layout: specification
title: ASERT Difficulty Adjustment Algorithm (aserti3-2d)
date: 2020-08-17
category: spec
activation: 1605441600
version: 0.6.3
author: freetrader, Jonathan Toomim, Calin Culianu, Mark Lundeberg, Tobias Ruck
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.
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:
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 intime_delta
is the difference, in signed integer seconds, between theideal_block_time
is a constant: 600 seconds, the targetedheight_delta
is the difference in block height between the currenthalflife
is a constant parameter sometimes referred to asnext_target
is the integer value of the target computed for the blockThe 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.
The ASERT algorithm will be activated according to the top-level upgrade spec [3].
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.
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
).
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.
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.
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.
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, © 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.
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.
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.
Choice of name
The specific algorithm name ‘aserti3-2d’ was chosen based on:
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.
Test vectors suitable for validating further implementations of the aserti3-2d
algorithm are available at:
alternatively 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
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:
[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
This specification is dual-licensed under the Creative Commons CC0 1.0 Universal and
GNU All-Permissive licenses.