Summary

Activation of a new new difficulty adjustment algorithm ‘aserti3-2d’ (or ‘ASERT’ for short) for the November 2020 Bitcoin Cash upgrade. Activation will be based on MTP, with the last pre-fork block used as the anchor block.

Motivation

  • To eliminate periodic oscillations in difficulty and hashrate
  • To reduce the difference in profitability between steady miners and those who switch to mining other blockchains.
  • To maintain average block intervals close to the 10 minute target.
  • To bring the average transaction confirmation time close to target time.

Technical background

The November 2017 Bitcoin Cash upgrade introduced a simple moving average as difficulty adjustment algorithm. This change unfortunately introduced daily periodic difficulty oscillations, which resulted in long confirmation times followed by a burst of rapid blocks. This harms the user experience of Bitcoin Cash, and punishes steady hashrate miners.

Research into the family of difficulty algorithms based on an exponential moving average (EMA) resulted in ASERT (Absolutely Scheduled Exponentially Rising Targets) [1], which has been developed by Mark Lundeberg in 2019 and fully described by him in 2020. An equivalent formula was independently discovered in 2018 by Jacob Eliosoff and in 2020 by Werner et. al [6].

ASERT does not have the same oscillations as the DAA introduced in the November 2017 upgrade and has a range of other attractive qualities such as robustness against singularities [15] without a need for additional rules, and absence of accumulation of rounding/approximation errors.

In extensive simulation against a range of other stable algorithms [2], an ASERT algorithm performed best across criteria that included:

  • Average block times closest to an ideal target time of 600 seconds.
  • Average transaction confirmation times closest to the target time.
  • Reducing the advantage of non-steady mining strategies, thereby maximizing the relative profitability of steady mining.

Specification

Terms and conventions

  • Fork block: The first block mined according to the new consensus rules.
  • Anchor block: The parent of the fork block.

Requirements

Target computation

The current block’s target bits are calculated by the following algorithm.

The aserti3-2d algorithm can be described by the following formula:

next_target = anchor_target * 2**((time_delta - ideal_block_time * (height_delta + 1)) / halflife)

where:

  • anchor_target is the unsigned 256 bit integer equivalent of the nBits value in the header of the anchor block.
  • time_delta is the difference, in signed integer seconds, between the timestamp in the header of the current block and the timestamp in the parent of the anchor block.
  • ideal_block_time is a constant: 600 seconds, the targeted average time between blocks.
  • height_delta is the difference in block height between the current block and the anchor block.
  • halflife is a constant parameter sometimes referred to as ‘tau’, with a value of 172800 (seconds) on mainnet.
  • next_target is the integer value of the target computed for the block after the current block.

The algorithm below implements the above formula using fixed-point integer arithmetic and a cubic polynomial approximation to the 2^x term.

The ‘target’ values used as input and output are the compact representations of actual 256-bit integer targets as specified for the ‘nBits’ field in the block header.

Python-code, uses Python 3 syntax:

def next_target_aserti3_2d(
    anchor_height: int,       # height of the anchor block.
    anchor_parent_time: int,  # timestamp (nTime) of the parent of the anchor block.
    anchor_bits: int,         # 'nBits' value of the anchor block.
    current_height: int,      # height of the current block.
    current_time: int,        # timestamp of the current block.
) -> int:                     # 'target' nBits of the current block.
    ideal_block_time = 600    # in seconds
    halflife = 172_800        # 2 days (in seconds)
    radix = 2**16             # 16 bits for decimal part of fixed-point integer arithmetic
    max_bits = 0x1d00_ffff    # maximum target in nBits representation
    max_target = bits_to_target(max_bits)  # maximum target as integer

    anchor_target = bits_to_target(anchor_bits)
    time_delta = current_time - anchor_parent_time
    height_delta = current_height - anchor_height  # can be negative
    # `//` is truncating division (int.__floordiv__) - see note 3 below
    exponent = time_delta - ideal_block_time * (height_delta + 1) // halflife

    # Compute equivalent of `num_shifts = math.floor(exponent / 2**16)`
    num_shifts = exponent >> 16

    exponent = exponent - num_shifts * radix
    factor = ((195_766_423_245_049 * exponent +
               971_821_376 * exponent**2 +
               5_127 * exponent**3 +
               2**47) >> 48) + radix
    next_target = anchor_target * factor

    # Calculate `next_target = math.floor(next_target * 2**factor)`
    if num_shifts < 0:
        next_target >>= -num_shifts
    else:
        # Implementations should be careful of overflow here (see note 6 below).
        next_target <<= num_shifts

    next_target >>= 16
    if next_target == 0:
        return target_to_bits(1)   # hardest valid target

    if next_target > max_target:
        return max_bits            # limit on easiest target
    return target_to_bits(next_target)

Note 1: The reference implementations make use of signed integer arithmetic. Alternative implementations may use strictly unsigned integer arithmetic.

Note 2: All implementations should strictly avoid use of floating point arithmetic in the computation of the exponent.

Note 3: In the calculation of the exponent, truncating integer division [7, 10] must be used, as indicated by the // division operator (int.__floordiv__).

Note 5: The convenience functions bits_to_target() and target_to_bits() are assumed to be available for conversion between compact ‘nBits’ and unsigned 256-bit integer representations of targets. Examples of such functions are available in the C++ and Python3 reference implementations.

Note 6: If a limited-width integer type is used for current_target, then the << operator may cause an overflow exception or silent discarding of most-significant bits. Implementations must detect and handle such cases to correctly emulate the behaviour of an unlimited-width calculation. Note that if the result at this point would exceed radix * max_target then max_bits may be returned immediately.

Note 7: The polynomial approximation that computes factor must be performed with 64 bit unsigned integer arithmetic or better. It will overflow a signed 64 bit integer. Since exponent is signed, it may be necessary to cast it to unsigned 64 bit integer. In languages like Java where long is always signed, an unsigned shift >>> 48 must be used to divide by 2^48.

Activation

The ASERT algorithm will be activated according to the top-level upgrade spec [3].

Anchor block

ASERT requires the choice of an anchor block to schedule future target computations.

The first block with an MTP that is greater/equal to the upgrade activation time will be used as the anchor block for subsequent ASERT calculations.

This corresponds to the last block mined under the pre-ASERT DAA rules.

Note 1: The anchor block is the block whose height and target (nBits) are used as the ‘absolute’ basis for ASERT’s scheduled target. The timestamp (nTime) of the anchor block’s parent is used.

Note 2: The height, timestamp, and nBits of this block are not known ahead of the upgrade. Implementations MUST dynamically determine it across the upgrade. Once the network upgrade has been consolidated by sufficient chain work or a checkpoint, implementations can simply hard-code the known height, nBits and associated (parent) timestamp this anchor block. Implementations MAY also hard-code other equivalent representations, such as an nBits value and a time offset from the genesis block.

REQ-ASERT-TESTNET-DIFF-RESET (testnet difficulty reset)

On testnet, an additional rule will be included: Any block with a timestamp that is more than 1200 seconds after its parent’s timestamp must use an nBits value of max_bits (0x1d00ffff).

Rationale and commentary on requirements / design decisions

  1. Choice of anchor block determination

    Choosing an anchor block that is far enough in the past would result in slightly simpler coding requirements but would create the possibility of a significant difficulty adjustment at the upgrade.

    The last block mined according to the old DAA was chosen since this block is the most proximal anchor and allows for the smoothest transition to the new algorithm.

  2. Avoidance of floating point calculations

    Compliance with IEEE-754 floating point arithmetic is not generally guaranteed by programming languages on which a new DAA needs to be implemented. This could result in floating point calculations yielding different results depending on compilers, interpreters or hardware.

    It is therefore highly advised to perform all calculations purely using integers and highly specific operators to ensure identical difficulty targets are enforced across all implementations.

  3. Choice of half-life

    A half-life of 2 days (halflife = 2 * 24 * 3600), equivalent to an e^x-based time constant of 2 * 144 * ln(2) or aserti3-415.5, was chosen because it reaches near-optimal performance in simulations by balancing the need to buffer against statistical noise and the need to respond rapidly to swings in price or hashrate, while also being easy for humans to understand: For every 2 days ahead of schedule a block’s timestamp becomes, the difficulty doubles.

  4. Choice of approximation polynomial

    The DAA is part of a control system feedback loop that regulates hashrate, and the exponential function and its integer approximation comprise its transfer function. As such, standard guidelines for ensuring control system stability apply. Control systems tend to be far more sensitive to differential nonlinearity (DNL) than integral nonlinearity (INL) in their transfer functions. Our requirements were to have a transfer function that was (a) monotonic, (b) contained no abrupt changes, (c) had precision and differential nonlinearity that was better than our multi-block statistical noise floor, (d) was simple to implement, and (e) had integral nonlinearity that was no worse than our single-block statistical noise floor.

    A simple, fast to compute cubic approximation of 2^x for 0 <= x < 1 was found to satisfy all of these requirements. It maintains an absolute error margin below 0.013% over this range [8]. In order to address the full (-infinity, +infinity) domain of the exponential function, we found the 2**(x + n) = 2**n * 2**x identity to be of use. Our cubic approximation gives the exactly correct values f(0) == 1 and f(1) == 2, which allows us to use this identity without concern for discontinuities at the edges of the approximation’s domain.

    First, there is the issue of DNL. Our goal was to ensure that our algorithm added no more than 25% as much noise as is inherent in our dataset. Our algorithm is effectively trying to estimate the characteristic hashrate over the recent past, using a 2-day (~288-block) half-life. Our expected exponential distribution of block intervals has a standard deviation (stddev) of 600 seconds. Over a 2-day half-life, our noise floor in our estimated hashrate should be about sqrt(1 / 288) * 600 seconds, or 35.3 seconds. Our chosen approximation method is able to achieve precision of 3 seconds in most circumstances, limited in two places by 16-bit operations: 172800 sec / 65536 = 2.6367 sec Our worst-case precision is 8 seconds, and is limited by the worst-case 15-bit precision of the nBits value. This 8 second worst-case is not within the scope of this work to address, as it would require a change to the block header. Our worst-case step size is 0.00305%,[11] due to the worst-case 15-bit nBits mantissa issue. Outside the 15-bit nBits mantissa range, our approximation has a worst-case precision of 0.0021%. Overall, we considered this to be satisfactory DNL performance.

    Second, there is the issue of INL. Simulation testing showed that difficulty and hashrate regulation performance was remarkably insensitive to integral non-linearity. We found that even the use of f(x) = 1 + x as an approximation of 2**x in the aserti1 algorithm was satisfactory when coupled with the 2**(x + n) = 2^n * 2^x identity, despite having 6% worst-case INL.[12][13] An approximation with poor INL will still show good hashrate regulation ability, but will have a different amount of drift for a given change in hashrate depending on where in the [0, 1) domain our exponent (modulo 1) lies. With INL of +/- 1%, for any given difficulty (or target), a block’s timestamp might end up being 1% of 172800 seconds ahead of or behind schedule. However, out of an abundance of caution, and because achieving higher precision was easy, we chose to aim for INL that would be comparable to or less than the typical drift that can be caused by one block. Out of a 2-day half-life window, one block’s variance comprises: 600 / 172800 = 0.347% Our cubic approximation’s INL performance is better than 0.013%,[14] which exceeds that requirement by a comfortable margin.

  5. Conversion of difficulty bits (nBits) to 256-bit target representations

    As there are few calculations in ASERT which involve 256-bit integers and the algorithm is executed infrequently, it was considered unnecessary to require more complex operations such as doing arithmetic directly on the compact target representations (nBits) that are the inputs/output of the difficulty algorithm.

    Furthermore, 256-bit (or even bignum) arithmetic is available in existing implementation and used within the previous DAA. Performance impacts are negligible.

  6. Choice of 16-bits of precision for fixed-point math

    The nBits format is comprised of 8 bits of base_256 exponent, followed by a 24-bit mantissa. The mantissa must have a value of at least 0x008000, which means that the worst-case scenario gives the mantissa only 15 bits of precision. The choice of 16-bit precision in our fixed point math ensures that overall precision is limited by this 15-bit nBits limit.

  7. Choice of name

    The specific algorithm name ‘aserti3-2d’ was chosen based on:

    • the ‘i’ refers to the integer-only arithmetic
    • the ‘3’ refers to the cubic approximation of the exponential
    • the ‘2d’ refers to the 2-day (172800 second) halflife

Implementation advice

Implementations must not make any rounding errors during their calculations. Rounding must be done exactly as specified in the algorithm. In practice, to guarantee that, you likely need to use integer arithmetic exclusively.

Implementations which use signed integers and use bit-shifting must ensure that the bit-shifting is arithmetic.

Note 1: In C++ compilers, right shifting negative signed integers is formally unspecified behavior until C++20 when it will become standard [5]. In practice, C/C++ compilers commonly implement arithmetic bit shifting for signed numbers. Implementers are advised to verify good behavior through compile-time assertions or unit tests.

Reference implementations

Test vectors

Test vectors suitable for validating further implementations of the aserti3-2d algorithm are available at:

https://gitlab.com/bitcoin-cash-node/bchn-sw/qa-assets/-/tree/master/test_vectors/aserti3-2d

and alternatively at:

https://download.bitcoincashnode.org/misc/data/asert/test_vectors

Acknowledgements

Thanks to Mark Lundeberg for granting permission to publish the ASERT paper [1], Jonathan Toomim for developing the initial Python and C++ implementations, upgrading the simulation framework [9] and evaluating the various difficulty algorithms.

Thanks to Jacob Eliosoff, Tom Harding and Scott Roberts for evaluation work on the families of EMA and other algorithms considered as replacements for the Bitcoin Cash DAA, and thanks to the following for review and their valuable suggestions for improvement:

  • Andrea Suisani (sickpig)
  • BigBlockIfTrue
  • Fernando Pellicioni
  • imaginary_username
  • mtrycz
  • Jochen Hoenicke
  • John Nieri (emergent_reasons)
  • Tom Zander

References

[1] “Static difficulty adjustments, with absolutely scheduled exponentially rising targets (DA-ASERT) – v2”, Mark B. Lundeberg, July 31, 2020

[2] “BCH upgrade proposal: Use ASERT as the new DAA”, Jonathan Toomim, 8 July 2020

[3] Bitcoin Cash November 15, 2020 Upgrade Specification.

[4] https://en.wikipedia.org/wiki/Arithmetic_shift

[5] https://en.cppreference.com/w/cpp/language/operator_arithmetic

[6] “Unstable Throughput: When the Difficulty Algorithm Breaks”, Sam M. Werner, Dragos I. Ilie, Iain Stewart, William J. Knottenbelt, June 2020

[7] “Different kinds of integer division”, Harry Garrood, blog, 2018

[8] Error in a cubic approximation of 2^x for 0 <= x < 1

[9] Jonathan Toomim adaptation of kyuupichan’s difficulty algorithm simulator: https://github.com/jtoomim/difficulty/tree/comparator

[10] “The Euclidean definition of the functions div and mod”, Raymond T. Boute, 1992, ACM Transactions on Programming Languages and Systems (TOPLAS). 14. 127-144. 10.1145/128861.128862

[11] http://toom.im/bch/aserti3_step_size.html

[12] f(x) = (1 + x)/2^x for 0<x<1, WolframAlpha.

[13] https://github.com/zawy12/difficulty-algorithms/issues/62#issuecomment-647060200

[14] http://toom.im/bch/aserti3_approx_error.html

[15] https://github.com/zawy12/difficulty-algorithms/issues/62#issuecomment-646187957

License

This specification is dual-licensed under the Creative Commons CC0 1.0 Universal and GNU All-Permissive licenses.