Dagger Hashimoto

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Contents

• Introduction
• DAG Generation
  • Parameters
  • Dagger graph building
• Light Client Evaluation
• Double Buffer of DAGs
• Hashimoto
• Mining and Verifying
• Appendix
  • Lehmer Random Number Generator
  • Modular Exponentiation as a Hash Function
  • More Efficient Cache-based Evaluation Algos

Introduction

Dagger Hashimoto was a precursor research implementation and specification for the mining algorithm for Ethereum 1.0, while it has been superceded by Ethash.

The information in this article shall be kept for historical purposes.

Dagger Hashimoto aims to simultaneously satisfy two goals:

1. ASIC-resistance: the benefit from creating specialized hardware for the algorithm should be as small as possible, ideally to the point that even in an economy where ASICs have been developed the speedup is sufficiently small that it is still marginally profitable for users on ordinary computers to mine with spare CPU power.
2. Light client verifiability: a block should be relatively 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).

Dagger Hashimoto builds on two key pieces of previous work:

• Hashimoto, an algorithm by Thaddeus Dryja which intends to achieve ASIC resistance by being IO-bound, ie. making memory reads the limiting factor in the mining process. The theory is that RAM is in principle inherently a much more generic ingredient than computation, and billions of dollars of research already go into optimizing it for different use cases which often involve near-random access patterns (hence "random access memory"); hence, existing RAM is likely to be moderately close to optimal for evaluating the algorithm. Hashimoto uses the blockchain as a source of data, simultaneously satisfying (1) and (3) above.
• Dagger, an algorithm by Vitalik Buterin which uses directed acyclic graphs to simultaneously achieve memory-hard computation but memory-easy validation. The core principle is that each individual nonce only requires a small portion of a large total data tree, and recomputing the subtree for each nonce is prohibitive for mining - hence the need to store the tree - but okay for a single nonce's worth of verification. Dagger was meant to be an alternative to existing memory-hard algorithms like Scrypt, which are memory-hard but are also very hard to verify when their memory-hardness is increased to genuinely secure levels. However, Dagger was proven to be vulnerable to shared memory hardware acceleration by Sergio Lerner and was then dropped in favor of other avenues of research.

Approaches that were tried between Dagger and Dagger Hashimoto but are currently not our primary focus include:

• "Blockchain-based proof of work" - a proof of work function that involves running contracts taken from the blockchain. The approach was abandoned because it was long-range attack vulnerabilities, since attackers can create forks and populate them with contracts that they have a secret fast "trapdoor" execution mechanism for.
• "Random circuit" - a proof of work function developed largely by Vlad Zamfir that involves generating a new program every 1000 nonces - essentially, choosing a new hash function each time, faster than even FPGAs can reconfigure. The approach was temporarily put aside because it was difficult to see what mechanism one can use to generate random programs that would be general enough so that specialization gains would be low; however, we see no fundamental reasons why the concept cannot be made to work.

The difference between Dagger Hashimoto and Hashimoto is that, instead of using the blockchain as a data source, Dagger Hashimoto uses a custom-generated 1 GB data set, which updates based on block data every N blocks. The data set is generated using the Dagger algorithm, allowing for the efficient calculation of a subset specific to every nonce for the light client verification algorithm. The difference between Dagger Hashimoto and Dagger is that, unlike in the original Dagger, the dataset used to query the block is semi-permanent, only being updated at occasional intervals (eg. once per week). This means that the portion of the effort that goes toward generating the dataset is close to zero, so Sergio Lerner's arguments regarding shared memory speedups become negligible.

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:

NUM_BITS = 512

def encode_int(x):
    "Encode an integer x as a string of 64 characters using a big-endian scheme"
    o = ''
    for _ in range(NUM_BITS / 8):
        o = chr(x % 256) + o
        x //= 256
    return o

def decode_int(s):
    "Unencode an integer x from a string using a big-endian scheme"
    x = 0
    for c in s:
        x *= 256
        x += ord(c)
    return x

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:

from pyethereum import utils
def sha3(x):
    if isinstance(x, (int, long)):
        x = encode_int(x)
    return decode_int(utils.sha3(x))

def dbl_sha3(x):
    if isinstance(x, (int, long)):
        x = encode_int(x)
    return decode_int(utils.sha3(utils.sha3(x)))    

Parameters

The parameters used for the algorithm are:

SAFE_PRIME_512 = 2**512 - 38117     # Largest Safe Prime less than 2**512

params = {
      "n": 4000055296 * 8 // NUM_BITS,  # Size of the dataset (4 Gigabytes); MUST BE MULTIPLE OF 65536
      "n_inc": 65536,                   # Increment in value of n per period; MUST BE MULTIPLE OF 65536
                                        # with epochtime=20000 gives 882 MB growth per year
      "cache_size": 2500,               # Size of the light client's cache (can be chosen by light
                                        # client; not part of the algo spec)
      "diff": 2**14,                    # Difficulty (adjusted during block evaluation)
      "epochtime": 100000,              # Length of an epoch in blocks (how often the dataset is updated)
      "k": 1,                           # Number of parents of a node
      "w": w,                          # Used for modular exponentiation hashing
      "accesses": 200,                  # Number of dataset accesses during hashimoto
      "P": SAFE_PRIME_512               # Safe Prime for hashing and random number generation
}

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 off at 1 GB and grows by 441 MB per year.

Dagger graph building

The dagger graph building primitive is defined as follows:

def produce_dag(params, seed, length):
    P = params["P"]
    picker = init = pow(sha3(seed), params["w"], P)
    o = [init]
    for i in range(1, length):
        x = picker = (picker * init) % P
        for _ in range(params["k"]):
            x ^= o[x % i]
        o.append(pow(x, params["w"], P))
    return o

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 is used to further hash the result.

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

Light Client Evaluation

The intent of the above graph construction is to allow each individual node in the graph can 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:

def quick_calc(params, seed, p):
    w, P = params["w"], params["P"]
    cache = {}

    def quick_calc_cached(p):
        if p in cache:
            pass
        elif p == 0:
            cache[p] = pow(sha3(seed), w, P)
        else:
            x = pow(sha3(seed), (p + 1) * w, P)
            for _ in range(params["k"]):
                x ^= quick_calc_cached(x % p)
            cache[p] = pow(x, w, P)
        return cache[p]

    return quick_calc_cached(p)

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. The client does not use the latest DAG produced, but the previous one. 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 and thus 51% attack risks within those few minutes before all data is recomputed.

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

def get_prevhash(n):
    from pyethereum.blocks import GENESIS_PREVHASH 
    from pyethreum import chain_manager
    if num <= 0:
        return hash_to_int(GENESIS_PREVHASH)
    else:
        prevhash = chain_manager.index.get_block_by_number(n - 1)
        return decode_int(prevhash)

def get_seedset(params, block):
    seedset = {}
    seedset["back_number"] = block.number - (block.number % params["epochtime"])
    seedset["back_hash"] = get_prevhash(seedset["back_number"])
    seedset["front_number"] = max(seedset["back_number"] - params["epochtime"], 0)
    seedset["front_hash"] = get_prevhash(seedset["front_number"])
    return seedset

def get_dagsize(params, block):
    return params["n"] + (block.number // params["epochtime"]) * params["n_inc"]

def get_daggerset(params, block):
    dagsz = get_dagsize(params, block)
    seedset = get_seedset(params, block)
    if seedset["front_hash"] <= 0:
        # No back buffer is possible, just make front buffer
        return {"front": {"dag": produce_dag(params, seedset["front_hash"], dagsz), 
                          "block_number": 0}}
    else:
        return {"front": {"dag": produce_dag(params, seedset["front_hash"], dagsz),
                          "block_number": seedset["front_number"]},
                "back": {"dag": produce_dag(params, seedset["back_hash"], dagsz),
                         "block_number": seedset["back_number"]}}

Hashimoto

The idea behind the original Hashimoto is to use the blockchain as a dataset, performing a computation which 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:

def orig_hashimoto(prev_hash, merkle_root, list_of_transactions, nonce):
    hash_output_A = sha256(prev_hash + merkle_root + nonce) 
    txid_mix = 0
    for i in range(64):
        shifted_A = hash_output_A >> i 
        transaction = shifted_A % len(list_of_transactions) 
        txid_mix ^= list_of_transactions[transaction] << i 
    return txid_max ^ (nonce << 192)

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

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

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 PoW 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:

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

Mining and Verifying

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

def mine(daggerset, params, block):
    from random import randint
    nonce = randint(0, 2**64)
    while 1:
        result = hashimoto(daggerset, get_dagsize(params, block),
                           params, decode_int(block.prevhash), nonce)
        if result * params["diff"] < 2**256:
            break
        nonce += 1
        if nonce >= 2**64:
            nonce = 0 
    return nonce

Here is the verification algorithm:

def verify(daggerset, params, block, nonce):
    result = hashimoto(daggerset, get_dagsize(params, block),
                       params, decode_int(block.prevhash), nonce)
    return result * params["diff"] < 2**256

Light-client friendly verification:

def light_verify(params, header, nonce):
    seedset = get_seedset(params, block)
    result = quick_hashimoto(seedset["front_hash"], get_dagsize(params, block),
                             params, decode_int(block.prevhash), nonce)
    return result * params["diff"] < 2**256

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

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

xᵐ mod P ≡ 1

Given these definitions, we have:

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 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:

xP-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 assuance, 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 exponentation 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 Algos

def quick_calc(params, seed, p):
    cache = produce_dag(params, seed, params["cache_size"])
    return quick_calc_cached(cache, params, p)

def quick_calc_cached(cache, params, p):
    P = params["P"]
    if p < len(cache):
        return cache[p]
    else:
        x = pow(cache[0], p + 1, P)
        for _ in range(params["k"]):
            x ^= quick_calc_cached(cache, params, x % p)
        return pow(x, params["w"], P)

def quick_hashimoto(seed, dagsize, params, header, nonce):
    cache = produce_dag(params, seed, params["cache_size"])
    return quick_hashimoto_cached(cache, dagsize, params, header, nonce)

def quick_hashimoto_cached(cache, dagsize, params, header, nonce):
    m = dagsize // 2
    mask = 2**64 - 1
    mix = sha3(encode_int(nonce) + header)
    for _ in range(params["accesses"]):
        mix ^= quick_calc_cached(cache, params, m + (mix & mask) % m)
    return dbl_sha3(mix)

---

Special thanks to feedback from:

• Tim Hughes
• Matthew Wampler-Doty
• Thaddeus Dryja