Clarify that Jacobian coordinates are the optimization, not the Legendre symbol

Author: sipa

bip-0340.mediawiki

@@ -80,7 +80,7 @@ Using the first option would be slightly more efficient for verification (around
 # Implicitly choosing the Y coordinate that is even<ref>Since ''p'' is odd, negation modulo ''p'' will map even numbers to odd numbers and the other way around. This means that for a valid X coordinate, one of the corresponding Y coordinates will be even, and the other will be odd.</ref>.
 # Implicitly choosing the Y coordinate that is a quadratic residue (i.e. has a square root modulo ''p'').
 
-The second option offers the greatest compatibility with existing key generation systems, where the standard 33-byte compressed public key format consists of a byte indicating the oddness of the Y coordinate, plus the full X coordinate. To avoid gratuitous incompatibilities, we pick that option for ''P'', and thus our X-only public keys become equivalent to a compressed public key that is the X-only key prefixed by the byte 0x02. For consistency, the same is done for ''R''<ref>An earlier version of this draft used the third option instead, based on a belief that this would in general trade signing efficiency for verification efficiency. Determining if a Y coordinate is a quadratic residue can be done by computing the Legendre symbol directly in Jacobian coordinates (a common optimization), while determining oddness needs a conversion to affine coordinates first, which needs a modular inversion. As modular inverses and Legendre symbols have similar [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-August/018081.html performance] in practice, this trade-off is not worth it.</ref>.
+The second option offers the greatest compatibility with existing key generation systems, where the standard 33-byte compressed public key format consists of a byte indicating the oddness of the Y coordinate, plus the full X coordinate. To avoid gratuitous incompatibilities, we pick that option for ''P'', and thus our X-only public keys become equivalent to a compressed public key that is the X-only key prefixed by the byte 0x02. For consistency, the same is done for ''R''<ref>An earlier version of this draft used the third option instead, based on a belief that this would in general trade signing efficiency for verification efficiency. When using Jacobian coordinates, a common optimization in ECC implementations, it is possible to determine if a Y coordinate is a quadratic residue by computing the Legendre symbol, without converting to affine coordinates first (which needs a modular inversion). As modular inverses and Legendre symbols have similar [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-August/018081.html performance] in practice, this trade-off is not worth it.</ref>.
 
 Despite halving the size of the set of valid public keys, implicit Y coordinates are not a reduction in security. Informally, if a fast algorithm existed to compute the discrete logarithm of an X-only public key, then it could also be used to compute the discrete logarithm of a full public key: apply it to the X coordinate, and then optionally negate the result. This shows that breaking an X-only public key can be at most a small constant term faster than breaking a full one.<ref>This can be formalized by a simple reduction that reduces an attack on Schnorr signatures with implicit Y coordinates to an attack to Schnorr signatures with explicit Y coordinates. The reduction works by reencoding public keys and negating the result of the hash function, which is modeled as random oracle, whenever the challenge public key has an explicit Y coordinate that is odd. A proof sketch can be found [https://medium.com/blockstream/reducing-bitcoin-transaction-sizes-with-x-only-pubkeys-f86476af05d7 here].</ref>.