Can Schnorr aggregate signatures be embedded within other Schnorr aggregate signatures?
Schnorr signatures are a type of digital signature used in various blockchain and cryptocurrency applications. They are designed to provide parties with a secure and efficient way to verify the authenticity and integrity of messages without revealing their contents. One of the key features of Schnorr signatures is that they can combine the signatures of multiple parties into a single shared signature, known as a composite signature. In this article, we will explore whether it is possible to combine aggregate signatures within other aggregate signatures using Schnorr protocols.
Understanding Schnorr Signatures
Schnorr signatures are based on the concept of “blind signing,” which allows parties to sign messages without revealing their contents to others. They consist of three parts: a private key (p), a public key (P), and a hash function (h). The public value P is derived from the private key p using the Schnorr hash function h(P) = r + p^k, where k is an integer parameter.
Aggregate Signatures
Schnorr aggregate signatures are used to combine the signatures of multiple parties into a single shared signature. An aggregate signature consists of a list of signatures (s_i), and the Schnorr hash function of each signature is combined using the hash function hAggregate(s_i).
Can aggregate signatures be embedded inside other aggregate signatures?
In theory, it is possible to embed aggregate signatures inside other aggregate signatures using Schnorr protocols. However, there are some limitations and potential problems that need to be considered.
One way to combine aggregate signatures is to use a combination of the Schnorr hash function and the new hash function hAggregate2(s_i). The idea is to create a new public value P2 by combining two private keys p1 and p2 using the Schnorr hash function: P2 = hAggregate2(P1, P2). We can then derive an aggregate signature s2 from this public value using the same Schnorr hash function.
However, there are potential problems to consider:
- Security: Nesting aggregate signatures can introduce new security risks, such as increased complexity and vulnerability to attacks. Verifying a nested aggregate signature or extracting data can be more difficult.
- Performance: Generating and verifying nested aggregate signatures can be computationally expensive, especially for large inputs. This may make them less practical for real-world applications.
Example Use Case
To illustrate this concept, let’s look at an example use case where we have two parties (Alice and Bob) who want to sign a message together using Schnorr signatures. We create a nested aggregate signature inside another aggregate signature.
Suppose Alice has a private key p1 = e1^p2^k1, where e1 is the public signature value. She wants to combine her signature with Bob’s signature to form an aggregate signature s_nested(s1, s2), where s1 and s2 are individual Schnorr signatures.
We create a new public value P2 by combining Alice’s private key p1 = e1^p2^k1 with Bob’s private key p2: P2 = hAggregate2(P1, P2) = e1^hAggregate(s1, s2).
Then, we derive the aggregate signature s_nested using the Schnorr hash function: s_nested = hAggregate(s1, s2)
The public value of this nested aggregate signature is P3 = e1^P2.
Conclusion
In summary, it is possible to embed aggregate signatures inside other aggregate signatures using Schnorr protocols. However, there are potential security and performance issues that need to be considered when creating such nested signatures. It is essential to carefully evaluate the trade-offs and limitations of this approach before applying it to a real-world scenario.
Recommendations
- Carefully consider the security risks and potential vulnerabilities that nested aggregate signatures introduce.