# Dagger-Hashimoto

Dagger-Hashimoto was the original research implementation and specification for Ethereum's mining algorithm. Dagger-Hashimoto was superseded by Ethash. Mining was switched off completely at The Merge on 15th September 2022. Since then, Ethereum has been secured using a proof-of-stake mechanism instead. This page is for historical interest - the information here is no longer relevant for post-Merge Ethereum.

## Prerequisites

To better understand this page, we recommend you first read up on proof-of-work consensus, mining, and mining algorithms.

## Dagger-Hashimoto

Dagger-Hashimoto aims to satisfy two goals:

**ASIC-resistance**: the benefit from creating specialized hardware for the algorithm should be as small as possible**Light client verifiability**: a block should be efficiently verifiable by a light client.

With an additional modification, we also specify how to fulfill a third goal if desired, but at the cost of additional complexity:

**Full chain storage**: mining should require storage of the complete blockchain state (due to the irregular structure of the Ethereum state trie, we anticipate that some pruning will be possible, particularly of some often-used contracts, but we want to minimize this).

## DAG Generation

The code for the algorithm will be defined in Python below. First, we give `encode_int`

for marshaling unsigned ints of specified precision to strings. Its inverse is also given:

1NUM_BITS = 51223def encode_int(x):4 "Encode an integer x as a string of 64 characters using a big-endian scheme"5 o = ''6 for _ in range(NUM_BITS / 8):7 o = chr(x % 256) + o8 x //= 2569 return o1011def decode_int(s):12 "Unencode an integer x from a string using a big-endian scheme"13 x = 014 for c in s:15 x *= 25616 x += ord(c)17 return x18HammasiNusxa olish

We next assume that `sha3`

is a function that takes an integer and outputs an integer, and `dbl_sha3`

is a double-sha3 function; if converting this reference code into an implementation use:

1from pyethereum import utils2def sha3(x):3 if isinstance(x, (int, long)):4 x = encode_int(x)5 return decode_int(utils.sha3(x))67def dbl_sha3(x):8 if isinstance(x, (int, long)):9 x = encode_int(x)10 return decode_int(utils.sha3(utils.sha3(x)))11HammasiNusxa olish

### Parameters

The parameters used for the algorithm are:

1SAFE_PRIME_512 = 2**512 - 38117 # Largest Safe Prime less than 2**51223params = {4 "n": 4000055296 * 8 // NUM_BITS, # Size of the dataset (4 Gigabytes); MUST BE MULTIPLE OF 655365 "n_inc": 65536, # Increment in value of n per period; MUST BE MULTIPLE OF 655366 # with epochtime=20000 gives 882 MB growth per year7 "cache_size": 2500, # Size of the light client's cache (can be chosen by light8 # client; not part of the algo spec)9 "diff": 2**14, # Difficulty (adjusted during block evaluation)10 "epochtime": 100000, # Length of an epoch in blocks (how often the dataset is updated)11 "k": 1, # Number of parents of a node12 "w": w, # Used for modular exponentiation hashing13 "accesses": 200, # Number of dataset accesses during hashimoto14 "P": SAFE_PRIME_512 # Safe Prime for hashing and random number generation15}16HammasiNusxa olish

`P`

in this case is a prime chosen such that `log₂(P)`

is just slightly less than 512, which corresponds to the 512 bits we have been using to represent our numbers. Note that only the latter half of the DAG actually needs to be stored, so the de-facto RAM requirement starts at 1 GB and grows by 441 MB per year.

### Dagger graph building

The dagger graph building primitive is defined as follows:

1def produce_dag(params, seed, length):2 P = params["P"]3 picker = init = pow(sha3(seed), params["w"], P)4 o = [init]5 for i in range(1, length):6 x = picker = (picker * init) % P7 for _ in range(params["k"]):8 x ^= o[x % i]9 o.append(pow(x, params["w"], P))10 return o11HammasiNusxa olish

Essentially, it starts off a graph as a single node, `sha3(seed)`

, and from there starts sequentially adding on other nodes based on random previous nodes. When a new node is created, a modular power of the seed is computed to randomly select some indices less than `i`

(using `x % i`

above), and the values of the nodes at those indices are used in a calculation to generate a new a value for `x`

, which is then fed into a small proof of work function (based on XOR) to ultimately generate the value of the graph at index `i`

. The rationale behind this particular design is to force sequential access of the DAG; the next value of the DAG that will be accessed cannot be determined until the current value is known. Finally, modular exponentiation hashes the result further.

This algorithm relies on several results from number theory. See the appendix below for a discussion.

## Light client evaluation

The above graph construction intends to allow each node in the graph to be reconstructed by computing a subtree of only a small number of nodes and requiring only a small amount of auxiliary memory. Note that with k=1, the subtree is only a chain of values going up to the first element in the DAG.

The light client computing function for the DAG works as follows:

1def quick_calc(params, seed, p):2 w, P = params["w"], params["P"]3 cache = {}45 def quick_calc_cached(p):6 if p in cache:7 pass8 elif p == 0:9 cache[p] = pow(sha3(seed), w, P)10 else:11 x = pow(sha3(seed), (p + 1) * w, P)12 for _ in range(params["k"]):13 x ^= quick_calc_cached(x % p)14 cache[p] = pow(x, w, P)15 return cache[p]1617 return quick_calc_cached(p)18HammasiNusxa olish

Essentially, it is simply a rewrite of the above algorithm that removes the loop of computing the values for the entire DAG and replaces the earlier node lookup with a recursive call or a cache lookup. Note that for `k=1`

the cache is unnecessary, although a further optimization actually precomputes the first few thousand values of the DAG and keeps that as a static cache for computations; see the appendix for a code implementation of this.

## Double buffer of DAGs

In a full client, a *double buffer* of 2 DAGs produced by the above formula is used. The idea is that DAGs are produced every `epochtime`

number of blocks according to the params above. Instead of the client using the latest DAG produced, it uses the one previous. The benefit of this is that it allows the DAGs to be replaced over time without needing to incorporate a step where miners must suddenly recompute all of the data. Otherwise, there is the potential for an abrupt temporary slowdown in chain processing at regular intervals and dramatically increasing centralization. Thus 51% attack risks within those few minutes before all data are recomputed.

The algorithm used to generate the set of DAGs used to compute the work for a block is as follows:

1def get_prevhash(n):2 from pyethereum.blocks import GENESIS_PREVHASH3 from pyethereum import chain_manager4 if num <= 0:5 return hash_to_int(GENESIS_PREVHASH)6 else:7 prevhash = chain_manager.index.get_block_by_number(n - 1)8 return decode_int(prevhash)910def get_seedset(params, block):11 seedset = {}12 seedset["back_number"] = block.number - (block.number % params["epochtime"])13 seedset["back_hash"] = get_prevhash(seedset["back_number"])14 seedset["front_number"] = max(seedset["back_number"] - params["epochtime"], 0)15 seedset["front_hash"] = get_prevhash(seedset["front_number"])16 return seedset1718def get_dagsize(params, block):19 return params["n"] + (block.number // params["epochtime"]) * params["n_inc"]2021def get_daggerset(params, block):22 dagsz = get_dagsize(params, block)23 seedset = get_seedset(params, block)24 if seedset["front_hash"] <= 0:25 # No back buffer is possible, just make front buffer26 return {"front": {"dag": produce_dag(params, seedset["front_hash"], dagsz),27 "block_number": 0}}28 else:29 return {"front": {"dag": produce_dag(params, seedset["front_hash"], dagsz),30 "block_number": seedset["front_number"]},31 "back": {"dag": produce_dag(params, seedset["back_hash"], dagsz),32 "block_number": seedset["back_number"]}}33HammasiNusxa olish

## Hashimoto

The idea behind the original Hashimoto is to use the blockchain as a dataset, performing a computation that selects N indices from the blockchain, gathers the transactions at those indices, performs an XOR of this data, and returns the hash of the result. Thaddeus Dryja's original algorithm, translated to Python for consistency, is as follows:

1def orig_hashimoto(prev_hash, merkle_root, list_of_transactions, nonce):2 hash_output_A = sha256(prev_hash + merkle_root + nonce)3 txid_mix = 04 for i in range(64):5 shifted_A = hash_output_A >> i6 transaction = shifted_A % len(list_of_transactions)7 txid_mix ^= list_of_transactions[transaction] << i8 return txid_max ^ (nonce << 192)9Nusxa olish

Unfortunately, while Hashimoto is considered RAM hard, it relies on 256-bit arithmetic, which has considerable computational overhead. However, Dagger-Hashimoto only uses the least significant 64 bits when indexing its dataset to address this issue.

1def hashimoto(dag, dagsize, params, header, nonce):2 m = dagsize / 23 mix = sha3(encode_int(nonce) + header)4 for _ in range(params["accesses"]):5 mix ^= dag[m + (mix % 2**64) % m]6 return dbl_sha3(mix)7Nusxa olish

The use of double SHA3 allows for a form of zero-data, near-instant pre-verification, verifying only that a correct intermediate value was provided. This outer layer of proof-of-work is highly ASIC-friendly and fairly weak, but exists to make DDoS even more difficult since that small amount of work must be done in order to produce a block that will not be rejected immediately. Here is the light-client version:

1def quick_hashimoto(seed, dagsize, params, header, nonce):2 m = dagsize // 23 mix = sha3(nonce + header)4 for _ in range(params["accesses"]):5 mix ^= quick_calc(params, seed, m + (mix % 2**64) % m)6 return dbl_sha3(mix)7Nusxa olish

## Mining and verifying

Now, let us put it all together into the mining algorithm:

1def mine(daggerset, params, block):2 from random import randint3 nonce = randint(0, 2**64)4 while 1:5 result = hashimoto(daggerset, get_dagsize(params, block),6 params, decode_int(block.prevhash), nonce)7 if result * params["diff"] < 2**256:8 break9 nonce += 110 if nonce >= 2**64:11 nonce = 012 return nonce13HammasiNusxa olish

Here is the verification algorithm:

1def verify(daggerset, params, block, nonce):2 result = hashimoto(daggerset, get_dagsize(params, block),3 params, decode_int(block.prevhash), nonce)4 return result * params["diff"] < 2**2565Nusxa olish

Light-client friendly verification:

1def light_verify(params, header, nonce):2 seedset = get_seedset(params, block)3 result = quick_hashimoto(seedset["front_hash"], get_dagsize(params, block),4 params, decode_int(block.prevhash), nonce)5 return result * params["diff"] < 2**2566Nusxa olish

Also, note that Dagger-Hashimoto imposes additional requirements on the block header:

- For two-layer verification to work, a block header must have both the nonce and the middle value pre-sha3
- Somewhere, a block header must store the sha3 of the current seedset

## Further reading

*Know of a community resource that helped you? Edit this page and add it!*

## Appendix

As noted above, the RNG used for DAG generation relies on some results from number theory. First, we provide assurance that the Lehmer RNG that is the basis for the `picker`

variable has a wide period. Second, we show that `pow(x,3,P)`

will not map `x`

to `1`

or `P-1`

provided `x ∈ [2,P-2]`

to start. Finally, we show that `pow(x,3,P)`

has a low collision rate when treated as a hashing function.

### Lehmer random number generator

While the `produce_dag`

function does not need to produce unbiased random numbers, a potential threat is that `seed**i % P`

only takes on a handful of values. This could provide an advantage to miners recognizing the pattern over those that do not.

To avoid this, a result from number theory is appealed to. A *Safe Prime* is defined to be a prime `P`

such that `(P-1)/2`

is also prime. The *order* of a member `x`

of the multiplicative group `ℤ/nℤ`

is defined to be the minimal `m`

such that

1xᵐ mod P ≡ 1

Observation 1. Let

`x`

be a member of the multiplicative group`ℤ/Pℤ`

for a safe prime`P`

. If`x mod P ≠ 1 mod P`

and`x mod P ≠ P-1 mod P`

, then the order of`x`

is either`P-1`

or`(P-1)/2`

.

*Proof*. Since `P`

is a safe prime, then by [Lagrange's Theorem][lagrange] we have that the order of `x`

is either `1`

, `2`

, `(P-1)/2`

, or `P-1`

.

The order of `x`

cannot be `1`

, since by Fermat's Little Theorem we have:

1xP-1 mod P ≡ 1

Hence `x`

must be a multiplicative identity of `ℤ/nℤ`

, which is unique. Since we assumed that `x ≠ 1`

by assumption, this is not possible.

The order of `x`

cannot be `2`

unless `x = P-1`

, since this would violate that `P`

is prime.

From the above proposition, we can recognize that iterating `(picker * init) % P`

will have a cycle length of at least `(P-1)/2`

. This is because we selected `P`

to be a safe prime approximately equal to be a higher power of two, and `init`

is in the interval `[2,2**256+1]`

. Given the magnitude fo `P`

, we should never expect a cycle from modular exponentiation.

When we are assigning the first cell in the DAG (the variable labeled `init`

), we compute `pow(sha3(seed) + 2, 3, P)`

. At first glance, this does not guarantee that the result is neither `1`

nor `P-1`

. However, since `P-1`

is a safe prime, we have the following additional assurance, which is a corollary of Observation 1:

Observation 2. Let

`x`

be a member of the multiplicative group`ℤ/Pℤ`

for a safe prime`P`

, and let`w`

be a natural number. If`x mod P ≠ 1 mod P`

and`x mod P ≠ P-1 mod P`

, as well as`w mod P ≠ P-1 mod P`

and`w mod P ≠ 0 mod P`

, then`xʷ mod P ≠ 1 mod P`

and`xʷ mod P ≠ P-1 mod P`

### Modular exponentiation as a hash function

For certain values of `P`

and `w`

, the function `pow(x, w, P)`

may have many collisions. For instance, `pow(x,9,19)`

only takes on values `{1,18}`

.

Given that `P`

is prime, then an appropriate `w`

for a modular exponentiation hashing function can be chosen using the following result:

Observation 3. Let

`P`

be a prime;`w`

and`P-1`

are relatively prime if and only if for all`a`

and`b`

in`ℤ/Pℤ`

:`aʷ mod P ≡ bʷ mod P`

if and only if`a mod P ≡ b mod P`

Thus, given that `P`

is prime and `w`

is relatively prime to `P-1`

, we have that `|{pow(x, w, P) : x ∈ ℤ}| = P`

, implying that the hashing function has the minimal collision rate possible.

In the special case that `P`

is a safe prime as we have selected, then `P-1`

only has factors 1, 2, `(P-1)/2`

and `P-1`

. Since `P`

> 7, we know that 3 is relatively prime to `P-1`

, hence `w=3`

satisfies the above proposition.

## More efficient cache-based evaluation algorithm

1def quick_calc(params, seed, p):2 cache = produce_dag(params, seed, params["cache_size"])3 return quick_calc_cached(cache, params, p)45def quick_calc_cached(cache, params, p):6 P = params["P"]7 if p < len(cache):8 return cache[p]9 else:10 x = pow(cache[0], p + 1, P)11 for _ in range(params["k"]):12 x ^= quick_calc_cached(cache, params, x % p)13 return pow(x, params["w"], P)1415def quick_hashimoto(seed, dagsize, params, header, nonce):16 cache = produce_dag(params, seed, params["cache_size"])17 return quick_hashimoto_cached(cache, dagsize, params, header, nonce)1819def quick_hashimoto_cached(cache, dagsize, params, header, nonce):20 m = dagsize // 221 mask = 2**64 - 122 mix = sha3(encode_int(nonce) + header)23 for _ in range(params["accesses"]):24 mix ^= quick_calc_cached(cache, params, m + (mix & mask) % m)25 return dbl_sha3(mix)26HammasiNusxa olish