Merge pull request bitcoin#148 from OrfeasLitos/link-implicit-y-proof-sketch

Link to proof sketch of security of implicit Y

Author: sipa

bip-schnorr.mediawiki

@@ -81,7 +81,7 @@ expensive conversion to affine coordinates first. This would even be the case if
 
 For ''P'' the speed of signing and verification does not significantly differ between any of the three options because affine coordinates of the point have to be computed anyway. For consistency reasons we choose the same option as for ''R''. The signing algorithm ensures that the signature is valid under the correct public key by negating the secret key if necessary.
 
-Implicit Y coordinates are not a reduction in security when expressed as the number of elliptic curve operations an attacker is expected to perform to compute the secret key. An attacker can normalize any given public key to a point whose Y coordinate is a square by negating the point if necessary. This is just a subtraction of field elements and not an elliptic curve operation<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 not a square.</ref>.
+Implicit Y coordinates are not a reduction in security when expressed as the number of elliptic curve operations an attacker is expected to perform to compute the secret key. An attacker can normalize any given public key to a point whose Y coordinate is a square by negating the point if necessary. This is just a subtraction of field elements and not an elliptic curve operation<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 not a square. A proof sketch can be found [here](https://medium.com/blockstream/reducing-bitcoin-transaction-sizes-with-x-only-pubkeys-f86476af05d7)</ref>.
 
 '''Tagged Hashes''' Cryptographic hash functions are used for multiple purposes in the specification below and in Bitcoin in general. To make sure hashes used in one context can't be reinterpreted in another one, hash functions can be tweaked with a context-dependent tag name, in such a way that collisions across contexts can be assumed to be infeasible. Such collisions obviously can not be ruled out completely, but only for schemes using tagging with a unique name. As for other schemes collisions are at least less likely with tagging than without.