fix BIP links

1 files changed, 5 insertions, 5 deletions

diff --git a/bip-0340.mediawiki b/bip-0340.mediawiki
index ad9e60e..9e0a73e 100644
--- a/bip-0340.mediawiki
+++ b/bip-0340.mediawiki
@@ -31,7 +31,7 @@ transactions. These are [https://www.secg.org/sec1-v2.pdf standardized], but hav
 compared to [http://publikationen.ub.uni-frankfurt.de/opus4/files/4280/schnorr.pdf Schnorr signatures] over the same curve:
 
 * '''Provable security''': Schnorr signatures are provably secure. In more detail, they are ''strongly unforgeable under chosen message attack (SUF-CMA)''<ref>Informally, this means that without knowledge of the secret key but given valid signatures of arbitrary messages, it is not possible to come up with further valid signatures.</ref> [https://www.di.ens.fr/~pointche/Documents/Papers/2000_joc.pdf in the random oracle model assuming the hardness of the elliptic curve discrete logarithm problem (ECDLP)] and [http://www.neven.org/papers/schnorr.pdf in the generic group model assuming variants of preimage and second preimage resistance of the used hash function]<ref>A detailed security proof in the random oracle model, which essentially restates [https://www.di.ens.fr/~pointche/Documents/Papers/2000_joc.pdf the original security proof by Pointcheval and Stern] more explicitly, can be found in [https://eprint.iacr.org/2016/191 a paper by Kiltz, Masny and Pan]. All these security proofs assume a variant of Schnorr signatures that use ''(e,s)'' instead of ''(R,s)'' (see Design above). Since we use a unique encoding of ''R'', there is an efficiently computable bijection that maps ''(R,s)'' to ''(e,s)'', which allows to convert a successful SUF-CMA attacker for the ''(e,s)'' variant to a successful SUF-CMA attacker for the ''(R,s)'' variant (and vice-versa). Furthermore, the proofs consider a variant of Schnorr signatures without key prefixing (see Design above), but it can be verified that the proofs are also correct for the variant with key prefixing. As a result, all the aforementioned security proofs apply to the variant of Schnorr signatures proposed in this document.</ref>. In contrast, the [https://nbn-resolving.de/urn:nbn:de:hbz:294-60803 best known results for the provable security of ECDSA] rely on stronger assumptions.
-* '''Non-malleability''': The SUF-CMA security of Schnorr signatures implies that they are non-malleable. On the other hand, ECDSA signatures are inherently malleable<ref>If ''(r,s)'' is a valid ECDSA signature for a given message and key, then ''(r,n-s)'' is also valid for the same message and key. If ECDSA is restricted to only permit one of the two variants (as Bitcoin does through a policy rule on the network), it can be [https://nbn-resolving.de/urn:nbn:de:hbz:294-60803 proven] non-malleable under stronger than usual assumptions.</ref>; a third party without access to the secret key can alter an existing valid signature for a given public key and message into another signature that is valid for the same key and message. This issue is discussed in [bip-0062.mediawiki BIP62] and [bip-0146.mediawiki BIP146].
+* '''Non-malleability''': The SUF-CMA security of Schnorr signatures implies that they are non-malleable. On the other hand, ECDSA signatures are inherently malleable<ref>If ''(r,s)'' is a valid ECDSA signature for a given message and key, then ''(r,n-s)'' is also valid for the same message and key. If ECDSA is restricted to only permit one of the two variants (as Bitcoin does through a policy rule on the network), it can be [https://nbn-resolving.de/urn:nbn:de:hbz:294-60803 proven] non-malleable under stronger than usual assumptions.</ref>; a third party without access to the secret key can alter an existing valid signature for a given public key and message into another signature that is valid for the same key and message. This issue is discussed in [[bip-0062.mediawiki|BIP62]] and [[bip-0146.mediawiki|BIP146]].
 * '''Linearity''': Schnorr signatures provide a simple and efficient method that enables multiple collaborating parties to produce a signature that is valid for the sum of their public keys. This is the building block for various higher-level constructions that improve efficiency and privacy, such as multisignatures and others (see Applications below).
 
 For all these advantages, there are virtually no disadvantages, apart
@@ -42,7 +42,7 @@ made:
 * '''Signature encoding''': Instead of using [https://en.wikipedia.org/wiki/X.690#DER_encoding DER]-encoding for signatures (which are variable size, and up to 72 bytes), we can use a simple fixed 64-byte format.
 * '''Public key encoding''': Instead of using ''compressed'' 33-byte encodings of elliptic curve points which are common in Bitcoin today, public keys in this proposal are encoded as 32 bytes.
 * '''Batch verification''': The specific formulation of ECDSA signatures that is standardized cannot be verified more efficiently in batch compared to individually, unless additional witness data is added. Changing the signature scheme offers an opportunity to address this.
-* '''Completely specified''': To be safe for usage in consensus systems, the verification algorithm must be completely specified at the byte level. This guarantees that nobody can construct a signature that is valid to some verifiers but not all. This is traditionally not a requirement for digital signature schemes, and the lack of exact specification for the DER parsing of ECDSA signatures has caused problems for Bitcoin [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2015-July/009697.html in the past], needing [https://github.com/bitcoin/bips/blob/master/bip-0066.mediawiki BIP66] to address it. In this document we aim to meet this property by design. For batch verification, which is inherently non-deterministic as the verifier can choose their batches, this property implies that the outcome of verification may only differ from individual verifications with negligible probability, even to an attacker who intentionally tries to make batch- and non-batch verification differ.
+* '''Completely specified''': To be safe for usage in consensus systems, the verification algorithm must be completely specified at the byte level. This guarantees that nobody can construct a signature that is valid to some verifiers but not all. This is traditionally not a requirement for digital signature schemes, and the lack of exact specification for the DER parsing of ECDSA signatures has caused problems for Bitcoin [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2015-July/009697.html in the past], needing [[bip-0066.mediawiki|BIP66]] to address it. In this document we aim to meet this property by design. For batch verification, which is inherently non-deterministic as the verifier can choose their batches, this property implies that the outcome of verification may only differ from individual verifications with negligible probability, even to an attacker who intentionally tries to make batch- and non-batch verification differ.
 
 By reusing the same curve and hash function as Bitcoin uses for ECDSA, we are able to retain existing mechanisms for choosing secret and public keys, and we avoid introducing new assumptions about the security of elliptic curves and hash functions.
 
@@ -62,11 +62,11 @@ encodings and operations.
 
 Since we would like to avoid the fragility that comes with short hashes, the ''e'' variant does not provide significant advantages. We choose the ''R''-option, which supports batch verification. 
 
-'''Key prefixing''' Using the verification rule above directly makes Schnorr signatures vulnerable to "related-key attacks" in which a third party can convert a signature ''(R, s)'' for public key ''P'' into a signature ''(R, s + a⋅hash(R || m))'' for public key ''P + a⋅G'' and the same message ''m'', for any given additive tweak ''a'' to the signing key. This would render signatures insecure when keys are generated using [bip-0032.mediawiki#public-parent-key--public-child-key BIP32's unhardened derivation] and other methods that rely on additive tweaks to existing keys such as Taproot.
+'''Key prefixing''' Using the verification rule above directly makes Schnorr signatures vulnerable to "related-key attacks" in which a third party can convert a signature ''(R, s)'' for public key ''P'' into a signature ''(R, s + a⋅hash(R || m))'' for public key ''P + a⋅G'' and the same message ''m'', for any given additive tweak ''a'' to the signing key. This would render signatures insecure when keys are generated using [[bip-0032.mediawiki#public-parent-key--public-child-key|BIP32's unhardened derivation]] and other methods that rely on additive tweaks to existing keys such as Taproot.
 
 To protect against these attacks, we choose ''key prefixed''<ref>A limitation of committing to the public key (rather than to a short hash of it, or not at all) is that it removes the ability for public key recovery or verifying signatures against a short public key hash. These constructions are generally incompatible with batch verification.</ref> Schnorr signatures; changing the equation to ''s⋅G = R + hash(R || P || m)⋅P''. [https://eprint.iacr.org/2015/1135.pdf It can be shown] that key prefixing protects against related-key attacks with additive tweaks. In general, key prefixing increases robustness in multi-user settings, e.g., it seems to be a requirement for proving the MuSig multisignature scheme secure (see Applications below).
 
-We note that key prefixing is not strictly necessary for transaction signatures as used in Bitcoin currently, because signed transactions indirectly commit to the public keys already, i.e., ''m'' contains a commitment to ''pk''. However, this indirect commitment should not be relied upon because it may change with proposals such as SIGHASH_NOINPUT ([bip-0118.mediawiki BIP118]), and would render the signature scheme unsuitable for other purposes than signing transactions, e.g., [https://bitcoin.org/en/developer-reference#signmessage signing ordinary messages].
+We note that key prefixing is not strictly necessary for transaction signatures as used in Bitcoin currently, because signed transactions indirectly commit to the public keys already, i.e., ''m'' contains a commitment to ''pk''. However, this indirect commitment should not be relied upon because it may change with proposals such as SIGHASH_NOINPUT ([[bip-0118.mediawiki|BIP118]]), and would render the signature scheme unsuitable for other purposes than signing transactions, e.g., [https://bitcoin.org/en/developer-reference#signmessage signing ordinary messages].
 
 '''Encoding R and public key point P''' There exist several possibilities for encoding elliptic curve points:
 # Encoding the full X and Y coordinates of ''P'' and ''R'', resulting in a 64-byte public key and a 96-byte signature.
@@ -141,7 +141,7 @@ The algorithm ''PubKey(sk)'' is defined as:
 
 Note that we use a very different public key format (32 bytes) than the ones used by existing systems (which typically use elliptic curve points as public keys, or 33-byte or 65-byte encodings of them). A side effect is that ''PubKey(sk) = PubKey(bytes(n - int(sk))'', so every public key has two corresponding secret keys.
 
-As an alternative to generating keys randomly, it is also possible and safe to repurpose existing key generation algorithms for ECDSA in a compatible way. The secret keys constructed by such an algorithm can be used as ''sk'' directly. The public keys constructed by such an algorithm (assuming they use the 33-byte compressed encoding) need to be converted by dropping the first byte. Specifically, [https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki BIP32] and schemes built on top of it remain usable.
+As an alternative to generating keys randomly, it is also possible and safe to repurpose existing key generation algorithms for ECDSA in a compatible way. The secret keys constructed by such an algorithm can be used as ''sk'' directly. The public keys constructed by such an algorithm (assuming they use the 33-byte compressed encoding) need to be converted by dropping the first byte. Specifically, [[bip-0032.mediawiki|BIP32]] and schemes built on top of it remain usable.
 
 ==== Default Signing ====