Address Tim's comments

1 files changed, 3 insertions, 3 deletions

diff --git a/bip-taproot.mediawiki b/bip-taproot.mediawiki
index a553791..5f8537c 100644
--- a/bip-taproot.mediawiki
+++ b/bip-taproot.mediawiki
@@ -65,7 +65,7 @@ In the text below, ''hash<sub>tag</sub>(m)'' is a shorthand for ''SHA256(SHA256(
 A Taproot output is a SegWit output (native or P2SH-nested, see [https://github.com/bitcoin/bips/blob/master/bip-0141.mediawiki BIP141]) with version number 1, and a 32-byte witness program.
 The following rules only apply when such an output is being spent. Any other outputs, including version 1 outputs with lengths other than 32 bytes, remain unencumbered.
 
-* Let ''q'' be the 32-byte array containing the witness program (second push in scriptPubKey or P2SH redeemScript) which represents a public key according to bip-schnorr.<ref>'''Why is the public key directly included in the output?''' While typical earlier constructions store a hash of a script or a public key in the output, this is rather wasteful when a public key is always involved. To guarantee batch verifiability, ''q'' must be known to every verifier, and thus only revealing its hash as an output would imply adding an additional 32 bytes to the witness. Furthermore, to maintain [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2016-January/012198.html 128-bit collision security] for outputs, a 256-bit hash would be required anyway, which is comparable in size (and thus in cost for senders) to revealing the public key directly. While the usage of public key hashes is often said to protect against ECDLP breaks or quantum computers, this protection is very weak at best: transactions are not protected while being confirmed, and a very [https://twitter.com/pwuille/status/1108097835365339136 large portion] of the currency's supply is not under such protection regardless. Actual resistance to such systems can be introduced by relying on different cryptographic assumptions, but this proposal focuses on improvements that do not change the security model. Note that using P2SH-wrapped outputs only have 80-bit collision security. This is considered low, and is relevant whenever the output includes data from more than a single party (public keys, hashes, ...). </ref>.
+* Let ''q'' be the 32-byte array containing the witness program (second push in scriptPubKey or P2SH redeemScript) which represents a public key according to bip-schnorr <ref>'''Why is the public key directly included in the output?''' While typical earlier constructions store a hash of a script or a public key in the output, this is rather wasteful when a public key is always involved. To guarantee batch verifiability, ''q'' must be known to every verifier, and thus only revealing its hash as an output would imply adding an additional 32 bytes to the witness. Furthermore, to maintain [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2016-January/012198.html 128-bit collision security] for outputs, a 256-bit hash would be required anyway, which is comparable in size (and thus in cost for senders) to revealing the public key directly. While the usage of public key hashes is often said to protect against ECDLP breaks or quantum computers, this protection is very weak at best: transactions are not protected while being confirmed, and a very [https://twitter.com/pwuille/status/1108097835365339136 large portion] of the currency's supply is not under such protection regardless. Actual resistance to such systems can be introduced by relying on different cryptographic assumptions, but this proposal focuses on improvements that do not change the security model. Note that using P2SH-wrapped outputs only have 80-bit collision security. This is considered low, and is relevant whenever the output includes data from more than a single party (public keys, hashes, ...). </ref>.
 * Fail if the witness stack has 0 elements.
 * If there are at least two witness elements, and the first byte of the last element is 0x50<ref>'''Why is the first byte of the annex <code>0x50</code>?''' Like the <code>0xc0</code>-<code>0xc1</code> constants, <code>0x50</code> is chosen as it could not be confused with a valid P2WPKH or P2WSH spending. As the control block's initial byte's lowest bit is used to indicate the public key's Y quadratic residuosity, each script version needs two subsequence byte values that are both not yet used in P2WPKH or P2WSH spending. To indicate the annex, only an "unpaired" available byte is necessary like <code>0x50</code>. This choice maximizes the available options for future script versions.</ref>, this last element is called ''annex'' ''a''<ref>'''What is the purpose of the annex?''' The annex is a reserved space for future extensions, such as indicating the validation costs of computationally expensive new opcodes in a way that is recognizable without knowing the outputs being spent. Until the meaning of this field is defined by another softfork, users SHOULD NOT include <code>annex</code> in transactions, or it may lead to PERMANENT FUND LOSS.</ref> and is removed from the witness stack. The annex (or the lack of thereof) is always covered by the transaction digest and contributes to transaction weight, but is otherwise ignored during taproot validation.
 * If there is exactly one element left in the witness stack, key path spending is used:
@@ -87,7 +87,7 @@ The following rules only apply when such an output is being spent. Any other out
 **** If ''k<sub>j</sub> &ge; e<sub>j</sub>'': ''k<sub>j+1</sub> = hash<sub>TapBranch</sub>(e<sub>j</sub> || k<sub>j</sub>)''.
 ** Let ''t = hash<sub>TapTweak</sub>(bytes(P) || k<sub>m</sub>) = hash<sub>TapTweak</sub>(2 + (c[0] & 1) || c[1:33] || k<sub>m</sub>)''.
 ** If ''t &ge; 0xFFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141'' (order of secp256k1), fail.
-** Let ''Q = point(q) if (c[0] & 1) = 1 and -point(q) otherwise''
+** Let ''Q = point(q) if (c[0] & 1) = 1 and -point(q) otherwise''. Fail if this point is not on the curve.
 ** If ''Q &ne; P + int(t)G'', fail.
 ** Execute the script, according to the applicable script rules<ref>'''What are the applicable script rules in script path spends?''' Bip-tapscript specifies validity rules that apply if the leaf version is ''0xc0'', but future proposals can introduce rules for other leaf versions.</ref>, using the witness stack elements excluding the script ''s'', the control block ''c'', and the annex ''a'' if present, as initial stack.
 
@@ -210,7 +210,7 @@ The function <code>taproot_output_script</code> returns a byte array with the sc
 
 [[File:bip-taproot/tree.png|frame|This diagram shows the hashing structure to obtain the tweak from an internal key ''P'' and a Merkle tree consisting of 5 script leaves. ''A'', ''B'', ''C'' and ''E'' are ''TapLeaf'' hashes similar to ''D'' and ''AB'' is a ''TapBranch'' hash. Note that when ''CDE'' is computed ''E'' is hashed first because ''E'' is less than ''CD''.]]
 
-'''Spending using the internal key''' A Taproot output can be spent with the private key corresponding to the <code>internal_pubkey</code>. To do so, a witness stack consists of a single element: a bip-schnorr signature on the signature hash as defined above, with the private key tweaked by the same <code>t</code> in the above snippet. In the code below, <code>internal_privkey</code> has a method <code>pubkey_gen</code> that returns a public key according to bip-schnorr and a boolean indicating the quadratic residuosity of the Y coordinate of the underlying point.
+'''Spending using the internal key''' A Taproot output can be spent with the private key corresponding to the <code>internal_pubkey</code>. To do so, a witness stack consists of a single element: a bip-schnorr signature on the signature hash as defined above, with the private key tweaked by the same <code>t</code> as in the above snippet. In the code below, <code>internal_privkey</code> has a method <code>pubkey_gen</code> that returns a public key according to bip-schnorr and a boolean indicating the quadratic residuosity of the Y coordinate of the underlying point.
 See the code below:
 
 <source lang="python">