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To: Bitcoin dev list <bitcoin-dev@lists.linuxfoundation.org>
From: Pieter Wuille <bitcoin-dev@wuille.net>
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Subject: [bitcoin-dev] Revisiting squaredness tiebreaker for R point in
	BIP340
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Hello all,

The current BIP340 draft[1] uses two different tiebreakers for conveying th=
e Y coordinate of points: for the R point inside signatures squaredness is =
used, while for public keys evenness is used. Originally both used squaredn=
ess, but it was changed[2] for public keys after observing this results in =
additional complexity for compatibility with existing systems.

The reason for choosing squaredness as tiebreaker was performance: in non-b=
atch signature validation, the recomputed R point must be verified to have =
the correct sign, to guarantee consistency with batch validation. Whether t=
he Y coordinate is square can be computed directly in Jacobian coordinates,=
 while determining evenness requires a conversion to affine coordinates fir=
st.

This argument of course relies on the assumption that determining whether t=
he Y coordinate is square can be done more efficiently than a conversion to=
 affine coordinates. It appears now that this assumption is incorrect, and =
the justification for picking the squaredness tiebreaking doesn't really ex=
ist. As it comes with other trade-offs (it slows down signing, and is a les=
s conventional choice), it would seem that we should reconsider the option =
of having the R point use the evenness tiebreaker (like public keys).

It is late in the process, but I feel I owe this explanation so that at lea=
st the possibility of changing can be discussed with all information. On th=
e upside, this was discovered in the context of looking into a cool improve=
ment to libsecp256k1[5], which makes things faster in general, but specific=
ally benefits the evenness variant.


# 1. What happened?

Computing squaredness is done through the Jacobi symbol (same inventor, but=
 unrelated to Jacobian coordinates). Computing evenness requires converting=
 points to affine coordinates first, and that needs a modular inverse. The =
assumption that Jacobi symbols are faster to compute than inverses was base=
d on:

* A (possibly) mistaken belief about the theory: fast algorithms for both J=
acobi symbols and inverses are internally based on variants of the same ext=
ended GCD algorithm[3]. Since an inverse needs to extract a full big intege=
r out of the transition steps made in the extgcd algorithm, while the Jacob=
i symbol just extracts a single bit, it had seemed that any advances applic=
able to one would be applicable to the other, but inverses would always nee=
d additional work on top. It appears however that a class of extgcd algorit=
hms exists (LSB based ones) that cannot be used for Jacobi calculations wit=
hout losing efficiency. Recent developments[4] and a proposed implementatio=
n in libsecp256k1[5] by Peter Dettman show that using this, inverses in som=
e cases can in fact be faster than Jacobi symbols.

* A broken benchmark. This belief was incorrectly confirmed by a broken ben=
chmark[6] in libsecp256k1 for the libgmp-based Jacobi symbol calculation an=
d modular inverse. The benchmark was repeatedly testing the same constant i=
nput, which apparently was around 2.5x faster than the average speed. It is=
 a variable-time algorithm, so a good variation of inputs matters. This mis=
take had me (and probably others) convinced for years that Jacobi symbols w=
ere amazingly fast, while in reality they were always very close in perform=
ance to inverses.


# 2. What is the actual impact of picking evenness instead?

It is hard to make very generic statements here, as BIP340 will hopefully b=
e used for a long time, and hardware advancements and algorithmic improveme=
nts may change the balance. That said, performance on current hardware with=
 optimized algorithms is the best approximation we have.

The numbers below give the expected performance change from squareness to e=
venness, for single BIP340 validation, and for signing. Positive numbers me=
an evenness is faster. Batch validation is not impacted at all.

In the short term, for block validation in Bitcoin Core, the numbers for ma=
ster-nogmp are probably the most relevant (as Bitcoin Core uses libsecp256k=
1 without libgmp, to reduce consensus-critical dependencies). If/when [5] g=
ets merged, safegcd-nogmp will be what matters. On a longer time scale, the=
 gmp numbers may be more relevant, as the Jacobi implementation there is ce=
rtainly closer to the state of the art.

* i7-7820HQ: (verify) (sign)
  - master-nogmp: -0.3% +16.1%
  - safegcd-nogmp: +6.7% +17.1%
  - master-gmp: +0.6% +7.7%
  - safegcd-gmp: +1.6% +8.6%

* Cortex-A53: (verify) (sign)
  - master-nogmp: -0.3% +15.7%
  - safegcd-nogmp: +7.5% +16.9%
  - master-gmp: +0.3% +4.1%
  - safegcd-gmp: 0.0% +3.5%

* EPYC 7742: (verify) (sign)
  - master-nogmp: -0.3% +16.8%
  - safegcd-nogmp: +8.6% +18.4%
  - master-gmp: 0.0% +7.4%
  - safegcd-gmp: +2.3% +7.8%

In well optimized cryptographic code speedups as large as a couple percent =
are difficult to come by, so we would usually consider changes of this magn=
itude relevant. Note however that while the percentages for signing speed a=
re larger, they are not what is unexpected here. The choice for the square =
tiebreaker was intended to improve verification speed at the cost of signin=
g speed. As it turns out that it doesn't actually benefit verification spee=
d, this is a bad trade-off.


# 3. How big a change is it

* In the BIP:
  - Changing both invocations of `has_square_y` to `has_even_y`.
  - Changing the `lift_x_square_y` invocation to `lift_x_even_y`.
  - Applying the same change to the test vector generation code, and the re=
sulting test vectors.
* In the libsecp256k1:
  - An 8-line patch to the proposed BIP340 implementation[7]: see [8]
* In Bitcoin Core:
  - Similarly small changes to the Python test reimplementation[9]
* Duplicating these changes in other draft implementations that may already=
 exist.
* Review for all the above.


# 4. Conclusion

We discovered that the justification for using squaredness tiebreakers in B=
IP340 is based on a misunderstanding, and recent developments show that it =
may in fact be a somewhat worse choice than the alternative. It is a relati=
vely simple change to address this, but that has be weighed against the imp=
act of changing the standard at this stage.

Thoughts?


# 5. References

  [1] https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki#design
  [2] https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-February=
/017639.html
  [3] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
  [4] https://gcd.cr.yp.to/safegcd-20190413.pdf
  [5] https://github.com/bitcoin-core/secp256k1/pull/767
  [6] https://github.com/bitcoin-core/secp256k1/pull/797
  [7] https://github.com/bitcoin-core/secp256k1/pull/558
  [8] https://github.com/sipa/secp256k1/commit/822311ca230a48d2c373f3e48b91=
b2a59e1371d6
  [9] https://github.com/bitcoin/bitcoin/pull/17977


Cheers,

--
Pieter