}
	return calcDatasetSize(epoch)
}

// calcDatasetSize calculates the dataset size for epoch. The dataset size grows linearly,
// however, we always take the highest prime below the linearly growing threshold in order
// to reduce the risk of accidental regularities leading to cyclic behavior.
func calcDatasetSize(epoch int) uint64 {
	size := datasetInitBytes + datasetGrowthBytes*uint64(epoch) - mixBytes
	for !new(big.Int).SetUint64(size / mixBytes).ProbablyPrime(1) { // Always accurate for n < 2^64
		size -= 2 * mixBytes
	}
	return size
}

// hasher is a repetitive hasher allowing the same hash data structures to be
// reused between hash runs instead of requiring new ones to be created.
type hasher func(dest []byte, data []byte)

// makeHasher creates a repetitive hasher, allowing the same hash data structures to
// be reused between hash runs instead of requiring new ones to be created. The returned
// function is not thread safe!
func makeHasher(h hash.Hash) hasher {
	// sha3.state supports Read to get the sum, use it to avoid the overhead of Sum.
	// Read alters the state but we reset the hash before every operation.
	type readerHash interface {
		hash.Hash
		Read([]byte) (int, error)
	}
	rh, ok := h.(readerHash)
	if !ok {
		panic("can't find Read method on hash")
	}
	outputLen := rh.Size()
	return func(dest []byte, data []byte) {
		rh.Reset()
		rh.Write(data)
		rh.Read(dest[:outputLen])
	}
}

// seedHash is the seed to use for generating a verification cache and the mining
// dataset.
func seedHash(block uint64) []byte {
	seed := make([]byte, 32)
	if block < epochLength {
		return seed
	}
	keccak256 := makeHasher(sha3.NewLegacyKeccak256())
	for i := 0; i < int(block/epochLength); i++ {
		keccak256(seed, seed)
	}
	return seed
}

// generateCache creates a verification cache of a given size for an input seed.
// The cache production process involves first sequentially filling up 32 MB of
// memory, then performing two passes of Sergio Demian Lerner's RandMemoHash
// algorithm from Strict Memory Hard Hashing Functions (2014). The output is a
// set of 524288 64-byte values.
// This method places the result into dest in machine byte order.
func generateCache(dest []uint32, epoch uint64, seed []byte) {
	// Print some debug logs to allow analysis on low end devices
	logger := log.New("epoch", epoch)

	start := time.Now()
	defer func() {
		elapsed := time.Since(start)

		logFn := logger.Debug
		if elapsed > 3*time.Second {
			logFn = logger.Info
		}
		logFn("Generated ethash verification cache", "elapsed", common.PrettyDuration(elapsed))
	}()
	// Convert our destination slice to a byte buffer
	header := *(*reflect.SliceHeader)(unsafe.Pointer(&dest))
	header.Len *= 4
	header.Cap *= 4
	cache := *(*[]byte)(unsafe.Pointer(&header))

	// Calculate the number of theoretical rows (we'll store in one buffer nonetheless)
	size := uint64(len(cache))
	rows := int(size) / hashBytes

	// Start a monitoring goroutine to report progress on low end devices
	var progress uint32

	done := make(chan struct{})
	defer close(done)

	go func() {
		for {
			select {
			case <-done:
				return
			case <-time.After(3 * time.Second):
				logger.Info("Generating ethash verification cache", "percentage", atomic.LoadUint32(&progress)*100/uint32(rows)/4, "elapsed", common.PrettyDuration(time.Since(start)))
			}
		}
	}()
	// Create a hasher to reuse between invocations
	keccak512 := makeHasher(sha3.NewLegacyKeccak512())

	// Sequentially produce the initial dataset
	keccak512(cache, seed)
	for offset := uint64(hashBytes); offset < size; offset += hashBytes {
		keccak512(cache[offset:], cache[offset-hashBytes:offset])
		atomic.AddUint32(&progress, 1)
	}
	// Use a low-round version of randmemohash
	temp := make([]byte, hashBytes)

	for i := 0; i < cacheRounds; i++ {
		for j := 0; j < rows; j++ {
			var (
				srcOff = ((j - 1 + rows) % rows) * hashBytes
				dstOff = j * hashBytes
				xorOff = (binary.LittleEndian.Uint32(cache[dstOff:]) % uint32(rows)) * hashBytes
			)
			bitutil.XORBytes(temp, cache[srcOff:srcOff+hashBytes], cache[xorOff:xorOff+hashBytes])
			keccak512(cache[dstOff:], temp)

			atomic.AddUint32(&progress, 1)
		}
	}
	// Swap the byte order on big endian systems and return
	if !isLittleEndian() {
		swap(cache)
	}
}

// swap changes the byte order of the buffer assuming a uint32 representation.
func swap(buffer []byte) {
	for i := 0; i < len(buffer); i += 4 {
		binary.BigEndian.PutUint32(buffer[i:], binary.LittleEndian.Uint32(buffer[i:]))
	}
}

// fnv is an algorithm inspired by the FNV hash, which in some cases is used as
// a non-associative substitute for XOR. Note that we multiply the prime with
// the full 32-bit input, in contrast with the FNV-1 spec which multiplies the
// prime with one byte (octet) in turn.
func fnv(a, b uint32) uint32 {
	return a*0x01000193 ^ b
}

// fnvHash mixes in data into mix using the ethash fnv method.
func fnvHash(mix []uint32, data []uint32) {
	for i := 0; i < len(mix); i++ {
		mix[i] = mix[i]*0x01000193 ^ data[i]
	}
}

// generateDatasetItem combines data from 256 pseudorandomly selected cache nodes,
// and hashes that to compute a single dataset node.
func generateDatasetItem(cache []uint32, index uint32, keccak512 hasher) []byte {
	// Calculate the number of theoretical rows (we use one buffer nonetheless)
	rows := uint32(len(cache) / hashWords)

	// Initialize the mix
	mix := make([]byte, hashBytes)

	binary.LittleEndian.PutUint32(mix, cache[(index%rows)*hashWords]^index)
	for i := 1; i < hashWords; i++ {
		binary.LittleEndian.PutUint32(mix[i*4:], cache[(index%rows)*hashWords+uint32(i)])
	}
	keccak512(mix, mix)

	// Convert the mix to uint32s to avoid constant bit shifting
	intMix := make([]uint32, hashWords)
	for i := 0; i < len(intMix); i++ {
		intMix[i] = binary.LittleEndian.Uint32(mix[i*4:])
	}
	// fnv it with a lot of random cache nodes based on index
	for i := uint32(0); i < datasetParents; i++ {
		parent := fnv(index^i, intMix[i%16]) % rows
		fnvHash(intMix, cache[parent*hashWords:])
	}
	// Flatten the uint32 mix into a binary one and return
	for i, val := range intMix {
		binary.LittleEndian.PutUint32(mix[i*4:], val)
	}
	keccak512(mix, mix)
	return mix
}

// generateDataset generates the entire ethash dataset for mining.
// This method places the result into dest in machine byte order.
func generateDataset(dest []uint32, epoch uint64, cache []uint32) {
	// Print some debug logs to allow analysis on low end devices
	logger := log.New("epoch", epoch)

	start := time.Now()
	defer func() {
		elapsed := time.Since(start)

		logFn := logger.Debug
		if elapsed > 3*time.Second {
			logFn = logger.Info
		}
		logFn("Generated ethash verification cache", "elapsed", common.PrettyDuration(elapsed))
	}()

	// Figure out whether the bytes need to be swapped for the machine
	swapped := !isLittleEndian()

	// Convert our destination slice to a byte buffer
	header := *(*reflect.SliceHeader)(unsafe.Pointer(&dest))
	header.Len *= 4
	header.Cap *= 4
	dataset := *(*[]byte)(unsafe.Pointer(&header))

	// Generate the dataset on many goroutines since it takes a while
	threads := runtime.NumCPU()
	size := uint64(len(dataset))

	var pend sync.WaitGroup
	pend.Add(threads)

	var progress uint32
	for i := 0; i < threads; i++ {
		go func(id int) {
			defer pend.Done()

			// Create a hasher to reuse between invocations
			keccak512 := makeHasher(sha3.NewLegacyKeccak512())

			// Calculate the data segment this thread should generate
			batch := uint32((size + hashBytes*uint64(threads) - 1) / (hashBytes * uint64(threads)))
			first := uint32(id) * batch
			limit := first + batch
			if limit > uint32(size/hashBytes) {
				limit = uint32(size / hashBytes)
			}
			// Calculate the dataset segment
			percent := uint32(size / hashBytes / 100)
			for index := first; index < limit; index++ {
				item := generateDatasetItem(cache, index, keccak512)
				if swapped {
					swap(item)
				}
				copy(dataset[index*hashBytes:], item)

				if status := atomic.AddUint32(&progress, 1); status%percent == 0 {
					logger.Info("Generating DAG in progress", "percentage", uint64(status*100)/(size/hashBytes), "elapsed", common.PrettyDuration(time.Since(start)))
				}
			}
		}(i)
	}
	// Wait for all the generators to finish and return
	pend.Wait()
}

// hashimoto aggregates data from the full dataset in order to produce our final
// value for a particular header hash and nonce.
func hashimoto(hash []byte, nonce uint64, size uint64, lookup func(index uint32) []uint32) ([]byte, []byte) {
	// Calculate the number of theoretical rows (we use one buffer nonetheless)
	rows := uint32(size / mixBytes)

	// Combine header+nonce into a 64 byte seed
	seed := make([]byte, 40)
	copy(seed, hash)
	binary.LittleEndian.PutUint64(seed[32:], nonce)

	seed = crypto.Keccak512(seed)
	seedHead := binary.LittleEndian.Uint32(seed)

	// Start the mix with replicated seed
	mix := make([]uint32, mixBytes/4)
	for i := 0; i < len(mix); i++ {
		mix[i] = binary.LittleEndian.Uint32(seed[i%16*4:])
	}
	// Mix in random dataset nodes
	temp := make([]uint32, len(mix))

	for i := 0; i < loopAccesses; i++ {
		parent := fnv(uint32(i)^seedHead, mix[i%len(mix)]) % rows
		for j := uint32(0); j < mixBytes/hashBytes; j++ {
			copy(temp[j*hashWords:], lookup(2*parent+j))
		}
		fnvHash(mix, temp)
	}
	// Compress mix
	for i := 0; i < len(mix); i += 4 {
		mix[i/4] = fnv(fnv(fnv(mix[i], mix[i+1]), mix[i+2]), mix[i+3])
	}
	mix = mix[:len(mix)/4]

	digest := make([]byte, common.HashLength)
	for i, val := range mix {
		binary.LittleEndian.PutUint32(digest[i*4:], val)
	}
	return digest, crypto.Keccak256(append(seed, digest...))
}

// hashimotoLight aggregates data from the full dataset (using only a small
// in-memory cache) in order to produce our final value for a particular header
// hash and nonce.
func hashimotoLight(size uint64, cache []uint32, hash []byte, nonce uint64) ([]byte, []byte) {
	keccak512 := makeHasher(sha3.NewLegacyKeccak512())

	lookup := func(index uint32) []uint32 {
		rawData := generateDatasetItem(cache, index, keccak512)

		data := make([]uint32, len(rawData)/4)
		for i := 0; i < len(data); i++ {
			data[i] = binary.LittleEndian.Uint32(rawData[i*4:])
		}
		return data
	}
	return hashimoto(hash, nonce, size, lookup)
}

// hashimotoFull aggregates data from the full dataset (using the full in-memory
// dataset) in order to produce our final value for a particular header hash and
// nonce.
func hashimotoFull(dataset []uint32, hash []byte, nonce uint64) ([]byte, []byte) {
	lookup := func(index uint32) []uint32 {
		offset := index * hashWords
		return dataset[offset : offset+hashWords]
	}
	return hashimoto(hash, nonce, uint64(len(dataset))*4, lookup)
}

const maxEpoch = 2048

// datasetSizes is a lookup table for the ethash dataset size for the first 2048
// epochs (i.e. 61440000 blocks).
var datasetSizes = [maxEpoch]uint64{
	1073739904, 1082130304, 1090514816, 1098906752, 1107293056,
	1115684224, 1124070016, 1132461952, 1140849536, 1149232768,
	1157627776, 1166013824, 1174404736, 1182786944, 1191180416,
	1199568512, 1207958912, 1216345216, 1224732032, 1233124736,
	1241513344, 1249902464, 1258290304, 1266673792, 1275067264,
	1283453312, 1291844992, 1300234112, 1308619904, 1317010048,
	1325397376, 1333787776, 1342176128, 1350561664, 1358954368,
	1367339392, 1375731584, 1384118144, 1392507008, 1400897408,
	1409284736, 1417673344, 1426062464, 1434451072, 1442839168,
	1451229056, 1459615616, 1468006016, 1476394112, 1484782976,
	1493171584, 1501559168, 1509948032, 1518337664, 1526726528,
	1535114624, 1543503488, 1551892096, 1560278656, 1568669056,
	1577056384, 1585446272, 1593831296, 1602219392, 1610610304,
	1619000192, 1627386752, 1635773824, 1644164224, 1652555648,
	1660943488, 1669332608, 1677721216, 1686109312, 1694497664,
	1702886272, 1711274624, 1719661184, 1728047744, 1736434816,
	1744829056, 1753218944, 1761606272, 1769995904, 1778382464,
	1786772864, 1795157888, 1803550592, 1811937664, 1820327552,
	1828711552, 1837102976, 1845488768, 1853879936, 1862269312,
	1870656896, 1879048064, 1887431552, 1895825024, 1904212096,
	1912601216, 1920988544, 1929379456, 1937765504, 1946156672,
	1954543232, 1962932096, 1971321728, 1979707264, 1988093056,
	1996487552, 2004874624, 2013262208, 2021653888, 2030039936,
	2038430848, 2046819968, 2055208576, 2063596672, 2071981952,
	2080373632, 2088762752, 2097149056, 2105539712, 2113928576,
	2122315136, 2130700672, 2139092608, 2147483264, 2155872128,
	2164257664, 2172642176, 2181035392, 2189426048, 2197814912,
	2206203008, 2214587264, 2222979712, 2231367808, 2239758208,
	2248145024, 2256527744, 2264922752, 2273312128, 2281701248,
	2290086272, 2298476672, 2306867072, 2315251072, 2323639168,
	2332032128, 2340420224, 2348808064, 2357196416, 2365580416,
	2373966976, 2382363008, 2390748544, 2399139968, 2407530368,
	2415918976, 2424307328, 2432695424, 2441084288, 2449472384,
	2457861248, 2466247808, 2474637184, 2483026816, 2491414144,
	2499803776, 2508191872, 2516582272, 2524970368, 2533359232,
	2541743488, 2550134144, 2558525056, 2566913408, 2575301504,
	2583686528, 2592073856, 2600467328, 2608856192, 2617240448,
	2625631616, 2634022016, 2642407552, 2650796416, 2659188352,
	2667574912, 2675965312, 2684352896, 2692738688, 2701130624,
	2709518464, 2717907328, 2726293376, 2734685056, 2743073152,
	2751462016, 2759851648, 2768232832, 2776625536, 2785017728,
	2793401984, 2801794432, 2810182016, 2818571648, 2826959488,
	2835349376, 2843734144, 2852121472, 2860514432, 2868900992,
	2877286784, 2885676928, 2894069632, 2902451584, 2910843008,
	2919234688, 2927622784, 2936011648, 2944400768, 2952789376,
	2961177728, 2969565568, 2977951616, 2986338944, 2994731392,
	3003120256, 3011508352, 3019895936, 3028287104, 3036675968,
	3045063808, 3053452928, 3061837696, 3070228352, 3078615424,
	3087003776, 3095394944, 3103782272, 3112173184, 3120562048,
	3128944768, 3137339264, 3145725056, 3154109312, 3162505088,
	3170893184, 3179280256, 3187669376, 3196056704, 3204445568,
	3212836736, 3221224064, 3229612928, 3238002304, 3246391168,
	3254778496, 3263165824, 3271556224, 3279944576, 3288332416,
	3296719232, 3305110912, 3313500032, 3321887104, 3330273152,
	3338658944, 3347053184, 3355440512, 3363827072, 3372220288,
	3380608384, 3388997504, 3397384576, 3405774208, 3414163072,
	3422551936, 3430937984, 3439328384, 3447714176, 3456104576,
	3464493952, 3472883584, 3481268864, 3489655168, 3498048896,
	3506434432, 3514826368, 3523213952, 3531603584, 3539987072,
	3548380288, 3556763264, 3565157248, 3573545344, 3581934464,
	3590324096, 3598712704, 3607098752, 3615488384, 3623877248,
	3632265856, 3640646528, 3649043584, 3657430144, 3665821568,
	3674207872, 3682597504, 3690984832, 3699367808, 3707764352,
	3716152448, 3724541056, 3732925568, 3741318016, 3749706368,
	3758091136, 3766481536, 3774872704, 3783260032, 3791650432,
	3800036224, 3808427648, 3816815488, 3825204608, 3833592704,
	3841981568, 3850370432, 3858755968, 3867147904, 3875536256,
	3883920512, 3892313728, 3900702592, 3909087872, 3917478784,
	3925868416, 3934256512, 3942645376, 3951032192, 3959422336,
	3967809152, 3976200064, 3984588416, 3992974976, 4001363584,
	4009751168, 4018141312, 4026530432, 4034911616, 4043308928,
	4051695488, 4060084352, 4068472448, 4076862848, 4085249408,
	4093640576, 4102028416, 4110413696, 4118805632, 4127194496,
	4135583104, 4143971968, 4152360832, 4160746112, 4169135744,
	4177525888, 4185912704, 4194303616, 4202691968, 4211076736,
	4219463552, 4227855488, 4236246656, 4244633728, 4253022848,
	4261412224, 4269799808, 4278184832, 4286578048, 4294962304,
	4303349632, 4311743104, 4320130432, 4328521088, 4336909184,
	4345295488, 4353687424, 4362073472, 4370458496, 4378852736,
	4387238528, 4395630208, 4404019072, 4412407424, 4420790656,
	4429182848, 4437571456, 4445962112, 4454344064, 4462738048,
	4471119232, 4479516544, 4487904128, 4496289664, 4504682368,
	4513068416, 4521459584, 4529846144, 4538232704, 4546619776,
	4555010176, 4563402112, 4571790208, 4580174464, 4588567936,
	4596957056, 4605344896, 4613734016, 4622119808, 4630511488,
	4638898816, 4647287936, 4655675264, 4664065664, 4672451968,
	4680842624, 4689231488, 4697620352, 4706007424, 4714397056,
	4722786176, 4731173248, 4739562368, 4747951744, 4756340608,
	4764727936, 4773114496, 4781504384, 4789894784, 4798283648,
	4806667648, 4815059584, 4823449472, 4831835776, 4840226176,
	4848612224, 4857003392, 4865391488, 4873780096, 4882169728,
	4890557312, 4898946944, 4907333248, 4915722368, 4924110976,
	4932499328, 4940889728, 4949276032, 4957666432, 4966054784,
	4974438016, 4982831488, 4991221376, 4999607168, 5007998848,
	5016386432, 5024763776, 5033164672, 5041544576, 5049941888,
	5058329728, 5066717056, 5075107456, 5083494272, 5091883904,
	5100273536, 5108662144, 5117048192, 5125436032, 5133827456,
	5142215296, 5150605184, 5158993024, 5167382144, 5175769472,
	5184157568, 5192543872, 5200936064, 5209324928, 5217711232,
	5226102656, 5234490496, 5242877312, 5251263872, 5259654016,
	5268040832, 5276434304, 5284819328, 5293209728, 5301598592,
	5309986688, 5318374784, 5326764416, 5335151488, 5343542144,
	5351929472, 5360319872, 5368706944, 5377096576, 5385484928,
	5393871232, 5402263424, 5410650496, 5419040384, 5427426944,
	5435816576, 5444205952, 5452594816, 5460981376, 5469367936,
	5477760896, 5486148736, 5494536832, 5502925952, 5511315328,
	5519703424, 5528089984, 5536481152, 5544869504, 5553256064,
	5561645696, 5570032768, 5578423936, 5586811264, 5595193216,
	5603585408, 5611972736, 5620366208, 5628750464, 5637143936,
	5645528192, 5653921408, 5662310272, 5670694784, 5679082624,
	5687474048, 5695864448, 5704251008, 5712641408, 5721030272,
	5729416832, 5737806208, 5746194304, 5754583936, 5762969984,
	5771358592, 5779748224, 5788137856, 5796527488, 5804911232,
	5813300608, 5821692544, 5830082176, 5838468992, 5846855552,
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	16928207488, 16936592768, 16944987776, 16953375616, 16961763968,
	16970152832, 16978540928, 16986929536, 16995319168, 17003704448,
	17012096896, 17020481152, 17028870784, 17037262208, 17045649536,
	17054039936, 17062426496, 17070814336, 17079205504, 17087592064,
	17095978112, 17104369024, 17112759424, 17121147776, 17129536384,
	17137926016, 17146314368, 17154700928, 17163089792, 17171480192,
	17179864192, 17188256896, 17196644992, 17205033856, 17213423488,
	17221811072, 17230198912, 17238588032, 17246976896, 17255360384,
	17263754624, 17272143232, 17280530048, 17288918912, 17297309312,
	17305696384, 17314085504, 17322475136, 17330863744, 17339252096,
	17347640192, 17356026496, 17364413824, 17372796544, 17381190016,
	17389583488, 17397972608, 17406360704, 17414748544, 17423135872,
	17431527296, 17439915904, 17448303232, 17456691584, 17465081728,
	17473468288, 17481857408, 17490247552, 17498635904, 17507022464,
	17515409024, 17523801728, 17532189824, 17540577664, 17548966016,
	17557353344, 17565741184, 17574131584, 17582519168, 17590907008,
	17599296128, 17607687808, 17616076672, 17624455808, 17632852352,
	17641238656, 17649630848, 17658018944, 17666403968, 17674794112,
	17683178368, 17691573376, 17699962496, 17708350592, 17716739968,
	17725126528, 17733517184, 17741898112, 17750293888, 17758673024,
	17767070336, 17775458432, 17783848832, 17792236928, 17800625536,
	17809012352, 17817402752, 17825785984, 17834178944, 17842563968,
	17850955648, 17859344512, 17867732864, 17876119424, 17884511872,
	17892900224, 17901287296, 17909677696, 17918058112, 17926451072,
	17934843776, 17943230848, 17951609216, 17960008576, 17968397696,
	17976784256, 17985175424, 17993564032, 18001952128, 18010339712,
	18018728576, 18027116672, 18035503232, 18043894144, 18052283264,
	18060672128, 18069056384, 18077449856, 18085837184, 18094225792,
	18102613376, 18111004544, 18119388544, 18127781248, 18136170368,
	18144558976, 18152947328, 18161336192, 18169724288, 18178108544,
	18186498944, 18194886784, 18203275648, 18211666048, 18220048768,
	18228444544, 18236833408, 18245220736}

// cacheSizes is a lookup table for the ethash verification cache size for the
// first 2048 epochs (i.e. 61440000 blocks).
var cacheSizes = [maxEpoch]uint64{
	16776896, 16907456, 17039296, 17170112, 17301056, 17432512, 17563072,
	17693888, 17824192, 17955904, 18087488, 18218176, 18349504, 18481088,
	18611392, 18742336, 18874304, 19004224, 19135936, 19267264, 19398208,
	19529408, 19660096, 19791424, 19922752, 20053952, 20184896, 20315968,
	20446912, 20576576, 20709184, 20840384, 20971072, 21102272, 21233216,
	21364544, 21494848, 21626816, 21757376, 21887552, 22019392, 22151104,
	22281536, 22412224, 22543936, 22675264, 22806464, 22935872, 23068096,
	23198272, 23330752, 23459008, 23592512, 23723968, 23854912, 23986112,
	24116672, 24247616, 24378688, 24509504, 24640832, 24772544, 24903488,
	25034432, 25165376, 25296704, 25427392, 25558592, 25690048, 25820096,
	25951936, 26081728, 26214208, 26345024, 26476096, 26606656, 26737472,
	26869184, 26998208, 27131584, 27262528, 27393728, 27523904, 27655744,
	27786688, 27917888, 28049344, 28179904, 28311488, 28441792, 28573504,
	28700864, 28835648, 28966208, 29096768, 29228608, 29359808, 29490752,
	29621824, 29752256, 29882816, 30014912, 30144448, 30273728, 30406976,
	30538432, 30670784, 30799936, 30932672, 31063744, 31195072, 31325248,
	31456192, 31588288, 31719232, 31850432, 31981504, 32110784, 32243392,
	32372672, 32505664, 32636608, 32767808, 32897344, 33029824, 33160768,
	33289664, 33423296, 33554368, 33683648, 33816512, 33947456, 34076992,
	34208704, 34340032, 34471744, 34600256, 34734016, 34864576, 34993984,
	35127104, 35258176, 35386688, 35518528, 35650624, 35782336, 35910976,
	36044608, 36175808, 36305728, 36436672, 36568384, 36699968, 36830656,
	36961984, 37093312, 37223488, 37355072, 37486528, 37617472, 37747904,
	37879232, 38009792, 38141888, 38272448, 38403392, 38535104, 38660672,
	38795584, 38925632, 39059264, 39190336, 39320768, 39452096, 39581632,
	39713984, 39844928, 39974848, 40107968, 40238144, 40367168, 40500032,
	40631744, 40762816, 40894144, 41023552, 41155904, 41286208, 41418304,
	41547712, 41680448, 41811904, 41942848, 42073792, 42204992, 42334912,
	42467008, 42597824, 42729152, 42860096, 42991552, 43122368, 43253696,
	43382848, 43515712, 43646912, 43777088, 43907648, 44039104, 44170432,
	44302144, 44433344, 44564288, 44694976, 44825152, 44956864, 45088448,
	45219008, 45350464, 45481024, 45612608, 45744064, 45874496, 46006208,
	46136768, 46267712, 46399424, 46529344, 46660672, 46791488, 46923328,
	47053504, 47185856, 47316928, 47447872, 47579072, 47710144, 47839936,
	47971648, 48103232, 48234176, 48365248, 48496192, 48627136, 48757312,
	48889664, 49020736, 49149248, 49283008, 49413824, 49545152, 49675712,
	49807168, 49938368, 50069056, 50200256, 50331584, 50462656, 50593472,
	50724032, 50853952, 50986048, 51117632, 51248576, 51379904, 51510848,
	51641792, 51773248, 51903296, 52035136, 52164032, 52297664, 52427968,
	52557376, 52690112, 52821952, 52952896, 53081536, 53213504, 53344576,
	53475776, 53608384, 53738816, 53870528, 54000832, 54131776, 54263744,
	54394688, 54525248, 54655936, 54787904, 54918592, 55049152, 55181248,
	55312064, 55442752, 55574336, 55705024, 55836224, 55967168, 56097856,
	56228672, 56358592, 56490176, 56621888, 56753728, 56884928, 57015488,
	57146816, 57278272, 57409216, 57540416, 57671104, 57802432, 57933632,
	58064576, 58195264, 58326976, 58457408, 58588864, 58720192, 58849984,
	58981696, 59113024, 59243456, 59375552, 59506624, 59637568, 59768512,
	59897792, 60030016, 60161984, 60293056, 60423872, 60554432, 60683968,
	60817216, 60948032, 61079488, 61209664, 61341376, 61471936, 61602752,
	61733696, 61865792, 61996736, 62127808, 62259136, 62389568, 62520512,
	62651584, 62781632, 62910784, 63045056, 63176128, 63307072, 63438656,
	63569216, 63700928, 63831616, 63960896, 64093888, 64225088, 64355392,
	64486976, 64617664, 64748608, 64879424, 65009216, 65142464, 65273792,
	65402816, 65535424, 65666752, 65797696, 65927744, 66060224, 66191296,
	66321344, 66453056, 66584384, 66715328, 66846656, 66977728, 67108672,
	67239104, 67370432, 67501888, 67631296, 67763776, 67895104, 68026304,
	68157248, 68287936, 68419264, 68548288, 68681408, 68811968, 68942912,
	69074624, 69205568, 69337024, 69467584, 69599168, 69729472, 69861184,
	69989824, 70122944, 70253888, 70385344, 70515904, 70647232, 70778816,
	70907968, 71040832, 71171648, 71303104, 71432512, 71564992, 71695168,
	71826368, 71958464, 72089536, 72219712, 72350144, 72482624, 72613568,
	72744512, 72875584, 73006144, 73138112, 73268672, 73400128, 73530944,
	73662272, 73793344, 73924544, 74055104, 74185792, 74316992, 74448832,
	74579392, 74710976, 74841664, 74972864, 75102784, 75233344, 75364544,
	75497024, 75627584, 75759296, 75890624, 76021696, 76152256, 76283072,
	76414144, 76545856, 76676672, 76806976, 76937792, 77070016, 77200832,
	77331392, 77462464, 77593664, 77725376, 77856448, 77987776, 78118336,
	78249664, 78380992, 78511424, 78642496, 78773056, 78905152, 79033664,
	79166656, 79297472, 79429568, 79560512, 79690816, 79822784, 79953472,
	80084672, 80214208, 80346944, 80477632, 80608576, 80740288, 80870848,
	81002048, 81133504, 81264448, 81395648, 81525952, 81657536, 81786304,
	81919808, 82050112, 82181312, 82311616, 82443968, 82573376, 82705984,
	82835776, 82967744, 83096768, 83230528, 83359552, 83491264, 83622464,
	83753536, 83886016, 84015296, 84147776, 84277184, 84409792, 84540608,
	84672064, 84803008, 84934336, 85065152, 85193792, 85326784, 85458496,
	85589312, 85721024, 85851968, 85982656, 86112448, 86244416, 86370112,
	86506688, 86637632, 86769344, 86900672, 87031744, 87162304, 87293632,
	87424576, 87555392, 87687104, 87816896, 87947968, 88079168, 88211264,
	88341824, 88473152, 88603712, 88735424, 88862912, 88996672, 89128384,
	89259712, 89390272, 89521984, 89652544, 89783872, 89914816, 90045376,
	90177088, 90307904, 90438848, 90569152, 90700096, 90832832, 90963776,
	91093696, 91223744, 91356992, 91486784, 91618496, 91749824, 91880384,
	92012224, 92143552, 92273344, 92405696, 92536768, 92666432, 92798912,
	92926016, 93060544, 93192128, 93322816, 93453632, 93583936, 93715136,
	93845056, 93977792, 94109504, 94240448, 94371776, 94501184, 94632896,
	94764224, 94895552, 95023424, 95158208, 95287744, 95420224, 95550016,
	95681216, 95811904, 95943872, 96075328, 96203584, 96337856, 96468544,
	96599744, 96731072, 96860992, 96992576, 97124288, 97254848, 97385536,
	97517248, 97647808, 97779392, 97910464, 98041408, 98172608, 98303168,
	98434496, 98565568, 98696768, 98827328, 98958784, 99089728, 99220928,
	99352384, 99482816, 99614272, 99745472, 99876416, 100007104,
	100138048, 100267072, 100401088, 100529984, 100662592, 100791872,
	100925248, 101056064, 101187392, 101317952, 101449408, 101580608,
	101711296, 101841728, 101973824, 102104896, 102235712, 102366016,
	102498112, 102628672, 102760384, 102890432, 103021888, 103153472,
	103284032, 103415744, 103545152, 103677248, 103808576, 103939648,
	104070976, 104201792, 104332736, 104462528, 104594752, 104725952,
	104854592, 104988608, 105118912, 105247808, 105381184, 105511232,
	105643072, 105774784, 105903296, 106037056, 106167872, 106298944,
	106429504, 106561472, 106691392, 106822592, 106954304, 107085376,
	107216576, 107346368, 107478464, 107609792, 107739712, 107872192,
	108003136, 108131392, 108265408, 108396224, 108527168, 108657344,
	108789568, 108920384, 109049792, 109182272, 109312576, 109444928,
	109572928, 109706944, 109837888, 109969088, 110099648, 110230976,
	110362432, 110492992, 110624704, 110755264, 110886208, 111017408,
	111148864, 111279296, 111410752, 111541952, 111673024, 111803456,
	111933632, 112066496, 112196416, 112328512, 112457792, 112590784,
	112715968, 112852672, 112983616, 113114944, 113244224, 113376448,
	113505472, 113639104, 113770304, 113901376, 114031552, 114163264,
	114294592, 114425536, 114556864, 114687424, 114818624, 114948544,
	115080512, 115212224, 115343296, 115473472, 115605184, 115736128,
	115867072, 115997248, 116128576, 116260288, 116391488, 116522944,
	116652992, 116784704, 116915648, 117046208, 117178304, 117308608,
	117440192, 117569728, 117701824, 117833024, 117964096, 118094656,
	118225984, 118357312, 118489024, 118617536, 118749632, 118882112,
	119012416, 119144384, 119275328, 119406016, 119537344, 119668672,
	119798464, 119928896, 120061376, 120192832, 120321728, 120454336,
	120584512, 120716608, 120848192, 120979136, 121109056, 121241408,
	121372352, 121502912, 121634752, 121764416, 121895744, 122027072,
	122157632, 122289088, 122421184, 122550592, 122682944, 122813888,
	122945344, 123075776, 123207488, 123338048, 123468736, 123600704,
	123731264, 123861952, 123993664, 124124608, 124256192, 124386368,
	124518208, 124649024, 124778048, 124911296, 125041088, 125173696,
	125303744, 125432896, 125566912, 125696576, 125829056, 125958592,
	126090304, 126221248, 126352832, 126483776, 126615232, 126746432,
	126876608, 127008704, 127139392, 127270336, 127401152, 127532224,
	127663552, 127794752, 127925696, 128055232, 128188096, 128319424,
	128449856, 128581312, 128712256, 128843584, 128973632, 129103808,
	129236288, 129365696, 129498944, 129629888, 129760832, 129892288,
	130023104, 130154048, 130283968, 130416448, 130547008, 130678336,
	130807616, 130939456, 131071552, 131202112, 131331776, 131464384,
	131594048, 131727296, 131858368, 131987392, 132120256, 132250816,
	132382528, 132513728, 132644672, 132774976, 132905792, 133038016,
	133168832, 133299392, 133429312, 133562048, 133692992, 133823296,
	133954624, 134086336, 134217152, 134348608, 134479808, 134607296,
	134741056, 134872384, 135002944, 135134144, 135265472, 135396544,
	135527872, 135659072, 135787712, 135921472, 136052416, 136182848,
	136313792, 136444864, 136576448, 136707904, 136837952, 136970048,
	137099584, 137232064, 137363392, 137494208, 137625536, 137755712,
	137887424, 138018368, 138149824, 138280256, 138411584, 138539584,
	138672832, 138804928, 138936128, 139066688, 139196864, 139328704,
	139460032, 139590208, 139721024, 139852864, 139984576, 140115776,
	140245696, 140376512, 140508352, 140640064, 140769856, 140902336,
	141032768, 141162688, 141294016, 141426496, 141556544, 141687488,
	141819584, 141949888, 142080448, 142212544, 142342336, 142474432,
	142606144, 142736192, 142868288, 142997824, 143129408, 143258944,
	143392448, 143523136, 143653696, 143785024, 143916992, 144045632,
	144177856, 144309184, 144440768, 144570688, 144701888, 144832448,
	144965056, 145096384, 145227584, 145358656, 145489856, 145620928,
	145751488, 145883072, 146011456, 146144704, 146275264, 146407232,
	146538176, 146668736, 146800448, 146931392, 147062336, 147193664,
	147324224, 147455936, 147586624, 147717056, 147848768, 147979456,
	148110784, 148242368, 148373312, 148503232, 148635584, 148766144,
	148897088, 149028416, 149159488, 149290688, 149420224, 149551552,
	149683136, 149814976, 149943616, 150076352, 150208064, 150338624,
	150470464, 150600256, 150732224, 150862784, 150993088, 151125952,
	151254976, 151388096, 151519168, 151649728, 151778752, 151911104,
	152042944, 152174144, 152304704, 152435648, 152567488, 152698816,
	152828992, 152960576, 153091648, 153222976, 153353792, 153484096,
	153616192, 153747008, 153878336, 154008256, 154139968, 154270912,
	154402624, 154533824, 154663616, 154795712, 154926272, 155057984,
	155188928, 155319872, 155450816, 155580608, 155712064, 155843392,
	155971136, 156106688, 156237376, 156367424, 156499264, 156630976,
	156761536, 156892352, 157024064, 157155008, 157284416, 157415872,
	157545536, 157677248, 157810496, 157938112, 158071744, 158203328,
	158334656, 158464832, 158596288, 158727616, 158858048, 158988992,
	159121216, 159252416, 159381568, 159513152, 159645632, 159776192,
	159906496, 160038464, 160169536, 160300352, 160430656, 160563008,
	160693952, 160822208, 160956352, 161086784, 161217344, 161349184,
	161480512, 161611456, 161742272, 161873216, 162002752, 162135872,
	162266432, 162397888, 162529216, 162660032, 162790976, 162922048,
	163052096, 163184576, 163314752, 163446592, 163577408, 163707968,
	163839296, 163969984, 164100928, 164233024, 164364224, 164494912,
	164625856, 164756672, 164887616, 165019072, 165150016, 165280064,
	165412672, 165543104, 165674944, 165805888, 165936832, 166067648,
	166198336, 166330048, 166461248, 166591552, 166722496, 166854208,
	166985408, 167116736, 167246656, 167378368, 167508416, 167641024,
	167771584, 167903168, 168034112, 168164032, 168295744, 168427456,
	168557632, 168688448, 168819136, 168951616, 169082176, 169213504,
	169344832, 169475648, 169605952, 169738048, 169866304, 169999552,
	170131264, 170262464, 170393536, 170524352, 170655424, 170782016,
	170917696, 171048896, 171179072, 171310784, 171439936, 171573184,
	171702976, 171835072, 171966272, 172097216, 172228288, 172359232,
	172489664, 172621376, 172747712, 172883264, 173014208, 173144512,
	173275072, 173407424, 173539136, 173669696, 173800768, 173931712,
	174063424, 174193472, 174325696, 174455744, 174586816, 174718912,
	174849728, 174977728, 175109696, 175242688, 175374272, 175504832,
	175636288, 175765696, 175898432, 176028992, 176159936, 176291264,
	176422592, 176552512, 176684864, 176815424, 176946496, 177076544,
	177209152, 177340096, 177470528, 177600704, 177731648, 177864256,
	177994816, 178126528, 178257472, 178387648, 178518464, 178650176,
	178781888, 178912064, 179044288, 179174848, 179305024, 179436736,
	179568448, 179698496, 179830208, 179960512, 180092608, 180223808,
	180354752, 180485696, 180617152, 180748096, 180877504, 181009984,
	181139264, 181272512, 181402688, 181532608, 181663168, 181795136,
	181926592, 182057536, 182190016, 182320192, 182451904, 182582336,
	182713792, 182843072, 182976064, 183107264, 183237056, 183368384,
	183494848, 183631424, 183762752, 183893824, 184024768, 184154816,
	184286656, 184417984, 184548928, 184680128, 184810816, 184941248,
	185072704, 185203904, 185335616, 185465408, 185596352, 185727296,
	185859904, 185989696, 186121664, 186252992, 186383552, 186514112,
	186645952, 186777152, 186907328, 187037504, 187170112, 187301824,
	187429184, 187562048, 187693504, 187825472, 187957184, 188087104,
	188218304, 188349376, 188481344, 188609728, 188743616, 188874304,
	189005248, 189136448, 189265088, 189396544, 189528128, 189660992,
	189791936, 189923264, 190054208, 190182848, 190315072, 190447424,
	190577984, 190709312, 190840768, 190971328, 191102656, 191233472,
	191364032, 191495872, 191626816, 191758016, 191888192, 192020288,
	192148928, 192282176, 192413504, 192542528, 192674752, 192805952,
	192937792, 193068608, 193198912, 193330496, 193462208, 193592384,
	193723456, 193854272, 193985984, 194116672, 194247232, 194379712,
	194508352, 194641856, 194772544, 194900672, 195035072, 195166016,
	195296704, 195428032, 195558592, 195690304, 195818176, 195952576,
	196083392, 196214336, 196345792, 196476736, 196607552, 196739008,
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