Editing Schnorr signature

{{short description|Digital signature scheme }}
In [[cryptography]], a '''Schnorr signature''' is a [[digital signature]] produced by the '''Schnorr signature algorithm''' that was described by [[Claus P. Schnorr|Claus Schnorr]]. It is a digital signature scheme known for its simplicity, among the first whose security is based on the [[Computational complexity theory#Intractability|intractability]] of certain [[discrete logarithm]] problems. It is efficient and generates short signatures.<ref name="Seurin"/> It was covered by {{US patent|4995082}} which expired in February 2010.

==Algorithm==

===Choosing parameters===
*All users of the signature scheme agree on a [[group (mathematics)|group]] <math>G</math> of prime order <math>q</math> with generator <math>g</math> in which the [[discrete log]] problem is assumed to be hard.  Typically a [[Schnorr group]] is used.
*All users agree on a [[cryptographic hash function]] <math>H: \{0,1\}^* \rightarrow \mathbb Z/q\mathbb Z</math>.

===Notation===
In the following,
*Exponentiation stands for repeated application of the group operation
*Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation (as applicable)
*Subtraction stands for subtraction on the set of congruence classes
*<math>M \in \{0,1\}^*</math>, the set of finite bit strings
*<math>s, e, e_v, x, k \in \mathbb Z/q\mathbb Z</math>, the [[integers mod n|set of congruence classes modulo <math>q</math>]]
*<math>y, r, r_v \in G</math>.

===Key generation===
*Choose a private signing key <math>x</math> from the allowed set.
*The public verification key is <math>y = g^{-x}</math>.

===Signing===
To sign a message <math>M</math>:
*Choose a random <math>k</math> from the allowed set.
*Let <math>r = g^k</math>.
*Let <math>e = H(r \parallel M)</math>, where <math>\parallel</math> denotes concatenation and <math>r</math> is represented as a bit string.
*Let <math>s = k + xe</math>.

The signature is the pair, <math>(s, e)</math>.

Note that <math>s, e \in \mathbb Z/q\mathbb Z</math>; if <math>q < 2^{256}</math>, then the signature representation can fit into 64 bytes.

===Verifying===
*Let <math>r_v = g^s y^e</math>
*Let <math>e_v = H(r_v \parallel M)</math>

If <math>e_v = e</math> then the signature is verified.

===Proof of correctness===
It is relatively easy to see that <math>e_v = e</math> if the signed message equals the verified message:

<math>r_v = g^s y^e = g^{k + xe} g^{-xe} = g^k = r</math>, and hence <math>e_v = H(r_v \parallel M) = H(r \parallel M) = e</math>.

Public elements: <math>G</math>, <math>g</math>, <math>q</math>, <math>y</math>, <math>s</math>, <math>e</math>, <math>r</math>. Private elements: <math>k</math>, <math>x</math>.

This shows only that a correctly signed message will verify correctly; many other properties are required for a secure signature algorithm.

===Key leakage from nonce reuse===

Just as with the closely related signature algorithms [[Digital Signature Algorithm|DSA]], [[ECDSA]], and [[ElGamal signature scheme|ElGamal]], reusing the secret nonce value <math>k</math> on two Schnorr signatures of different messages will allow observers to recover the private key.<ref name="Tibouchi">{{cite web
| url=https://ecc2017.cs.ru.nl/slides/ecc2017-tibouchi.pdf
| title=Attacks on Schnorr signatures with biased nonces
| first=Mehdi
| last=Tibouchi
| work=ECC Workshop
| date=2017-11-13
| access-date=2023-02-06}}</ref> In the case of Schnorr signatures, this simply requires subtracting <math>s</math> values:

: <math>s' - s = (k' - k) + x (e' - e)</math>.

If <math>k'=k</math> but <math>e'\ne e</math> then <math>x</math> can be simply isolated. In fact, even slight biases in the value <math>k</math> or partial leakage of <math>k</math> can reveal the private key, after collecting sufficiently many signatures and solving the [[hidden number problem]].<ref name="Tibouchi"/>

===Security argument===
The signature scheme was constructed by applying the [[Fiat–Shamir heuristic|Fiat–Shamir transformation]]<ref>{{cite conference <!-- Citation bot no -->
| first1=Amos
| last1=Fiat
| authorlink1=Amos Fiat
| first2=Adi
| last2=Shamir
| authorlink2=Adi Shamir
| title=Advances in Cryptology
| chapter=How to Prove Yourself: Practical Solutions to Identification and Signature Problems
| conference=Conference on the Theory and Application of Cryptographic Techniques. Proceedings of CRYPTO '86|series=Lecture Notes in Computer Science
| date=1987
| volume=263
| pages=186–194
| doi=10.1007/3-540-47721-7_12
| editor=Andrew M. Odlyzko
| editor-link=Andrew Odlyzko
| isbn=978-3-540-18047-0 |s2cid=4838652 |doi-access=free}}</ref> to Schnorr's identification protocol.<ref>{{cite conference <!-- Citation bot no -->
| first=C. P.
| last=Schnorr
| author-link=Claus P. Schnorr
| title=Advances in Cryptology
| chapter=Efficient Identification and Signatures for Smart Cards
| conference=Conference on the Theory and Application of Cryptographic Techniques. Proceedings of CRYPTO '89
| series=Lecture Notes in Computer Science
| date=1990
| volume=435
| pages=239–252
| doi=10.1007/0-387-34805-0_22
| isbn=978-0-387-97317-3
| s2cid=5526090
| editor=Gilles Brassard
| editor-link=Gilles Brassard
| doi-access=free}}</ref><ref>{{cite journal
| url=http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/4280
| title=Efficient signature generation by smart cards 
| first=C. P.
| last=Schnorr
| author-link=Claus P. Schnorr
| journal=[[Journal of Cryptology]]
| volume=4
| pages=161–174
| year=1991
| issue=3 
| doi=10.1007/BF00196725| s2cid=10976365 
}}</ref> Therefore, (as per Fiat and Shamir's arguments), it is secure if <math>H</math> is modeled as a [[random oracle]].

Its security can also be argued in the [[generic group model]], under the assumption that <math>H</math> is "random-prefix preimage resistant" and "random-prefix second-preimage resistant".<ref name="Neven">{{cite web
| first1=Gregory
| last1=Neven
| first2=Nigel
| last2=Smart
| authorlink2=Nigel Smart (cryptographer)
| first3=Bogdan
| last3=Warinschi
| title=Hash Function Requirements for Schnorr Signatures
| url=http://www.neven.org/papers/schnorr.html
| publisher=IBM Research
| access-date=19 July 2012}}</ref>  In particular, <math>H</math> does ''not'' need to be [[Collision resistance|collision resistant]].

In 2012, Seurin<ref name="Seurin">{{Cite web
| url = https://eprint.iacr.org/2012/029
| title = On the Exact Security of Schnorr-Type Signatures in the Random Oracle Model
| last = Seurin
| first = Yannick
| date = 2012-01-12
| website = [[Cryptology ePrint Archive]]|publisher = International Association for Cryptologic Research
| access-date = 2023-02-06}}</ref> provided an exact proof of the Schnorr signature scheme. In particular, Seurin shows that the security proof using the [[forking lemma]] is the best possible result for any signature schemes based on one-way [[group homomorphism]]s including Schnorr-type signatures and the [[Guillou–Quisquater signature scheme]]s. Namely, under the [[Random_Oracle_Model,Discrete_Logs|ROMDL]] assumption, any algebraic reduction must lose a factor <math>f({\epsilon}_F)q_h </math> in its time-to-success ratio, where <math>f \le 1</math> is a function that remains close to 1 as long as "<math>{\epsilon}_F</math> is noticeably smaller than 1", where <math>{\epsilon}_F</math> is the probability of forging an error making at most <math>q_h</math> queries to the random oracle.

==Short Schnorr signatures==

The aforementioned process achieves a ''t''-bit security level with 4''t''-bit signatures. For example, a 128-bit security level would require 512-bit (64-byte) signatures. The security is limited by discrete logarithm attacks on the group, which have a complexity of the ''square-root'' of the group size.

In Schnorr's original 1991 paper, it was suggested that since collision resistance in the hash is not required, shorter hash functions may be just as secure, and indeed recent developments suggest that a ''t''-bit security level can be achieved with 3''t''-bit signatures.<ref name="Neven"/> Then, a 128-bit security level would require only 384-bit (48-byte) signatures, and this could be achieved by truncating the size of ''e'' until it is half the length of the ''s'' bitfield.

== Implementations ==
Schnorr signature is used by numerous products. A notable usage is the deterministic Schnorr's signature using the secp256k1 [[Elliptic-curve cryptography|elliptic curve]] for [[Bitcoin]] transaction signature after the Taproot update.<ref>{{Cite web |last=Wuille |first=Pieter |last2=Nick |first2=Jonas |last3=Ruffing |first3=Tim |title=BIP340: Schnorr Signatures for secp256k1 |url=https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki |access-date=2024-11-11 |website=GitHub |language=en}}</ref>

==See also==
* [[Digital Signature Algorithm|DSA]]
* [[EdDSA]]
* [[ElGamal signature scheme]]

==References==
{{reflist}}

==External links==
* [https://tools.ietf.org/html/rfc8235 RFC 8235]
* [https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki BIP 340: Schnorr Signatures for secp256k1]

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[[Category:Digital signature schemes]]