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Diffstat (limited to 'bip-0340.mediawiki')
| -rw-r--r-- | bip-0340.mediawiki | 67 |
1 files changed, 23 insertions, 44 deletions
diff --git a/bip-0340.mediawiki b/bip-0340.mediawiki index b4e5f60..9502a69 100644 --- a/bip-0340.mediawiki +++ b/bip-0340.mediawiki @@ -64,7 +64,7 @@ Since we would like to avoid the fragility that comes with short hashes, the ''e '''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). +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 which means that the public key is prefixed to the message in the challenge hash input. This changes 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]. @@ -78,16 +78,11 @@ Using the first option would be slightly more efficient for verification (around '''Implicit Y coordinates''' In order to support efficient verification and batch verification, the Y coordinate of ''P'' and of ''R'' cannot be ambiguous (every valid X coordinate has two possible Y coordinates). We have a choice between several options for symmetry breaking: # Implicitly choosing the Y coordinate that is in the lower half. # 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 (has a square root modulo the field size, or "is a square" for short)<ref>A product of two numbers is a square when either both or none of the factors are squares. As ''-1'' is not a square modulo secp256k1's field size ''p'', and the two Y coordinates corresponding to a given X coordinate are each other's negation, this means exactly one of the two must be a square.</ref>. +# Implicitly choosing the Y coordinate that is a quadratic residue (i.e. has a square root modulo ''p''). -In the case of ''R'' the third option is slower at signing time but a bit faster to verify, as it is possible to directly compute whether the Y coordinate is a square when the points are represented in -[https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates Jacobian coordinates] (a common optimization to avoid modular inverses -for elliptic curve operations). The two other options require a possibly -expensive conversion to affine coordinates first. This would even be the case if the sign or oddness were explicitly coded (option 2 in the list above). We therefore choose option 3. +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>. -For ''P'', using the third option comes at the cost of computing a Jacobi symbol (test for squaredness) at key generation or signing time, without avoiding a conversion to affine coordinates (as public keys will use affine coordinates anyway). We choose the second option, making public keys implicitly have an even Y coordinate, to maximize compatibility with existing key generation algorithms and infrastructure. - -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 even 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 odd. A proof sketch can be found [https://medium.com/blockstream/reducing-bitcoin-transaction-sizes-with-x-only-pubkeys-f86476af05d7 here].</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>. '''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. @@ -95,11 +90,11 @@ For example, without tagged hashing a BIP340 signature could also be valid for a This proposal suggests to include the tag by prefixing the hashed data with ''SHA256(tag) || SHA256(tag)''. Because this is a 64-byte long context-specific constant and the ''SHA256'' block size is also 64 bytes, optimized implementations are possible (identical to SHA256 itself, but with a modified initial state). Using SHA256 of the tag name itself is reasonably simple and efficient for implementations that don't choose to use the optimization. -'''Final scheme''' As a result, our final scheme ends up using public key ''pk'' which is the X coordinate of a point ''P'' on the curve whose Y coordinate is even and signatures ''(r,s)'' where ''r'' is the X coordinate of a point ''R'' whose Y coordinate is a square. The signature satisfies ''s⋅G = R + tagged_hash(r || pk || m)⋅P''. +'''Final scheme''' As a result, our final scheme ends up using public key ''pk'' which is the X coordinate of a point ''P'' on the curve whose Y coordinate is even and signatures ''(r,s)'' where ''r'' is the X coordinate of a point ''R'' whose Y coordinate is even. The signature satisfies ''s⋅G = R + tagged_hash(r || pk || m)⋅P''. === Specification === -The following conventions are used, with constants as defined for [https://www.secg.org/sec2-v2.pdf secp256k1]: +The following conventions are used, with constants as defined for [https://www.secg.org/sec2-v2.pdf secp256k1]. We note that adapting this specification to other elliptic curves is not straightforward and can result in an insecure scheme<ref>Among other pitfalls, using the specification with a curve whose order is not close to the size of the range of the nonce derivation function is insecure.</ref>. * Lowercase variables represent integers or byte arrays. ** The constant ''p'' refers to the field size, ''0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F''. ** The constant ''n'' refers to the curve order, ''0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141''. @@ -111,23 +106,17 @@ The following conventions are used, with constants as defined for [https://www.s ** [https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication Multiplication (⋅) of an integer and a point] refers to the repeated application of the group operation. * Functions and operations: ** ''||'' refers to byte array concatenation. -** The function ''x[i:j]'', where ''x'' is a byte array, returns a ''(j - i)''-byte array with a copy of the ''i''-th byte (inclusive) to the ''j''-th byte (exclusive) of ''x''. +** The function ''x[i:j]'', where ''x'' is a byte array and ''i, j ≥ 0'', returns a ''(j - i)''-byte array with a copy of the ''i''-th byte (inclusive) to the ''j''-th byte (exclusive) of ''x''. ** The function ''bytes(x)'', where ''x'' is an integer, returns the 32-byte encoding of ''x'', most significant byte first. ** The function ''bytes(P)'', where ''P'' is a point, returns ''bytes(x(P))''. ** The function ''int(x)'', where ''x'' is a 32-byte array, returns the 256-bit unsigned integer whose most significant byte first encoding is ''x''. -** The function ''is_square(x)'', where ''x'' is an integer, returns whether or not ''x'' is a quadratic residue modulo ''p''. Since ''p'' is prime, it is equivalent to the [https://en.wikipedia.org/wiki/Legendre_symbol Legendre symbol] ''(x / p) = x<sup>(p-1)/2</sup> mod p'' being equal to ''1''<ref>For points ''P'' on the secp256k1 curve it holds that ''y(P)<sup>(p-1)/2</sup> ≠ 0 mod p''.</ref>. -** The function ''has_square_y(P)'', where ''P'' is a point, is defined as ''not is_infinite(P) and is_square(y(P))''<ref>For points ''P'' on the secp256k1 curve it holds that ''has_square_y(P) = not has_square_y(-P)''.</ref>. -** The function ''has_even_y(x)'', where ''P'' is a point, returns ''y(P) mod 2 = 0''. -** The function ''lift_x_square_y(x)'', where ''x'' is an integer in range ''0..p-1'', returns the point ''P'' for which ''x(P) = x''<ref> - Given a candidate X coordinate ''x'' in the range ''0..p-1'', there exist either exactly two or exactly zero valid Y coordinates. If no valid Y coordinate exists, then ''x'' is not a valid X coordinate either, i.e., no point ''P'' exists for which ''x(P) = x''. The valid Y coordinates for a given candidate ''x'' are the square roots of ''c = x<sup>3</sup> + 7 mod p'' and they can be computed as ''y = ±c<sup>(p+1)/4</sup> mod p'' (see [https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus Quadratic residue]) if they exist, which can be checked by squaring and comparing with ''c''. -</ref> and ''has_square_y(P)''<ref> - If ''P := lift_x_square_y(x)'' does not fail, then ''y := y(P) = c<sup>(p+1)/4</sup> mod p'' is square. Proof: If ''lift_x_square_y'' does not fail, ''y'' is a square root of ''c'' and therefore the [https://en.wikipedia.org/wiki/Legendre_symbol Legendre symbol] ''(c / p)'' is ''c<sup>(p-1)/2</sup> = 1 mod p''. Because the Legendre symbol ''(y / p)'' is ''y<sup>(p-1)/2</sup> mod p = c<sup>((p+1)/4)((p-1)/2)</sup> mod p = 1<sup>((p+1)/4)</sup> mod p = 1 mod p'', ''y'' is square. -</ref>, or fails if no such point exists. The function ''lift_x_square_y(x)'' is equivalent to the following pseudocode: +** The function ''has_even_y(P)'', where ''P'' is a point, returns ''y(P) mod 2 = 0''. +** The function ''lift_x(x)'', where ''x'' is an integer in range ''0..p-1'', returns the point ''P'' for which ''x(P) = x''<ref> + Given a candidate X coordinate ''x'' in the range ''0..p-1'', there exist either exactly two or exactly zero valid Y coordinates. If no valid Y coordinate exists, then ''x'' is not a valid X coordinate either, i.e., no point ''P'' exists for which ''x(P) = x''. The valid Y coordinates for a given candidate ''x'' are the square roots of ''c = x<sup>3</sup> + 7 mod p'' and they can be computed as ''y = ±c<sup>(p+1)/4</sup> mod p'' (see [https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus Quadratic residue]) if they exist, which can be checked by squaring and comparing with ''c''.</ref> and ''has_even_y(P)'', or fails if no such point exists. The function ''lift_x(x)'' is equivalent to the following pseudocode: *** Let ''c = x<sup>3</sup> + 7 mod p''. *** Let ''y = c<sup>(p+1)/4</sup> mod p''. *** Fail if ''c ≠ y<sup>2</sup> mod p''. -*** Return the unique point ''P'' such that ''x(P) = x'' and ''y(P) = y'', or fail if no such point exists. -** The function ''lift_x_even_y(x)'', where ''x'' is an integer in range ''0..p-1'', returns the point ''P'' for which ''x(P) = x'' and ''has_even_y(P)'', or fails if no such point exists. If such a point does exist, it is always equal to either ''lift_x_square_y(x)'' or ''-lift_x_square_y(x)'', which suggests implementing it in terms of ''lift_x_square_y'', and optionally negating the result. +*** Return the unique point ''P'' such that ''x(P) = x'' and ''y(P) = y'' if ''y mod 2 = 0'' or ''y(P) = p-y'' otherwise. ** The function ''hash<sub>tag</sub>(x)'' where ''tag'' is a UTF-8 encoded tag name and ''x'' is a byte array returns the 32-byte hash ''SHA256(SHA256(tag) || SHA256(tag) || x)''. ==== Public Key Generation ==== @@ -158,13 +147,13 @@ The algorithm ''Sign(sk, m)'' is defined as: * Fail if ''d' = 0'' or ''d' ≥ n'' * Let ''P = d'⋅G'' * Let ''d = d' '' if ''has_even_y(P)'', otherwise let ''d = n - d' ''. -* Let ''t'' be the byte-wise xor of ''bytes(d)'' and ''H<sub>BIP340/aux</sub>(a)''<ref>The auxiliary random data is hashed (with a unique tag) as a precaution against situations where the randomness may be correlated with the private key itself. It is xored with the private key (rather than combined with it in a hash) to reduce the number of operations exposed to the actual secret key.</ref>. -* Let ''rand = hash<sub>BIP340/nonce</sub>(t || bytes(P) || m)''<ref>Including the [https://moderncrypto.org/mail-archive/curves/2020/001012.html public key as input to the nonce hash] helps ensure the robustness of the signing algorithm by preventing leakage of the secret key if the calculation of the public key ''P'' is performed incorrectly or maliciously, for example if it is left to the caller for performance reasons.</ref>. +* Let ''t'' be the byte-wise xor of ''bytes(d)'' and ''hash<sub>BIP0340/aux</sub>(a)''<ref>The auxiliary random data is hashed (with a unique tag) as a precaution against situations where the randomness may be correlated with the private key itself. It is xored with the private key (rather than combined with it in a hash) to reduce the number of operations exposed to the actual secret key.</ref>. +* Let ''rand = hash<sub>BIP0340/nonce</sub>(t || bytes(P) || m)''<ref>Including the [https://moderncrypto.org/mail-archive/curves/2020/001012.html public key as input to the nonce hash] helps ensure the robustness of the signing algorithm by preventing leakage of the secret key if the calculation of the public key ''P'' is performed incorrectly or maliciously, for example if it is left to the caller for performance reasons.</ref>. * Let ''k' = int(rand) mod n''<ref>Note that in general, taking a uniformly random 256-bit integer modulo the curve order will produce an unacceptably biased result. However, for the secp256k1 curve, the order is sufficiently close to ''2<sup>256</sup>'' that this bias is not observable (''1 - n / 2<sup>256</sup>'' is around ''1.27 * 2<sup>-128</sup>'').</ref>. * Fail if ''k' = 0''. * Let ''R = k'⋅G''. -* Let ''k = k' '' if ''has_square_y(R)'', otherwise let ''k = n - k' ''. -* Let ''e = int(hash<sub>BIP340/challenge</sub>(bytes(R) || bytes(P) || m)) mod n''. +* Let ''k = k' '' if ''has_even_y(R)'', otherwise let ''k = n - k' ''. +* Let ''e = int(hash<sub>BIP0340/challenge</sub>(bytes(R) || bytes(P) || m)) mod n''. * Let ''sig = bytes(R) || bytes((k + ed) mod n)''. * If ''Verify(bytes(P), m, sig)'' (see below) returns failure, abort<ref>Verifying the signature before leaving the signer prevents random or attacker provoked computation errors. This prevents publishing invalid signatures which may leak information about the secret key. It is recommended, but can be omitted if the computation cost is prohibitive.</ref>. * Return the signature ''sig''. @@ -190,17 +179,17 @@ Input: * A signature ''sig'': a 64-byte array The algorithm ''Verify(pk, m, sig)'' is defined as: -* Let ''P = lift_x_even_y(int(pk))''; fail if that fails. +* Let ''P = lift_x(int(pk))''; fail if that fails. * Let ''r = int(sig[0:32])''; fail if ''r ≥ p''. * Let ''s = int(sig[32:64])''; fail if ''s ≥ n''. -* Let ''e = int(hash<sub>BIP340/challenge</sub>(bytes(r) || bytes(P) || m)) mod n''. +* Let ''e = int(hash<sub>BIP0340/challenge</sub>(bytes(r) || bytes(P) || m)) mod n''. * Let ''R = s⋅G - e⋅P''. -* Fail if ''not has_square_y(R)'' or ''x(R) ≠ r''. +* Fail if ''not has_even_y(R)'' or ''x(R) ≠ r''. * Return success iff no failure occurred before reaching this point. For every valid secret key ''sk'' and message ''m'', ''Verify(PubKey(sk),m,Sign(sk,m))'' will succeed. -Note that the correctness of verification relies on the fact that ''lift_x_even_y'' always returns a point with an even Y coordinate. A hypothetical verification algorithm that treats points as public keys, and takes the point ''P'' directly as input would fail any time a point with odd Y is used. While it is possible to correct for this by negating points with odd Y coordinate before further processing, this would result in a scheme where every (message, signature) pair is valid for two public keys (a type of malleability that exists for ECDSA as well, but we don't wish to retain). We avoid these problems by treating just the X coordinate as public key. +Note that the correctness of verification relies on the fact that ''lift_x'' always returns a point with an even Y coordinate. A hypothetical verification algorithm that treats points as public keys, and takes the point ''P'' directly as input would fail any time a point with odd Y is used. While it is possible to correct for this by negating points with odd Y coordinate before further processing, this would result in a scheme where every (message, signature) pair is valid for two public keys (a type of malleability that exists for ECDSA as well, but we don't wish to retain). We avoid these problems by treating just the X coordinate as public key. ==== Batch Verification ==== @@ -213,26 +202,16 @@ Input: The algorithm ''BatchVerify(pk<sub>1..u</sub>, m<sub>1..u</sub>, sig<sub>1..u</sub>)'' is defined as: * Generate ''u-1'' random integers ''a<sub>2...u</sub>'' in the range ''1...n-1''. They are generated deterministically using a [https://en.wikipedia.org/wiki/Cryptographically_secure_pseudorandom_number_generator CSPRNG] seeded by a cryptographic hash of all inputs of the algorithm, i.e. ''seed = seed_hash(pk<sub>1</sub>..pk<sub>u</sub> || m<sub>1</sub>..m<sub>u</sub> || sig<sub>1</sub>..sig<sub>u</sub> )''. A safe choice is to instantiate ''seed_hash'' with SHA256 and use [https://tools.ietf.org/html/rfc8439 ChaCha20] with key ''seed'' as a CSPRNG to generate 256-bit integers, skipping integers not in the range ''1...n-1''. * For ''i = 1 .. u'': -** Let ''P<sub>i</sub> = lift_x_even_y(int(pk<sub>i</sub>))''; fail if it fails. +** Let ''P<sub>i</sub> = lift_x(int(pk<sub>i</sub>))''; fail if it fails. ** Let ''r<sub>i</sub> = int(sig<sub>i</sub>[0:32])''; fail if ''r<sub>i</sub> ≥ p''. ** Let ''s<sub>i</sub> = int(sig<sub>i</sub>[32:64])''; fail if ''s<sub>i</sub> ≥ n''. -** Let ''e<sub>i</sub> = int(hash<sub>BIP340/challenge</sub>(bytes(r<sub>i</sub>) || bytes(P<sub>i</sub>) || m<sub>i</sub>)) mod n''. -** Let ''R<sub>i</sub> = lift_x_square_y(r<sub>i</sub>)''; fail if ''lift_x_square_y(r<sub>i</sub>)'' fails. +** Let ''e<sub>i</sub> = int(hash<sub>BIP0340/challenge</sub>(bytes(r<sub>i</sub>) || bytes(P<sub>i</sub>) || m<sub>i</sub>)) mod n''. +** Let ''R<sub>i</sub> = lift_x(r<sub>i</sub>)''; fail if ''lift_x(r<sub>i</sub>)'' fails. * Fail if ''(s<sub>1</sub> + a<sub>2</sub>s<sub>2</sub> + ... + a<sub>u</sub>s<sub>u</sub>)⋅G ≠ R<sub>1</sub> + a<sub>2</sub>⋅R<sub>2</sub> + ... + a<sub>u</sub>⋅R<sub>u</sub> + e<sub>1</sub>⋅P<sub>1</sub> + (a<sub>2</sub>e<sub>2</sub>)⋅P<sub>2</sub> + ... + (a<sub>u</sub>e<sub>u</sub>)⋅P<sub>u</sub>''. * Return success iff no failure occurred before reaching this point. If all individual signatures are valid (i.e., ''Verify'' would return success for them), ''BatchVerify'' will always return success. If at least one signature is invalid, ''BatchVerify'' will return success with at most a negligible probability. -=== Optimizations === - -Many techniques are known for optimizing elliptic curve implementations. Several of them apply here, but are out of scope for this document. Two are listed below however, as they are relevant to the design decisions: - -'''Squareness testing''' The function ''is_square(x)'' is defined as above, but can be computed more efficiently using an [https://en.wikipedia.org/wiki/Jacobi_symbol#Calculating_the_Jacobi_symbol extended GCD algorithm]. - -'''Jacobian coordinates''' Elliptic Curve operations can be implemented more efficiently by using [https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates Jacobian coordinates]. Elliptic Curve operations implemented this way avoid many intermediate modular inverses (which are computationally expensive), and the scheme proposed in this document is in fact designed to not need any inversions at all for verification. When operating on a point ''P'' with Jacobian coordinates ''(x,y,z)'' which is not the point at infinity and for which ''x(P)'' is defined as ''x / z<sup>2</sup>'' and ''y(P)'' is defined as ''y / z<sup>3</sup>'': -* ''has_square_y(P)'' can be implemented as ''is_square(yz mod p)''. -* ''x(P) ≠ r'' can be implemented as ''(0 ≤ r < p) and (x ≠ z<sup>2</sup>r mod p)''. - == Applications == There are several interesting applications beyond simple signatures. @@ -242,7 +221,7 @@ While recent academic papers claim that they are also possible with ECDSA, conse By means of an interactive scheme such as [https://eprint.iacr.org/2018/068 MuSig], participants can aggregate their public keys into a single public key which they can jointly sign for. This allows ''n''-of-''n'' multisignatures which, from a verifier's perspective, are no different from ordinary signatures, giving improved privacy and efficiency versus ''CHECKMULTISIG'' or other means. -Moreover, Schnorr signatures are compatible with [https://web.archive.org/web/20031003232851/http://www.research.ibm.com/security/dkg.ps distributed key generation], which enables interactive threshold signatures schemes, e.g., the schemes described by [http://cacr.uwaterloo.ca/techreports/2001/corr2001-13.ps Stinson and Strobl (2001)] or [https://web.archive.org/web/20060911151529/http://theory.lcs.mit.edu/~stasio/Papers/gjkr03.pdf Genaro, Jarecki and Krawczyk (2003)]. These protocols make it possible to realize ''k''-of-''n'' threshold signatures, which ensure that any subset of size ''k'' of the set of ''n'' signers can sign but no subset of size less than ''k'' can produce a valid Schnorr signature. However, the practicality of the existing schemes is limited: most schemes in the literature have been proven secure only for the case ''k-1 < n/2'', are not secure when used concurrently in multiple sessions, or require a reliable broadcast mechanism to be secure. Further research is necessary to improve this situation. +Moreover, Schnorr signatures are compatible with [https://web.archive.org/web/20031003232851/http://www.research.ibm.com/security/dkg.ps distributed key generation], which enables interactive threshold signatures schemes, e.g., the schemes described by [http://cacr.uwaterloo.ca/techreports/2001/corr2001-13.ps Stinson and Strobl (2001)] or [https://web.archive.org/web/20060911151529/http://theory.lcs.mit.edu/~stasio/Papers/gjkr03.pdf Gennaro, Jarecki and Krawczyk (2003)]. These protocols make it possible to realize ''k''-of-''n'' threshold signatures, which ensure that any subset of size ''k'' of the set of ''n'' signers can sign but no subset of size less than ''k'' can produce a valid Schnorr signature. However, the practicality of the existing schemes is limited: most schemes in the literature have been proven secure only for the case ''k-1 < n/2'', are not secure when used concurrently in multiple sessions, or require a reliable broadcast mechanism to be secure. Further research is necessary to improve this situation. === Adaptor Signatures === |