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Ethash

This spec is REVISION 23. Whenever you substantively (ie. not clarifications) update the algorithm, please update the revision number in this sentence. Also, in all implementations please include a spec revision number

Ethash is the planned PoW algorithm for Ethereum 1.0. It is the latest version of Dagger-Hashimoto, although it can no longer appropriately be called that since many of the original features of both algorithms have been drastically changed in the last month of research and development. See https://github.com/ethereum/wiki/wiki/Dagger-Hashimoto for the original version.

The general route that the algorithm takes is as follows:

  1. There exists a seed which can be computed for each block by scanning through the block headers up until that point.
  2. From the seed, one can compute a 16 MB pseudorandom cache. Light clients store the cache.
  3. From the cache, we can generate a 1 GB dataset, with the property that each item in the dataset depends on only a small number of items from the cache. Full clients and miners store the dataset. The dataset grows linearly with time.
  4. Mining involves grabbing random slices of the dataset and hashing them together. Verification can be done with low memory by using the cache to regenerate the specific pieces of the dataset that you need, so you only need to store the cache.

The large dataset is updated once every 30000 blocks, so the vast majority of a miner's effort will be reading the dataset, not making changes to it.

See https://github.com/ethereum/wiki/wiki/Ethash-Design-Rationale for design rationale considerations for this algorithm.

Definitions

We employ the following definitions:

WORD_BYTES = 4                    # bytes in word
DATASET_BYTES_INIT = 2**30        # bytes in dataset at genesis
DATASET_BYTES_GROWTH = 2**23      # dataset growth per epoch
CACHE_BYTES_INIT = 2**24          # bytes in dataset at genesis
CACHE_BYTES_GROWTH = 2**17        # cache growth per epoch
CACHE_MULTIPLIER=1024             # Size of the DAG relative to the cache
EPOCH_LENGTH = 30000              # blocks per epoch
MIX_BYTES = 128                   # width of mix
HASH_BYTES = 64                   # hash length in bytes
DATASET_PARENTS = 256             # number of parents of each dataset element
CACHE_ROUNDS = 3                  # number of rounds in cache production
ACCESSES = 64                     # number of accesses in hashimoto loop

Parameters

The parameters for Ethash's cache and dataset depend on the block number. The cache size and dataset size both grow 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.

def get_cache_size(block_number):
    sz = CACHE_BYTES_INIT + CACHE_BYTES_GROWTH * (block_number // EPOCH_LENGTH)
    sz -= HASH_BYTES
    while not isprime(sz / HASH_BYTES):
        sz -= 2 * HASH_BYTES
    return sz

def get_full_size(block_number):
    sz = DATASET_BYTES_INIT + DATASET_BYTES_GROWTH * (block_number // EPOCH_LENGTH)
    sz -= MIX_BYTES
    while not isprime(sz / MIX_BYTES):
        sz -= 2 * MIX_BYTES
    return sz

Tables of dataset and cache size values are provided in the appendix.

Cache Generation

Now, we specify the function for producing a cache:

def mkcache(cache_size, seed):
    n = cache_size // HASH_BYTES

    # Sequentially produce the initial dataset
    o = [sha3_512(seed)]
    for i in range(1, n):
        o.append(sha3_512(o[-1]))

    # Use a low-round version of randmemohash
    for _ in range(CACHE_ROUNDS):
        for i in range(n):
            v = o[i][0] % n
            o[i] = sha3_512(map(xor, o[(i-1+n) % n], o[v]))

    return o

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.

Data aggregation function

We use an algorithm inspired by the FNV hash in some cases 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.

FNV_PRIME = 0x01000193

def fnv(v1, v2):
    return (v1 * FNV_PRIME ^ v2) % 2**32

Full dataset calculation

Each 64-byte item in the full 1 GB dataset is computed as follows:

def calc_dataset_item(cache, i):
    n = len(cache)
    r = HASH_BYTES // WORD_BYTES
    # initialize the mix
    mix = copy.copy(cache[i % n])
    mix[0] ^= i
    mix = sha3_512(mix)
    # fnv it with a lot of random cache nodes based on i
    for j in range(DATASET_PARENTS):
        cache_index = fnv(i ^ j, mix[j % r])
        mix = map(fnv, mix, cache[cache_index % n])
    return sha3_512(mix)

Essentially, we combine data from 256 pseudorandomly selected cache nodes, and hash that to compute the dataset node. The entire dataset is then generated by:

def calc_dataset(full_size, cache):
    return [calc_dataset_item(cache, i) for i in range(full_size // HASH_BYTES)]

Main Loop

Now, we specify the main "hashimoto"-like loop, where we aggregate data from the full dataset in order to produce our final value for a particular header and nonce. In the code below, header represents the SHA3-256 hash of the RLP representation of a truncated block header, that is, of a header excluding the fields mixHash and nonce. nonce is the eight bytes of a 64 bit unsigned integer in big-endian order. So nonce[::-1] is the eight-byte little-endian representation of that value:

def hashimoto(header, nonce, full_size, dataset_lookup):
    n = full_size / HASH_BYTES
    w = MIX_BYTES // WORD_BYTES
    mixhashes = MIX_BYTES / HASH_BYTES
    # combine header+nonce into a 64 byte seed
    s = sha3_512(header + nonce[::-1])
    # start the mix with replicated s
    mix = []
    for _ in range(MIX_BYTES / HASH_BYTES):
        mix.extend(s)
    # mix in random dataset nodes
    for i in range(ACCESSES):
        p = fnv(i ^ s[0], mix[i % w]) % (n // mixhashes) * mixhashes
        newdata = []
        for j in range(MIX_BYTES / HASH_BYTES):
            newdata.extend(dataset_lookup(p + j))
        mix = map(fnv, mix, newdata)
    # compress mix
    cmix = []
    for i in range(0, len(mix), 4):
        cmix.append(fnv(fnv(fnv(mix[i], mix[i+1]), mix[i+2]), mix[i+3]))
    return {
        "mix digest": serialize_hash(cmix),
        "result": serialize_hash(sha3_256(s+cmix))
    }

def hashimoto_light(full_size, cache, header, nonce):
    return hashimoto(header, nonce, full_size, lambda x: calc_dataset_item(cache, x))

def hashimoto_full(full_size, dataset, header, nonce):
    return hashimoto(header, nonce, full_size, lambda x: dataset[x])

Essentially, we maintain a "mix" 128 bytes wide, and repeatedly sequentially fetch 128 bytes from the full dataset and use the fnv function to combine it with the mix. 128 bytes of sequential access are used so that each round of the algorithm always fetches a full page from RAM, minimizing translation lookaside buffer misses which ASICs would theoretically be able to avoid.

If the output of this algorithm is below the desired target, then the nonce is valid. Note that the extra application of sha3_256 at the end ensures that there exists an intermediate nonce which can be provided to prove that at least a small amount of work was done; this quick outer PoW verification can be used for anti-DDoS purposes. It also serves to provide statistical assurance that the result is an unbiased, 256 bit number.

Mining

The mining algorithm is defined as follows:

def mine(full_size, dataset, header, difficulty):
    target = zpad(encode_int(2**256 // difficulty), 64)[::-1]
    from random import randint
    nonce = randint(0, 2**64)
    while hashimoto_full(full_size, dataset, header, nonce) > target:
        nonce = (nonce + 1) % 2**64
    return nonce

Defining the Seed Hash

In order to compute the seed hash that would be used to mine on top of a given block, we use the following algorithm:

 def get_seedhash(block):
     s = '\x00' * 32
     for i in range(block.number // EPOCH_LENGTH):
         s = serialize_hash(sha3_256(s))
     return s

Note that for smooth mining and verifying, we recommend pre-computing future seedhashes and datasets in a separate thread.

Appendix

The following code should be prepended if you are interested in running the above python spec as code.

import sha3, copy

# Assumes little endian bit ordering (same as Intel architectures)
def decode_int(s):
    return int(s[::-1].encode('hex'), 16) if s else 0

def encode_int(s):
    a = "%x" % s
    return '' if s == 0 else ('0' * (len(a) % 2) + a).decode('hex')[::-1]

def zpad(s, length):
    return s + '\x00' * max(0, length - len(s))

def serialize_hash(h):
    return ''.join([zpad(encode_int(x), 4) for x in h])
  
def deserialize_hash(h):
    return [decode_int(h[i:i+WORD_BYTES]) for i in range(0, len(h), WORD_BYTES)]
  
def hash_words(h, sz, x):
    if isinstance(x, list):
        x = serialize_hash(x)
    y = h(x)
    return deserialize_hash(y)

def serialize_cache(ds):
    return ''.join([serialize_hash(h) for h in ds])
  
serialize_dataset = serialize_cache

# sha3 hash function, outputs 64 bytes
def sha3_512(x):
    return hash_words(lambda v: sha3.sha3_512(v).digest(), 64, x)

def sha3_256(x):
    return hash_words(lambda v: sha3.sha3_256(v).digest(), 32, x)

def xor(a, b):
    return a ^ b

def isprime(x):
    for i in range(2, int(x**0.5)):
         if x % i == 0:
             return False
    return True

Data Sizes

The following lookup tables provide approximately 2048 tabulated epochs of data sizes and cache sizes. They were generated with the Mathematica function provided here:

def get_datasize(block_number):
    return data_sizes[block_number // EPOCH_LENGTH]

def get_cachesize(block_number):
    return cache_sizes[block_number // EPOCH_LENGTH]

data_sizes = [
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, 
5855247488, 5863636096, 5872024448, 5880411008, 5888799872, 
5897186432, 5905576832, 5913966976, 5922352768, 5930744704, 
5939132288, 5947522432, 5955911296, 5964299392, 5972688256, 
5981074304, 5989465472, 5997851008, 6006241408, 6014627968, 
6023015552, 6031408256, 6039796096, 6048185216, 6056574848, 
6064963456, 6073351808, 6081736064, 6090128768, 6098517632, 
6106906496, 6115289216, 6123680896, 6132070016, 6140459648, 
6148849024, 6157237376, 6165624704, 6174009728, 6182403712, 
6190792064, 6199176064, 6207569792, 6215952256, 6224345216, 
6232732544, 6241124224, 6249510272, 6257899136, 6266287744, 
6274676864, 6283065728, 6291454336, 6299843456, 6308232064, 
6316620928, 6325006208, 6333395584, 6341784704, 6350174848, 
6358562176, 6366951296, 6375337856, 6383729536, 6392119168, 
6400504192, 6408895616, 6417283456, 6425673344, 6434059136, 
6442444672, 6450837376, 6459223424, 6467613056, 6476004224, 
6484393088, 6492781952, 6501170048, 6509555072, 6517947008, 
6526336384, 6534725504, 6543112832, 6551500672, 6559888768, 
6568278656, 6576662912, 6585055616, 6593443456, 6601834112, 
6610219648, 6618610304, 6626999168, 6635385472, 6643777408, 
6652164224, 6660552832, 6668941952, 6677330048, 6685719424, 
6694107776, 6702493568, 6710882176, 6719274112, 6727662976, 
6736052096, 6744437632, 6752825984, 6761213824, 6769604224, 
6777993856, 6786383488, 6794770816, 6803158144, 6811549312, 
6819937664, 6828326528, 6836706176, 6845101696, 6853491328, 
6861880448, 6870269312, 6878655104, 6887046272, 6895433344, 
6903822208, 6912212864, 6920596864, 6928988288, 6937377152, 
6945764992, 6954149248, 6962544256, 6970928768, 6979317376, 
6987709312, 6996093824, 7004487296, 7012875392, 7021258624, 
7029652352, 7038038912, 7046427776, 7054818944, 7063207808, 
7071595136, 7079980928, 7088372608, 7096759424, 7105149824, 
7113536896, 7121928064, 7130315392, 7138699648, 7147092352, 
7155479168, 7163865728, 7172249984, 7180648064, 7189036672, 
7197424768, 7205810816, 7214196608, 7222589824, 7230975104, 
7239367552, 7247755904, 7256145536, 7264533376, 7272921472, 
7281308032, 7289694848, 7298088832, 7306471808, 7314864512, 
7323253888, 7331643008, 7340029568, 7348419712, 7356808832, 
7365196672, 7373585792, 7381973888, 7390362752, 7398750592, 
7407138944, 7415528576, 7423915648, 7432302208, 7440690304, 
7449080192, 7457472128, 7465860992, 7474249088, 7482635648, 
7491023744, 7499412608, 7507803008, 7516192384, 7524579968, 
7532967296, 7541358464, 7549745792, 7558134656, 7566524032, 
7574912896, 7583300992, 7591690112, 7600075136, 7608466816, 
7616854912, 7625244544, 7633629824, 7642020992, 7650410368, 
7658794112, 7667187328, 7675574912, 7683961984, 7692349568, 
7700739712, 7709130368, 7717519232, 7725905536, 7734295424, 
7742683264, 7751069056, 7759457408, 7767849088, 7776238208, 
7784626816, 7793014912, 7801405312, 7809792128, 7818179968, 
7826571136, 7834957184, 7843347328, 7851732352, 7860124544, 
7868512384, 7876902016, 7885287808, 7893679744, 7902067072, 
7910455936, 7918844288, 7927230848, 7935622784, 7944009344, 
7952400256, 7960786048, 7969176704, 7977565312, 7985953408, 
7994339968, 8002730368, 8011119488, 8019508096, 8027896192, 
8036285056, 8044674688, 8053062272, 8061448832, 8069838464, 
8078227328, 8086616704, 8095006592, 8103393664, 8111783552, 
8120171392, 8128560256, 8136949376, 8145336704, 8153726848, 
8162114944, 8170503296, 8178891904, 8187280768, 8195669632, 
8204058496, 8212444544, 8220834176, 8229222272, 8237612672, 
8246000768, 8254389376, 8262775168, 8271167104, 8279553664, 
8287944064, 8296333184, 8304715136, 8313108352, 8321497984, 
8329885568, 8338274432, 8346663296, 8355052928, 8363441536, 
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